Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Proof 2

Let $T$ be a finitely satisfiable $\mathcal L$-theory. There is a finitely satisfiable $\mathcal L$-theory $T'$ which contains $T$ as a subset such that for all $\mathcal L$-sentences $\phi$, either $\phi \in T'$ or $\neg\phi \in T'$. {{explain|Does this actually mean the same as the statement of the theorem in Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension? If so, replace it. If not, then it is a different proof altogether. If the two statements are equivalent, then this needs to be demonstrated.}}


Reflexive Closure of Strict Total Ordering is Total Ordering

Let $S$ be a set. Let $\prec$ be a strict total ordering of $S$. Let $\preceq$ be the reflexive closure of $\prec$. Then $\preceq$ is an total ordering of $S$.


Node of Rooted Tree is on Branch

Let $T$ be a rooted tree with root node $r_T$. Let $t$ be a node of $T$. Then there exists a branch $\Gamma$ of $T$ such that $t \in \Gamma$.


Leaf of Rooted Tree is on One Branch

Let $T$ be a rooted tree with root node $r_T$. Let $t$ be a leaf node of $T$. Then there exists a unique branch $\Gamma$ of $T$ such that $t \in \Gamma$.


Node of Rooted Tree with Multiple Children is on Multiple Branches

Let $T$ be a rooted tree with root node $r_T$. Let $t$ be a node of $T$. Let $t$ have more than one child nodes. Then $t$ is on more than one branch of $T$.


Branch of Finite Tree is Finite

Let $T$ be a finite rooted tree with root node $r_T$. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is a finite branch.


Same Dimensional Vector Spaces are Isomorphic

Let $K$ be a division ring. Let $V$, $W$ be finite dimensional $K$-vector spaces. Suppose that $\dim_K V = \dim_K W$. Then: :$V \cong W$ That is, $V$ and $W$ are isomorphic.


Five Lemma

Let $A$ be a commutative ring with unity. Let: ::$\begin{xy}\xymatrix@L+2mu@+1em{  M_1 \ar[r]^*{\alpha_1}      \ar[d]^*{\phi_1} &  M_2 \ar[r]^*{\alpha_2}      \ar[d]^*{\phi_2} &  M_3 \ar[r]^*{\alpha_3}      \ar[d]^*{\phi_3} &  M_4 \ar[r]^*{\alpha_4}      \ar[d]^*{\phi_4} &  M_5 \ar[d]^*{\phi_5} \\  N_1 \ar[r]_*{\beta_1} &  N_2 \ar[r]_*{\beta_2} &  N_3 \ar[r]_*{\beta_3} &  N_4 \ar[r]_*{\beta_4} &  N_5 }\end{xy}$ be a commutative diagram of $A$-modules. Suppose that the rows are exact. Then: :If $\phi_2$ and $\phi_4$ are surjective and $\phi_5$ is injective then $\phi_3$ is surjective. :If $\phi_2$ and $\phi_4$ are injective and $\phi_1$ is surjective then $\phi_3$ is injective.


Snake Lemma

Let $A$ be a commutative ring with unity. Let: ::$\begin{xy}\xymatrix@L+2mu@+1em{ &  M_1 \ar[r]_*{\alpha_1}      \ar[d]^*{\phi_1} &  M_2 \ar[r]_*{\alpha_2}      \ar[d]^*{\phi_2} &  M_3 \ar[d]^*{\phi_3}      \ar[r] &   0 \\  0 \ar[r] &  N_1 \ar[r]_*{\beta_1} &  N_2 \ar[r]_*{\beta_2} &  N_3 & }\end{xy}$ be a commutative diagram of $A$-modules. Suppose that the rows are exact. Then we have a commutative diagram: ::$\begin{xy}\xymatrix@L+2mu@+1em{ &  \ker \phi_1 \ar[r]_*{\tilde\alpha_1}      \ar[d]^*{\iota_1} &  \ker \phi_2 \ar[r]_*{\tilde\alpha_2}      \ar[d]^*{\iota_2} &  \ker \phi_3      \ar[d]^*{\iota_3} & \\ &  M_1 \ar[r]_*{\alpha_1}      \ar[d]^*{\phi_1} &  M_2 \ar[r]_*{\alpha_2}      \ar[d]^*{\phi_2} &  M_3 \ar[d]^*{\phi_3}      \ar[r] &   0 \\  0 \ar[r] &  N_1 \ar[r]_*{\beta_1}      \ar[d]^*{\pi_1} &  N_2 \ar[r]_*{\beta_2}      \ar[d]^*{\pi_2} &  N_3 \ar[d]^*{\pi_3} \\ &  \operatorname{coker} \phi_1 \ar[r]_*{\bar\beta_1} &  \operatorname{coker} \phi_2 \ar[r]_*{\bar\beta_2} &  \operatorname{coker} \phi_3 & }\end{xy}$ where: *For $i=1,2,3$, $\iota_i$ is the inclusion mapping *For $i=1,2,3$, $\pi_i$ is the canonical epimorphism *For $i = 1,2$, $\tilde\alpha_i = \left({ \alpha_i }\right)_{| \ker \phi_i}$ *For $i = 1,2$, $\bar\beta_i$ is defined by: ::$\forall\; n_i + \operatorname{im}\phi_i \in \operatorname{coker}\phi_i : \bar\beta_i\left({ n_i + \operatorname{im}\phi_i }\right) = \beta_i\left({ n_i }\right) + \operatorname{im} \phi_{i+1}$ Moreover there exists a morphism $\delta : \ker\phi_3 \to \operatorname{coker}\phi_1$ such that we have an exact sequence: ::$\begin{xy}\xymatrix@L+2mu@+1em{  \ker \phi_1 \ar[r]_*{\tilde\alpha_1} &  \ker \phi_2 \ar[r]_*{\tilde\alpha_2} &  \ker\phi_3 \ar[r]_*{\delta} &   \operatorname{coker}\phi_1 \ar[r]_*{\bar\beta_1} &  \operatorname{coker}\phi_2 \ar[r]_*{\bar\beta_2} &  \operatorname{coker}\phi_3 }\end{xy}$


Hall's Marriage Theorem/Finite Set

Let $\SS = \sequence {S_k}_{k \in I}$ be a finite indexed family of finite sets. For each $F \subseteq I$, let $\displaystyle Y_F = \bigcup_{k \mathop \in F} S_k$. Let $Y = Y_I$. Then the following are equivalent: :$(1): \quad \SS$ satisfies the '''marriage condition''': for each subset $F$ of $I$, $\card F \le \card {Y_F}$. :$(2): \quad$ There exists an injection $f: I \to Y$ such that $\forall k \in I: \map f k \in S_k$.


Connecting Homomorphism is Functorial

Let $A$ be a commutative ring with unity. Let: ::$\begin{xy}\xymatrix{ &&& M_1 \ar@{->}[rr]      \ar@{->}[dl]      \ar@{->}[dd]|!{[d];[d]}\hole && M_2 %      \ar@{->}[rr]      \ar@{->}[dl]      \ar@{->}[dd]|!{[d];[d]}\hole && M_3 \ar@{->}[dl]       \ar@{->}[dd]|!{[d];[d]}\hole       \ar@{->}[rr] && 0 \\ && M_1' \ar@{->}[rr]    \ar@{->}[dd] && M_2' \ar@{->}[rr]      \ar@{->}[dd] && M_3' \ar@{->}[dd]        \ar@{->}[rr] && 0 \\ & 0 \ar@{->}[rr]|!{[r];[r]}\hole && N_1 \ar@{->}[rr]|!{[r];[r]}\hole      \ar@{->}[dl]_{F’} && N_2 \ar@{->}[rr]|!{[r];[r]}\hole      \ar@{->}[dl] && N_3 \ar@{->}[dl]_{F’’} \\ % 0 \ar@{->}[rr] && N_1' \ar@{->}^(.65){e’}[rr] && N_2' \ar@{->}[rr] && N_3' }\end{xy}$ be a commutative diagram of $A$-modules. Suppose that the rows are exact. {{finish}}


Law of Cosines/Proof 3

=== Lemma: Right Triangle === {{:Law of Cosines/Right Triangle}} === Acute Triangle === {{:Law of Cosines/Proof 3/Acute Triangle}} === Obtuse Triangle === {{:Law of Cosines/Proof 3/Obtuse Triangle}}


Hall's Marriage Theorem/General Set

Let $\SS = \sequence {S_k}_{k \in I}$ be an indexed family of finite sets. For each $F \subseteq I$, let $\displaystyle Y_F = \bigcup_{k \mathop \in F} S_k$. Let $Y = Y_I$. Then the following are equivalent: :$(1): \quad \SS$ satisfies the '''marriage condition''': for each finite subset $F$ of $I$, $\card F \le \card {Y_F}$. :$(2): \quad$ There exists an injection $f: I \to Y$ such that $\forall k \in I: \map f k \in S_k$.


Branch of Finite Propositional Tableau is Finite

Let $T$ be a finite propositional tableau. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is a finite branch.


Tableau Confutation is Finished

Let $T$ be a tableau confutation. Then $T$ is a finished tableau.


Expression of Vector as Linear Combination from Basis is Unique/General Result

Let $V$ be a vector space over a division ring $R$. Let $B$ be a basis for $V$. Let $x \in V$. Then there is a unique finite subset $C$ of $R \times B$ such that: :$\displaystyle x = \sum_{\left({r, v}\right) \mathop \in C} r \cdot v$ :$\forall \left({r, v}\right) \in C: r \ne 0_R$


Axiom:Axiom of Choice for Finite Sets/Proof from Ordering Principle

Suppose that the Ordering Principle holds. Let $\SS$ be a non-empty set of finite non-empty sets. Then there exists a choice function for $\SS$.


Axiom:Axiom of Choice for Finite Sets/Proof from Hall's Marriage Theorem

Suppose that Hall's Marriage Theorem holds for all sets of finite sets. Let $\SS$ be a non-empty set of finite, non-empty sets. Then there exists a choice function for $\SS$.


Dependent Choice for Finite Sets

Let $\RR$ be a binary relation on a non-empty set $S$. For each $a \in S$, let $C_a = \set {b \in S: a \mathrel \RR b }$ Suppose that: :For all $a \in S$, $C_a$ is a non-empty finite set. Let $s \in S$. Then there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that: :$x_0 = s$ :$\forall n \in \N: x_n \mathrel \RR x_{n+1}$


Dependent Choice (Fixed First Element)

Let $\mathcal R$ be a binary relation on a non-empty set $S$. Suppose that: :$\forall a \in S: \exists b \in S: a \mathrel{\mathcal R} b$ that is, that $\mathcal R$ is a left-total relation (specifically a ''serial relation''). Let $s \in S$. Then there exists a sequence $\left\langle{x_n}\right\rangle_{n \in \N}$ in $S$ such that: :$x_0 = s$ :$\forall n \in \N: x_n \mathrel{\mathcal R} x_{n+1}$


Locally Finite Connected Graph is Countable

Let $G = \struct {V, E}$ be a graph which is connected and locally finite. Then $G$ has countably many vertices and countably many edges.


Law of Cosines/Right Triangle

Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that: : $a$ is opposite $A$ : $b$ is opposite $B$ : $c$ is opposite $C$. Let $\triangle ABC$ be a right triangle such that $\angle A$ is right. Then: :$c^2 = a^2 + b^2 - 2 a b \cos C$


Minimal Infinite Successor Set is Minimal

The minimal infinite successor set $\omega$ is a subset of every infinite successor set.


Law of Cosines/Proof 3/Acute Triangle

Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that: : $a$ is opposite $A$ : $b$ is opposite $B$ : $c$ is opposite $C$. Let $\triangle ABC$ be an acute triangle. Then: :$c^2 = a^2 + b^2 - 2a b \cos C$


Law of Cosines/Proof 3/Obtuse Triangle

Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that: : $a$ is opposite $A$ : $b$ is opposite $B$ : $c$ is opposite $C$. Let $\triangle ABC$ be an obtuse triangle such that $A$ is obtuse Then: :$c^2 = a^2 + b^2 - 2a b \cos C$


König's Lemma/Countable

Let $G = \struct {V, E}$ be a graph with countably infinitely many vertices which is connected and is locally finite. Then every vertex of $G$ lies on a path of infinite length.


Schröder Rule/Proof 2

Let $A$, $B$ and $C$ be relations on a set $S$. Then the following are equivalent statements: :$(1): \quad A \circ B \subseteq C$ :$(2): \quad A^{-1} \circ \overline{C} \subseteq \overline{B}$ :$(3): \quad \overline{C} \circ B^{-1} \subseteq \overline{A}$ where: : $\circ$ denotes relation composition : $A^{-1}$ denotes the inverse of $A$ : $\overline{A}$ denotes the complement of $A$.


Equivalence of Definitions of Logical Consistence

Let $\mathbf H$ be a countable set (either finite or infinite) of WFFs of propositional logic. The following statements are logically equivalent: :$(1): \quad$ $\mathbf H$ has a model. :$(2): \quad$ $\mathbf H$ is consistent for the proof system of propositional tableaus. :$(3): \quad$ $\mathbf H$ has no tableau confutation.


Paths from Vertex of Minimal Length form Tree

Let $G = \left({V, E}\right)$ be a simple graph. Let $r \in V$ be a vertex in $G$. Let $P$ be the set of finite open paths beginning at $r$ that are of minimal length to reach their endpoints. {{refactor|Extract the below sentence and write it as a definition of a "minimal length path".}} That is, an open path $p$ beginning at $r$ and ending at some vertex $s$ is in $P$ {{iff}} there is no path $q$ beginning at $r$ and ending at $s$ such that the length of $q$ is less than the length of $p$. Let $p, q \in P$. Let $\left\{{p, q}\right\} \in E'$ {{iff}} either: :$q$ is formed by extending $p$ with one edge and one vertex of $G$ or: :$p$ is formed by extending $q$ with one edge and one vertex of $G$. Then $T = \left({P, E'}\right)$ is a tree.


Finite Sequences in Set Form Acyclic Graph

Let $S$ be a set. Let $V$ be the set of finite sequences in $S$. Let $E$ be the set of unordered pairs $\{p, q\}$ of elements of $V$ such that either: : $q$ is formed by extending $p$ by one element or : $p$ is formed by extending $q$ by one element. That is: : $| \operatorname{Dom}(p) * \operatorname{Dom}(q) | = 1$, where $*$ is symmetric difference and : $p \restriction D = q \restriction D$, where $D = \operatorname{Dom}(p) \cap \operatorname{Dom}(q)$ Then $T = (V, E)$ is an acyclic graph.


Connected Vertices are Connected by Path

Let $G = \struct {V, E}$ be a simple graph. Let $x, y \in V$. Let there exist a walk $w: \N_n \to V$ from $x$ to $y$. Then there exists a subsequence $z_n$ of $w$ which is a path from $x$ to $y$.


Negation as Implication of Bottom

$p \implies \bot \dashv\vdash \neg p$


Clavius's Law/Formulation 1/Proof 2

From Peirce's Law: :$\left({p \implies q}\right) \implies p \vdash p$ follows Clavius's Law: :$\neg p \implies p \vdash p$


Clavius's Law implies Law of Excluded Middle

From Clavius's Law: :$\neg p \implies p \vdash p$ follows the Law of Excluded Middle: :$\vdash p \lor \neg p$


Parity of Integer equals Parity of its Square/Even

Let $p \in \Z$ be an integer. Let $p$ be even. Then $p^2$ is also even.


Parity of Integer equals Parity of its Square/Odd

Let $p \in \Z$ be an integer. Let $p$ be odd. Then $p^2$ is also odd.


Negation of Excluded Middle is False/Form 1

$\neg (p \lor \neg p) \vdash \bot$


Negation of Excluded Middle is False/Form 2

$\vdash \neg \neg (p \lor \neg p)$


Double Negation Elimination implies Law of Excluded Middle/Proof 2

Let the Law of Double Negation Elimination be supposed to hold: :$\neg \neg p \vdash p$ Then the Law of Excluded Middle likewise holds: :$\vdash p \lor \neg p$


Injection iff Left Cancellable/Necessary Condition

Let $f: Y \to Z$ be an injection. Then $f$ is left cancellable.


Injection iff Left Cancellable/Sufficient Condition

Let $f: Y \to Z$ be a mapping which is left cancellable. Then $f$ is an injection.


Peirce's Law implies Law of Excluded Middle

From Peirce's Law: :$\left({p \implies q}\right) \implies p \vdash p$ follows the Law of Excluded Middle: :$\vdash p \lor \neg p$


Pseudocomplemented Lattice is Bounded

Let $(L, \wedge, \vee, \preceq)$ be a pseudocomplemented lattice. Then $(L, \wedge, \vee, \preceq)$ is a bounded lattice.


Topology forms Complete Lattice

Let $\struct {X, \tau}$ be a topological space. Then $\struct {\tau, \subseteq}$ is a complete lattice.


Law of Excluded Middle implies Peirce's Law

From the Law of Excluded Middle follows Peirce's Law: :$\left({p \lor \neg p}\right) \vdash \left({\left({p \implies q}\right) \implies p}\right) \implies p$


Inverse of Composite Bijection/Proof 2

Let $f$ and $g$ be bijections. Then: :$\paren {g \circ f}^{-1} = f^{-1} \circ g^{-1}$ and $f^{-1} \circ g^{-1}$ is itself a bijection.


Sierpiński's Theorem

Let $\left({S, \tau}\right)$ be a compact connected Hausdorff space. Let $\left\{{F_n: n \in \N}\right\}$ be a pairwise disjoint closed cover of $S$. {{explain|In context it's obvious, but worth mentioning that $F_n$ is a finite cover as well?}} Then $F_n = S$ for some $n \in \N$.


Sierpiński's Theorem/Lemma 1

Let $\struct {S, \tau}$ be a compact connected Hausdorff space. Let $A$ be a closed, non-empty proper subset of $S$. Let $C$ be a component of $A$. Then: :$C \cap \partial A \ne \O$ where $\partial A$ denotes the boundary of $A$.


Quasicomponent of Compact Hausdorff Space is Connected

Let $\left({X, \tau}\right)$ be a compact Hausdorff space. Let $C$ be a quasicomponent of $\left({X, \tau}\right)$. Then $C$ is connected.


Quasicomponents and Components are Equal in Compact Hausdorff Space

Let $T = \left({S, \tau}\right)$ be a compact Hausdorff space. Then for each $A \subseteq S$: $A$ is a component of $S$ {{iff}} $A$ is a quasicomponent of $S$.


Derived Set in T1 Space is Closed

Let $\struct {X, \tau}$ be a $T_1$ space. Let $S \subseteq X$. Let $S'$ be the derived set of $S$. Then $S'$ is closed.


Multiplicative Inverse in Monoid of Integers Modulo m

Let $\struct {\Z_m, \times_m}$ be the multiplicative monoid of integers modulo $m$. Then: :$\eqclass k m \in \Z_m$ has an inverse in $\struct {\Z_m, \times_m}$ {{iff}}: :$k \perp m$


Element Commutes with Square in Group

Let $\left({G, \circ}\right)$ be a group. Let $x \in G$. Then $x$ commutes with $x \circ x$.


Element Commutes with Square in Semigroup

Let $\left({S, \circ}\right)$ be a semigroup. Let $x \in S$. Then $x$ commutes with $x \circ x$.


Subsets of Disjoint Sets are Disjoint

Let $S$ and $T$ be disjoint sets. Let $S' \subseteq S$ and $T' \subseteq T$. Then $S'$ and $T'$ are disjoint.


Network with Positive Integer Mapping is Multigraph

Let $N = \struct {V, E, w}$ be a network whose weights are all strictly positive integers. Then $N$ can be represented as a multigraph. Conversely, any multigraph can be expressed as a network whose weights are all strictly positive integers.


Complete Graph is Regular

Let $K_p$ be the complete graph of order $p$. Then $K_p$ is $p-1$-regular.


No Simple Graph is Perfect

Let $G$ be a simple graph whose order is $2$ or greater. Then $G$ is not perfect.


Tukey-Teichmüller Lemma

Let $\FF$ be a non-empty set of finite character. Then $\FF$ has an element which is maximal with respect to the set inclusion relation.


Circuit has Three Edges or More

Let $G$ be a simple graph. Let $C$ be a circuit in $G$. Then $C$ has at least $3$ edges.


Dirac's Theorem/Proof 1

If a connected graph $G$ has $n \ge 3$ vertices and the degree of each vertex is at least $\dfrac n 2$, then $G$ is Hamiltonian.


Dirac's Theorem/Proof 2


Graph of Cube is Hamiltonian

The graph of the cube is Hamiltonian.


Graph of Icosahedron is Hamiltonian

The graph of the icosahedron is Hamiltonian.


Path in Tree is Unique/Necessary Condition

Let $T$ be a tree. Then there is exactly one path between any two vertices.


Path in Tree is Unique/Sufficient Condition

Let $T$ be a graph. Let $T$ be such that between any two vertices there is exactly one path. Then $T$ is a tree.


Size of Tree is One Less than Order/Necessary Condition

Let $T$ be a tree of order $n$. Then the size of $T$ is $n-1$.


Size of Tree is One Less than Order/Sufficient Condition

Let $T$ be a connected simple graph of order $n$. Let the size of $T$ be $n-1$. Then $T$ is a tree.


Size of Tree is One Less than Order/Necessary Condition/Induction Step/Proof 1

Let the following hold: :A tree of order $k$ is of size $k - 1$. Then this holds: :A tree of order $k + 1$  is of size $k$.


Size of Tree is One Less than Order/Necessary Condition/Induction Step/Proof 2

Let the following hold: :For all $j \le k$, a tree of order $j$ is of size $j-1$. Then this holds: :A tree of order $k+1$  is of size $k$.


Kruskal's Algorithm produces Minimum Spanning Tree

Kruskal's Algorithm produces a minimum spanning tree.


Regular Graph is Tree iff Complete Graph of Order 2

Let $G$ be a non-edgeless regular graph. Then $G$ is a tree {{iff}} $G$ is $K_2$, the complete graph of order $2$.


Perimeter of Circle

The perimeter $C$ of a circle with radius $r$ is given by: : $C = 2 \pi r$


Euler's Criterion

Let $p$ be an odd prime. Let $a \not \equiv 0 \pmod p$. Then: {{begin-eqn}} {{eqn | l = a^{\frac {p-1} 2}       | o = \equiv       | r = 1       | rr= \pmod p       | c = {{iff}} $a$ is a quadratic residue of $p$ }} {{eqn | l=a ^{\frac {p-1} 2}       | o = \equiv       | r = -1       | rr= \pmod p       | c = {{iff}} $a$ is a quadratic non-residue of $p$. }} {{end-eqn}}


Value of Degree in Radians

The value of a degree in radians is given by: :$1 \degrees = \dfrac {\pi} {180} \radians \approx 0.01745 \ 32925 \ 19943 \ 29576 \ 92 \ldots \radians$ {{OEIS|A019685}}


Difference of Two Squares/Geometric Proof

:$\forall x, y \in \R: x^2 - y^2 = \paren {x + y} \paren {x - y}$


Area of Parallelogram/Square

The area of a square equals the product of one of its bases and the associated altitude.


Area of Parallelogram/Rectangle

The area of a rectangle equals the product of one of its bases and the associated altitude.


Area of Parallelogram/Parallelogram

Let $ABCD$ be a parallelogram whose adjacent sides are of length $a$ and $b$ enclosing an angle $\theta$. The area of $ABCD$ equals the product of one of its bases and the associated altitude: {{begin-eqn}} {{eqn | l = \map \Area {ABCD}       | r = b h       | c =  }} {{eqn | r = a b \sin \theta       | c =  }} {{end-eqn}} where: :$b$ is the side of $ABCD$ which has been chosen to be the base :$h$ is the altitude of $ABCD$ from $b$.


Perimeter of Rectangle

Let $ABCD$ be a rectangle whose side lengths are $a$ and $b$. The perimeter of $ABCD$ is $2 a + 2 b$.


Perimeter of Parallelogram

Let $ABCD$ be a parallelogram whose side lengths are $a$ and $b$. The perimeter of $ABCD$ is $2 a + 2 b$.


Perimeter of Triangle

Let $ABC$ be a triangle. Then the perimeter $P$ of $ABC$ is given by: :$P = a + b + c$ where $a, b, c$ are the lengths of the sides of $ABC$.


Area of Trapezoid

:410px Let $ABCD$ be a trapezoid: :whose parallel sides are of lengths $a$ and $b$ and :whose height is $h$. Then the area of $ABCD$ is given by: :$\Box ABCD = \dfrac {h \paren {a + b} } 2$


Perimeter of Trapezoid

:400px Let $ABCD$ be a trapezoid: :whose parallel sides are of lengths $a$ and $b$ :whose height is $h$. and :whose non-parallel sides are at angles $\theta$ and $\phi$ with the parallels. The perimeter $P$ of $ABCD$ is given by: :$P = a + b + h \paren {\csc \theta + \csc \phi}$ where $\csc$ denotes cosecant.


Perimeter of Regular Polygon

Let $P$ be a regular $n$-sided polygon whose side length is $b$. Then the perimeter $L$ of $P$ is given by: :$L = n b$


Commensurability is Transitive

Let $a$, $b$, $c$ be three real numbers. Let $a$ and $b$ be commensurable, and $b$ and $c$ be commensurable. Then $a$ and $c$ are commensurable.


Arc Length of Sector

Let $\mathcal C = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\mathcal C$ whose angle between $AB$ and $AC$ is $\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$\mathcal s = r \theta$ where: :$r = AB$ is the length of the radius of the circle :$\theta$ is measured in radians.


Length of Inradius of Triangle

Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the length of the inradius $r$ of $\triangle ABC$ is given by: :$r = \dfrac {\sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } } s$ where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.


Length of Circumradius of Triangle

Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the length of the circumradius $R$ of $\triangle ABC$ is given by: :$R = \dfrac {abc} {4 \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } }$ where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.


Area of Triangle in Terms of Inradius

Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\mathcal A$ of $\triangle ABC$ is given by: :$\mathcal A = r s$ where: :$r$ is the inradius of $\triangle ABC$ :$s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.


Area of Regular Polygon by Circumradius

Let $P$ be a regular $n$-gon. Let $C$ be a circumcircle of $P$. Let the radius of $C$ be $r$. Then the area $\mathcal A$ of $P$ is given by: :$\mathcal A = \dfrac 1 2 n r^2 \sin \dfrac {2 \pi} n$


Perimeter of Regular Polygon by Inradius

Let $P$ be a regular $n$-gon. Let $C$ be an incircle of $P$. Let the radius of $C$ be $r$. Then the perimeter $\mathcal P$ of $P$ is given by: :$\mathcal P = 2 n r \tan \dfrac \pi n$


Image of Interval by Continuous Function is Interval/Proof 1

Let $I$ be a real interval. Let $f: I \to \R$ be a continuous real function. Then the image of $f$ is a real interval.


Image of Interval by Continuous Function is Interval/Proof 2


Area of Isosceles Triangle

Let $\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $\theta$ be the angle of the apex $A$. Let $r$ be the length of a leg of $\triangle ABC$. Then the area $\mathcal A$ of $\triangle ABC$ is given by: :$\mathcal A = \dfrac 1 2 r^2 \sin \theta$


Area of Segment of Circle

Let $C$ be a circle of radius $r$. Let $S$ be a segment of $C$ such that its base subtends an angle of $\theta$ at the center of the circle. Then the area $\AA$ of $S$ is given by: :$\AA = \dfrac 1 2 r^2 \paren {\theta - \sin \theta}$ where $\theta$ is measured in radians.


Mills' Theorem

There exists a real number $A$ such that $\left\lfloor{A^{3^n} }\right\rfloor$ is a prime number for all $n \in \N_{>0}$, where: :$\left\lfloor{x}\right\rfloor$ denotes the floor function of $x$ :$\N$ denotes the set of all natural numbers.


Equation of Ellipse in Reduced Form

Let $K$ be an ellipse aligned in a cartesian plane in reduced form. Let: :the major axis of $K$ have length $2 a$ :the minor axis of $K$ have length $2 b$.


Area of Ellipse

Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$. The area $\mathcal A$ of $K$ is given by: : $\mathcal A = \pi a b$


Equation of Ellipse in Reduced Form/Cartesian Frame

The equation of $K$ is: :$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$


Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form

The equation of $K$ in parametric form is: :$x = a \cos \theta, y = b \sin \theta$


Perimeter of Ellipse

Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$. The perimeter $\PP$ of $K$ is given by: :$\ds \PP = 4 a \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \theta} \rd \theta$ where: :$k = \dfrac {\sqrt {a^2 - b^2} } a$ The definite integral: :$\displaystyle \PP = \int_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \theta} \rd \theta$ is the complete elliptic integral of the second kind.


Distance of Point from Origin in Cartesian Coordinates

Let $P = \tuple {x, y}$ be a point in the cartesian plane. Then $P$ is at a distance of $\sqrt {x^2 + y^2}$ from the origin.


Composite of Monomorphisms is Monomorphism

Let: :$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$ :$\struct {S_2, *_1, *_2, \ldots, *_n}$ :$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ be algebraic structures. Let: :$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$ :$\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ be monomorphisms. Then the composite of $\phi$ and $\psi$ is also a monomorphism.


Composite of Automorphisms is Automorphism

Let $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ be an algebraic structure. Let: :$\phi: \struct {S, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ :$\psi: \struct {S, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ be automorphisms. Then the composite of $\phi$ and $\psi$ is also an automorphism.


Composite of Endomorphisms is Endomorphism

Let $\left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ be an algebraic structure. Let: : $\phi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ : $\psi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ be endomorphisms. Then the composite of $\phi$ and $\psi$ is also an endomorphism.


Sine of Angle in Cartesian Plane

Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\sin \theta = \dfrac y r$ where $\sin$ denotes the sine of $\theta$.


Equivalence of Definitions of Sine of Angle

Let $\theta$ be an angle. {{TFAE|def = Sine of Angle|view = sine of $\theta$}}


Cardinality of Subset Relation on Power Set of Finite Set

Let $S$ be a set such that: :$\card S = n$ where $\card S$ denotes the cardinality of $S$. From Subset Relation on Power Set is Partial Ordering we have that $\struct {\powerset S, \subseteq}$ is an ordered set. The cardinality of $\subseteq$ as a relation is $3^n$.


Quotient Structure of Semigroup is Semigroup

Let $\RR$ be a congruence relation on a semigroup $\struct {S, \circ}$. Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a semigroup.


Quotient Structure of Monoid is Monoid

Let $\RR$ be a congruence relation on a monoid $\struct {S, \circ}$ with an identity $e$. Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a monoid.


Quotient Structure of Group is Group

Let $\RR$ be a congruence relation on a group $\struct {G, \circ}$. Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a group.


Quotient Structure of Abelian Group is Abelian Group

Let $\RR$ be a congruence relation on an abelian group $\struct {G, \circ}$. Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is an abelian group.


Quotient Structure is Similar to Structure

Let $\RR$ be a congruence relation on a algebraic structure $\struct {G, \circ}$. {{MissingLinks|"similar" structure?}} Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a similar structure to $\struct {G, \circ}$.


Shape of Secant Function

The nature of the secant function on the set of real numbers $\R$ is as follows: :$(1): \quad \sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$ :$(2): \quad \sec x$ is continuous and strictly decreasing on the intervals $\hointr {-\pi} {-\dfrac \pi 2}$ and $\hointl {-\dfrac \pi 2} 0$ :$(3): \quad \sec x \to + \infty$ as $x \to -\dfrac \pi 2^+$ :$(4): \quad \sec x \to + \infty$ as $x \to \dfrac \pi 2^-$ :$(5): \quad \sec x \to - \infty$ as $x \to \dfrac \pi 2^+$ :$(6): \quad \sec x \to - \infty$ as $x \to \dfrac {3 \pi} 2^-$


Shape of Cosecant Function

The nature of the cosecant function on the set of real numbers $\R$ is as follows: :$(1): \quad$ strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$ :$(2): \quad$ strictly increasing on the intervals $\hointr {\dfrac \pi 2} \pi$ and $\hointl \pi {\dfrac {3 \pi} 2}$ :$(3): \quad$ $\csc x \to +\infty$ as $x \to 0^+$ :$(4): \quad$ $\csc x \to +\infty$ as $x \to \pi^-$ :$(5): \quad$ $\csc x \to -\infty$ as $x \to \pi^+$ :$(6): \quad$ $\csc x \to -\infty$ as $x \to 2 \pi^-$


Integer Multiples under Multiplication form Semigroup

Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, \times}$ is a semigroup. If $\size n > 1$ then $\struct {n \Z, \times}$ has no identity.


Order of External Direct Product

Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures. Then the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$.


Congruence (Number Theory) is Congruence Relation

Congruence modulo $m$ is a congruence relation on $\left({\Z, +}\right)$.


Equivalence of Definitions of Cotangent of Angle

Let $\theta$ be an angle. {{TFAE|def = Cotangent of Angle|cotangent}}


Equivalence of Definitions of Cosecant of Angle

Let $\theta$ be an angle. {{TFAE|def = Cosecant of Angle|view = cosecant}}


Cosine of Angle in Cartesian Plane

Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\cos \theta = \dfrac x r$ where $\cos$ denotes the cosine of $\theta$.


Tangent of Angle in Cartesian Plane

Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\tan \theta = \dfrac y x$ where $\tan$ denotes the tangent of $\theta$.


Cotangent of Angle in Cartesian Plane

Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\cot \theta = \dfrac x y$ where $\cot$ denotes the cotangent of $\theta$.


Secant of Angle in Cartesian Plane

Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\sec \theta = \dfrac r x$ where $\sec$ denotes the secant of $\theta$.


Cosecant of Angle in Cartesian Plane

Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\csc \theta = \dfrac r x$ where $\csc$ denotes the secant of $\theta$.


Tangent is Sine divided by Cosine

Let $\theta$ be an angle such that $\cos \theta \ne 0$. Then: :$\tan \theta = \dfrac {\sin \theta} {\cos \theta}$ where $\tan$, $\sin$ and $\cos$ mean tangent, sine and cosine respectively.


Cotangent is Cosine divided by Sine

Let $\theta$ be an angle such that $\sin \theta \ne 0$. Then: :$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$ where $\cot$, $\sin$ and $\cos$ mean cotangent, sine and cosine respectively.


Cotangent is Reciprocal of Tangent

:$\cot \theta = \dfrac 1 {\tan \theta}$


Secant is Reciprocal of Cosine

:$\sec \theta = \dfrac 1 {\cos \theta}$


Cosecant is Reciprocal of Sine

:$\csc \theta = \dfrac 1 {\sin \theta}$


Particular Values of Sine Function

The following values of the sine function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Sine of Zero === {{:Sine of Zero is Zero}} === Sine of 15 Degrees === {{:Sine of 15 Degrees}} === Sine of 30 Degrees === {{:Sine of 30 Degrees}} === Sine of 45 Degrees === {{:Sine of 45 Degrees}} === Sine of 60 Degrees === {{:Sine of 60 Degrees}} === Sine of 75 Degrees === {{:Sine of 75 Degrees}} === Sine of Right Angle === {{:Sine of Right Angle}} === Sine of 105 Degrees === {{:Sine of 105 Degrees}} === Sine of 120 Degrees === {{:Sine of 120 Degrees}} === Sine of 135 Degrees === {{:Sine of 135 Degrees}} === Sine of 150 Degrees === {{:Sine of 150 Degrees}} === Sine of 165 Degrees === {{:Sine of 165 Degrees}} === Sine of Straight Angle === {{:Sine of Straight Angle}} === Sine of 195 Degrees === {{:Sine of 195 Degrees}} === Sine of 210 Degrees === {{:Sine of 210 Degrees}} === Sine of 225 Degrees === {{:Sine of 225 Degrees}} === Sine of 240 Degrees === {{:Sine of 240 Degrees}} === Sine of 255 Degrees === {{:Sine of 255 Degrees}} === Sine of Three Right Angles === {{:Sine of Three Right Angles}} === Sine of 285 Degrees === {{:Sine of 285 Degrees}} === Sine of 300 Degrees === {{:Sine of 300 Degrees}} === Sine of 315 Degrees === {{:Sine of 315 Degrees}} === Sine of 330 Degrees === {{:Sine of 330 Degrees}} === Sine of 345 Degrees === {{:Sine of 345 Degrees}} === Sine of Full Angle === {{:Sine of Full Angle}}


Positive-Term Generalized Sum Converges iff Supremum

Let $\struct {G, \circ, \le}$ be an abelian totally ordered group, considered under the order topology. Let $\set {x_i: i \in I}$ be an indexed set of positive elements of $G$. {{explain|What actually does "positive" mean in this context?}} Then: :the generalized sum $\displaystyle \sum \set {x_i: i \in I}$ converges to a point $x \in G$ {{iff}} :$x$ is the supremum of: ::$P := \displaystyle \set {\sum_{i \mathop \in F} x_i: \text{$F \subseteq I$ and $F$ is finite} }$


Half Angle Formulas/Sine

{{begin-eqn}} {{eqn | l = \sin \frac \theta 2       | r = +\sqrt {\frac {1 - \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$ }} {{eqn | l = \sin \frac \theta 2       | r = -\sqrt {\dfrac {1 - \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text {III}$ or quadrant $\text {IV}$ }} {{end-eqn}}


Half Angle Formulas/Cosine

{{begin-eqn}} {{eqn | l = \cos \frac \theta 2       | r = +\sqrt {\frac {1 + \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {IV}$ }} {{eqn | l = \cos \frac \theta 2       | r = -\sqrt {\frac {1 + \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {III}$ }} {{end-eqn}}


Half Angle Formulas/Tangent

{{begin-eqn}} {{eqn | l = \tan \frac \theta 2       | r = +\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }       | c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {III}$ }} {{eqn | l = \tan \frac \theta 2       | r = -\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }       | c = for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {IV}$ }} {{end-eqn}} where $\tan$ denotes tangent and $\cos$ denotes cosine. When $\theta = \paren {2 k + 1} \pi$, $\tan \dfrac \theta 2$ is undefined.


Kernel of Normal Operator is Kernel of Adjoint

Let $H$ be a Hilbert space. Let $A \in \map B H$ be a normal operator. Then: :$\ker A = \ker A^*$ where: :$\ker$ denotes kernel :$A^*$ denotes the adjoint of $A$.


Half Angle Formulas/Tangent/Corollary 1

:$\tan \dfrac \theta 2 = \dfrac {\sin \theta} {1 + \cos \theta}$


Half Angle Formulas/Tangent/Corollary 2

:$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$


Half Angle Formulas/Tangent/Corollary 3

:$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$


Bisection of Angle in Cartesian Plane

Let $\theta$ be the angular coordinate of a point $P$ in a polar coordinate plane. Let $QOR$ be a straight line that bisects the angle $\theta$. Then the angular coordinates of $Q$ and $R$ are $\dfrac \theta 2$ and $\pi + \dfrac \theta 2$.


Sine of 15 Degrees

:$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$


Sine of 30 Degrees

:$\sin 30 \degrees = \sin \dfrac \pi 6 = \dfrac 1 2$


Sine of 45 Degrees

:$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$


Sine of 60 Degrees

:$\sin 60 \degrees = \sin \dfrac \pi 3 = \dfrac {\sqrt 3} 2$


Sine of 75 Degrees

:$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$


Sine of Right Angle

:$\sin 90^\circ = \sin \dfrac \pi 2 = 1$


Sine of Angle plus Right Angle

: $\sin \paren {x + \dfrac \pi 2} = \cos x$


Sine of 105 Degrees

:$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$


Sine of 120 Degrees

:$\sin 120 \degrees = \sin \dfrac {2 \pi} 3 = \dfrac {\sqrt 3} 2$


Sine of 135 Degrees

:$\sin 135 \degrees = \sin \dfrac {3 \pi} 4 = \dfrac {\sqrt 2} 2$


Sine of 150 Degrees

:$\sin 150 \degrees = \sin \dfrac {5 \pi} 6 = \dfrac 1 2$


Sine of 165 Degrees

:$\sin 165 \degrees = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$


Sine of Straight Angle

:$\sin 180 \degrees = \sin \pi = 0$


Sine of Supplementary Angle

:$\sin \paren {\pi - \theta} = \sin \theta$ where $\sin$ denotes sine. That is, the sine of an angle equals its supplement.


Cosine of Supplementary Angle

:$\map \cos {\pi - \theta} = -\cos \theta$ where $\cos$ denotes cosine. That is, the cosine of an angle is the negative of its supplement.


Sine of Conjugate Angle

:$\map \sin {2 \pi - \theta} = -\sin \theta$ where $\sin$ denotes sine. That is, the sine of an angle is the negative of its conjugate.


Cosine of Conjugate Angle

:$\map \cos {2 \pi - \theta} = \cos \theta$ where $\cos$ denotes cosine. That is, the cosine of an angle equals its conjugate.


Sine of 195 Degrees

:$\sin 195 \degrees = \sin \dfrac {13 \pi} {12} = -\dfrac {\sqrt 6 - \sqrt 2} 4$


Sine of 210 Degrees

:$\sin 210 \degrees = \sin \dfrac {7 \pi} 6 = -\dfrac 1 2$


Sine of 225 Degrees

:$\sin 225 \degrees = \sin \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$


Sine of 240 Degrees

:$\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$


Sine of 255 Degrees

:$\sin 255^\circ = \sin \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$


Sine of Three Right Angles

:$\sin 270 \degrees = \sin \dfrac {3 \pi} 2 = -1$


Sine of 285 Degrees

:$\sin 285^\circ = \sin \dfrac {19 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$


Sine of 300 Degrees

:$\sin 300 \degrees = \sin \dfrac {5 \pi} 3 = -\dfrac {\sqrt 3} 2$


Sine of 315 Degrees

:$\sin 315 \degrees = \sin \dfrac {7 \pi} 4 = -\dfrac {\sqrt 2} 2$


Sine of 330 Degrees

:$\sin 330 \degrees = \sin \dfrac {11 \pi} 6 = -\dfrac 1 2$


Sine of 345 Degrees

:$\sin 345^\circ = \sin \dfrac {23 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$


Sine of Full Angle

:$\sin 360^\circ = \sin 2 \pi = 0$


Particular Values of Cosine Function

The following values of the cosine function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cosine of Zero === {{:Cosine of Zero is One}} === Cosine of 15 Degrees === {{:Cosine of 15 Degrees}} === Cosine of 30 Degrees === {{:Cosine of 30 Degrees}} === Cosine of 45 Degrees === {{:Cosine of 45 Degrees}} === Cosine of 60 Degrees === {{:Cosine of 60 Degrees}} === Cosine of 75 Degrees === {{:Cosine of 75 Degrees}} === Cosine of Right Angle === {{:Cosine of Right Angle}} === Cosine of 105 Degrees === {{:Cosine of 105 Degrees}} === Cosine of 120 Degrees === {{:Cosine of 120 Degrees}} === Cosine of 135 Degrees === {{:Cosine of 135 Degrees}} === Cosine of 150 Degrees === {{:Cosine of 150 Degrees}} === Cosine of 165 Degrees === {{:Cosine of 165 Degrees}} === Cosine of Straight Angle === {{:Cosine of Straight Angle}} === Cosine of 195 Degrees === {{:Cosine of 195 Degrees}} === Cosine of 210 Degrees === {{:Cosine of 210 Degrees}} === Cosine of 225 Degrees === {{:Cosine of 225 Degrees}} === Cosine of 240 Degrees === {{:Cosine of 240 Degrees}} === Cosine of 255 Degrees === {{:Cosine of 255 Degrees}} === Cosine of Three Right Angles === {{:Cosine of Three Right Angles}} === Cosine of 285 Degrees === {{:Cosine of 285 Degrees}} === Cosine of 300 Degrees === {{:Cosine of 300 Degrees}} === Cosine of 315 Degrees === {{:Cosine of 315 Degrees}} === Cosine of 330 Degrees === {{:Cosine of 330 Degrees}} === Cosine of 345 Degrees === {{:Cosine of 345 Degrees}} === Cosine of Full Angle === {{:Cosine of Full Angle}}


Cosine of 15 Degrees

:$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$


Cosine of 30 Degrees

:$\cos 30 \degrees = \cos \dfrac \pi 6 = \dfrac {\sqrt 3} 2$


Cosine of 45 Degrees

:$\cos 45 \degrees = \cos \dfrac \pi 4 = \dfrac {\sqrt 2} 2$


Cosine of 60 Degrees

:$\cos 60 \degrees = \cos \dfrac \pi 3 = \dfrac 1 2$


Cosine of 75 Degrees

:$\cos 75^\circ = \cos \dfrac {5 \pi}{12} = \dfrac {\sqrt 6 - \sqrt 2} 4$


Cosine of Right Angle

:$\cos 90^\circ = \cos \dfrac \pi 2 = 0$


Cosine of 105 Degrees

:$\cos 105 \degrees = \cos \dfrac {7 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$


Cosine of Angle plus Right Angle

:$\map \cos {x + \dfrac \pi 2} = -\sin x$


Cosine of 120 Degrees

:$\cos 120 \degrees = \cos \dfrac {2 \pi} 3 = -\dfrac 1 2$


Cosine of 135 Degrees

:$\cos 135 \degrees = \cos \dfrac {3 \pi} 4 = -\dfrac {\sqrt 2} 2$


Cosine of 150 Degrees

:$\cos 150 \degrees = \cos \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 2$


Cosine of 165 Degrees

:$\cos 165 \degrees = \cos \dfrac {11 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$


Cosine of Straight Angle

:$\cos 180^\circ = \cos \pi = - 1$


Cosine of 195 Degrees

:$\cos 195 \degrees = \cos \dfrac {13 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$


Cosine of 210 Degrees

:$\cos 210 \degrees = \cos \dfrac {7 \pi} 6 = -\dfrac {\sqrt 3} 2$


Cosine of 225 Degrees

:$\cos 225 \degrees = \cos \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$


Cosine of 240 Degrees

:$\cos 240 \degrees = \cos \dfrac {4 \pi} 3 = -\dfrac 1 2$


Cosine of 255 Degrees

:$\cos 255^\circ = \cos \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$


Cosine of Three Right Angles

:$\cos 270^\circ = \cos \dfrac {3 \pi} 2 = 0$


Cosine of 285 Degrees

:$\cos 285^\circ = \cos \dfrac {19 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$


Cosine of 300 Degrees

:$\cos 300 \degrees = \cos \dfrac {5 \pi} 3 = \dfrac 1 2$


Cosine of 315 Degrees

:$\cos 315 \degrees = \cos \dfrac {7 \pi} 4 = \dfrac {\sqrt 2} 2$


Cosine of 330 Degrees

:$\cos 330 \degrees = \cos \dfrac {11 \pi} 6 = \dfrac {\sqrt 3} 2$


Cosine of 345 Degrees

:$\cos 345 \degrees = \cos \dfrac {23 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$


Cosine of Full Angle

:$\cos 360 \degrees = \cos 2 \pi = 1$


Particular Values of Tangent Function

The following values of the tangent function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Tangent of Zero === {{:Tangent of Zero}} === Tangent of 15 Degrees === {{:Tangent of 15 Degrees}} === Tangent of 22.5 Degrees === {{:Tangent of 22.5 Degrees}} === Tangent of 30 Degrees === {{:Tangent of 30 Degrees}} === Tangent of 45 Degrees === {{:Tangent of 45 Degrees}} === Tangent of 60 Degrees === {{:Tangent of 60 Degrees}} === Tangent of 67.5 Degrees === {{:Tangent of 67.5 Degrees}} === Tangent of 75 Degrees === {{:Tangent of 75 Degrees}} === Tangent of Right Angle === {{:Tangent of Right Angle}} === Tangent of 105 Degrees === {{:Tangent of 105 Degrees}} === Tangent of 120 Degrees === {{:Tangent of 120 Degrees}} === Tangent of 135 Degrees === {{:Tangent of 135 Degrees}} === Tangent of 150 Degrees === {{:Tangent of 150 Degrees}} === Tangent of 165 Degrees === {{:Tangent of 165 Degrees}} === Tangent of Straight Angle === {{:Tangent of Straight Angle}} === Tangent of 195 Degrees === {{:Tangent of 195 Degrees}} === Tangent of 210 Degrees === {{:Tangent of 210 Degrees}} === Tangent of 225 Degrees === {{:Tangent of 225 Degrees}} === Tangent of 240 Degrees === {{:Tangent of 240 Degrees}} === Tangent of 255 Degrees === {{:Tangent of 255 Degrees}} === Tangent of Three Right Angles === {{:Tangent of Three Right Angles}} === Tangent of 285 Degrees === {{:Tangent of 285 Degrees}} === Tangent of 300 Degrees === {{:Tangent of 300 Degrees}} === Tangent of 315 Degrees === {{:Tangent of 315 Degrees}} === Tangent of 330 Degrees === {{:Tangent of 330 Degrees}} === Tangent of 345 Degrees === {{:Tangent of 345 Degrees}} === Tangent of Full Angle === {{:Tangent of Full Angle}}


Tangent of Zero

:$\tan 0 = 0$


Tangent of 15 Degrees

:$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$


Tangent of 30 Degrees

:$\tan 30^\circ = \tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$


Particular Values of Cotangent Function

The following values of the cotangent function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cotangent of Zero === {{:Cotangent of Zero}} === Cotangent of 15 Degrees === {{:Cotangent of 15 Degrees}} === Cotangent of 30 Degrees === {{:Cotangent of 30 Degrees}} === Cotangent of 45 Degrees === {{:Cotangent of 45 Degrees}} === Cotangent of 60 Degrees === {{:Cotangent of 60 Degrees}} === Cotangent of 75 Degrees === {{:Cotangent of 75 Degrees}} === Cotangent of Right Angle === {{:Cotangent of Right Angle}} === Cotangent of 105 Degrees === {{:Cotangent of 105 Degrees}} === Cotangent of 120 Degrees === {{:Cotangent of 120 Degrees}} === Cotangent of 135 Degrees === {{:Cotangent of 135 Degrees}} === Cotangent of 150 Degrees === {{:Cotangent of 150 Degrees}} === Cotangent of 165 Degrees === {{:Cotangent of 165 Degrees}} === Cotangent of Straight Angle === {{:Cotangent of Straight Angle}} === Cotangent of 195 Degrees === {{:Cotangent of 195 Degrees}} === Cotangent of 210 Degrees === {{:Cotangent of 210 Degrees}} === Cotangent of 225 Degrees === {{:Cotangent of 225 Degrees}} === Cotangent of 240 Degrees === {{:Cotangent of 240 Degrees}} === Cotangent of 255 Degrees === {{:Cotangent of 255 Degrees}} === Cotangent of Three Right Angles === {{:Cotangent of Three Right Angles}} === Cotangent of 285 Degrees === {{:Cotangent of 285 Degrees}} === Cotangent of 300 Degrees === {{:Cotangent of 300 Degrees}} === Cotangent of 315 Degrees === {{:Cotangent of 315 Degrees}} === Cotangent of 330 Degrees === {{:Cotangent of 330 Degrees}} === Cotangent of 345 Degrees === {{:Cotangent of 345 Degrees}} === Cotangent of Full Angle === {{:Cotangent of Full Angle}}


Particular Values of Secant Function

The following values of the secant function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Secant of Zero === {{:Secant of Zero}} === Secant of $15 \degrees$ === {{:Secant of 15 Degrees}} === Secant of $30 \degrees$ === {{:Secant of 30 Degrees}} === Secant of $45 \degrees$ === {{:Secant of 45 Degrees}} === Secant of $60 \degrees$ === {{:Secant of 60 Degrees}} === Secant of $75 \degrees$ === {{:Secant of 75 Degrees}} === Secant of Right Angle === {{:Secant of Right Angle}} === Secant of $105 \degrees$ === {{:Secant of 105 Degrees}} === Secant of $120 \degrees$ === {{:Secant of 120 Degrees}} === Secant of $135 \degrees$ === {{:Secant of 135 Degrees}} === Secant of $150 \degrees$ === {{:Secant of 150 Degrees}} === Secant of $165 \degrees$ === {{:Secant of 165 Degrees}} === Secant of Straight Angle === {{:Secant of Straight Angle}} === Secant of $195 \degrees$ === {{:Secant of 195 Degrees}} === Secant of $210 \degrees$ === {{:Secant of 210 Degrees}} === Secant of $225 \degrees$ === {{:Secant of 225 Degrees}} === Secant of $240 \degrees$ === {{:Secant of 240 Degrees}} === Secant of $255 \degrees$ === {{:Secant of 255 Degrees}} === Secant of Three Right Angles === {{:Secant of Three Right Angles}} === Secant of $285 \degrees$ === {{:Secant of 285 Degrees}} === Secant of $300 \degrees$ === {{:Secant of 300 Degrees}} === Secant of $315 \degrees$ === {{:Secant of 315 Degrees}} === Secant of $330 \degrees$ === {{:Secant of 330 Degrees}} === Secant of $345 \degrees$ === {{:Secant of 345 Degrees}} === Secant of Full Angle === {{:Secant of Full Angle}}


Particular Values of Cosecant Function

The following values of the cosecant function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cosecant of Zero === {{:Cosecant of Zero}} === Cosecant of 15 Degrees === {{:Cosecant of 15 Degrees}} === Cosecant of 30 Degrees === {{:Cosecant of 30 Degrees}} === Cosecant of 45 Degrees === {{:Cosecant of 45 Degrees}} === Cosecant of 60 Degrees === {{:Cosecant of 60 Degrees}} === Cosecant of 75 Degrees === {{:Cosecant of 75 Degrees}} === Cosecant of Right Angle === {{:Cosecant of Right Angle}} === Cosecant of 105 Degrees === {{:Cosecant of 105 Degrees}} === Cosecant of 120 Degrees === {{:Cosecant of 120 Degrees}} === Cosecant of 135 Degrees === {{:Cosecant of 135 Degrees}} === Cosecant of 150 Degrees === {{:Cosecant of 150 Degrees}} === Cosecant of 165 Degrees === {{:Cosecant of 165 Degrees}} === Cosecant of Straight Angle === {{:Cosecant of Straight Angle}} === Cosecant of 195 Degrees === {{:Cosecant of 195 Degrees}} === Cosecant of 210 Degrees === {{:Cosecant of 210 Degrees}} === Cosecant of 225 Degrees === {{:Cosecant of 225 Degrees}} === Cosecant of 240 Degrees === {{:Cosecant of 240 Degrees}} === Cosecant of 255 Degrees === {{:Cosecant of 255 Degrees}} === Cosecant of Three Right Angles === {{:Cosecant of Three Right Angles}} === Cosecant of 285 Degrees === {{:Cosecant of 285 Degrees}} === Cosecant of 300 Degrees === {{:Cosecant of 300 Degrees}} === Cosecant of 315 Degrees === {{:Cosecant of 315 Degrees}} === Cosecant of 330 Degrees === {{:Cosecant of 330 Degrees}} === Cosecant of 345 Degrees === {{:Cosecant of 345 Degrees}} === Cosecant of Full Angle === {{:Cosecant of Full Angle}}


Tangent of 45 Degrees

:$\tan 45 \degrees = \tan \dfrac \pi 4 = 1$


Tangent of 60 Degrees

:$\tan 60^\circ = \tan \dfrac \pi 3 = \sqrt 3$


Tangent of 75 Degrees

:$\tan 75^\circ = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$


Tangent of Complement equals Cotangent

:$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$ where $\tan$ and $\cot$ are tangent and cotangent respectively. That is, the cotangent of an angle is the tangent of its complement. This relation is defined wherever $\sin \theta \ne 0$.


Cotangent of Complement equals Tangent

:$\cot \left({\dfrac \pi 2 - \theta}\right) = \tan \theta$ for $\theta \ne \left({2 n + 1}\right) \dfrac \pi 2$ where $\cot$ and $\tan$ are cotangent and tangent respectively. That is, the tangent of an angle is the cotangent of its complement. This relation is defined wherever $\cos \theta \ne 0$.


Tangent of Angle plus Right Angle

: $\tan \left({x + \dfrac \pi 2}\right) = -\cot x$


Cotangent of Angle plus Right Angle

: $\cot \left({x + \dfrac \pi 2}\right) = -\tan x$


Tangent of Supplementary Angle

:$\map \tan {\pi - \theta} = -\tan \theta$ where $\tan$ denotes tangent. That is, the tangent of an angle is the negative of its supplement.


Cotangent of Supplementary Angle

:$\map \cot {\pi - \theta} = -\cot \theta$ where $\cot$ denotes tangent. That is, the cotangent of an angle is the negative of its supplement.


Tangent of Conjugate Angle

:$\map \tan {2 \pi - \theta} = -\tan \theta$ where $\tan$ denotes tangent. That is, the tangent of an angle is the negative of its conjugate.


Cotangent of Conjugate Angle

:$\map \cot {2 \pi - \theta} = -\cot \theta$ where $\cot$ denotes cotangent. That is, the cotangent of an angle is the negative of its conjugate.


Tangent of Right Angle

:$\tan 90^\circ = \tan \dfrac \pi 2$ is undefined


Tangent of 105 Degrees

:$\tan 105^\circ = \tan \dfrac {7 \pi} {12} = - \left({2 + \sqrt 3}\right)$


Tangent of 120 Degrees

:$\tan 120 \degrees = \tan \dfrac {2 \pi} 3 = -\sqrt 3$


Tangent of 135 Degrees

:$\tan 135^\circ = \tan \dfrac {3 \pi} 4 = - 1$


Tangent of 150 Degrees

:$\tan 150^\circ = \tan \dfrac {5 \pi} 6 = - \dfrac {\sqrt 3} 3$


Tangent of 165 Degrees

:$\tan 165^\circ = \tan \dfrac {11 \pi} {12} = - \left({2 - \sqrt 3}\right)$


Tangent of Straight Angle

:$\tan 180^\circ = \tan \pi = 0$


Tangent of 195 Degrees

:$\tan 195^\circ = \tan \dfrac {13 \pi} {12} = 2 - \sqrt 3$


Tangent of 210 Degrees

:$\tan 210 \degrees = \tan \dfrac {7 \pi} 6 = \dfrac {\sqrt 3} 3$


Tangent of 225 Degrees

:$\tan 225 \degrees = \tan \dfrac {5 \pi} 4 = 1$


Tangent of 240 Degrees

:$\tan 240 \degrees = \tan \dfrac {4 \pi} 3 = \sqrt 3$


Tangent of 255 Degrees

:$\tan 255^\circ = \tan \dfrac {17 \pi} {12} = 2 + \sqrt 3$


Tangent of Three Right Angles

:$\tan 270^\circ = \tan \dfrac {3 \pi} 2$ is undefined


Tangent of 285 Degrees

:$\tan 285^\circ = \tan \dfrac {19 \pi} {12} = -\left({2 + \sqrt 3}\right)$


Tangent of 300 Degrees

:$\tan 300^\circ = \tan \dfrac {5 \pi} 3 = -\sqrt 3$


Tangent of 315 Degrees

:$\tan 315^\circ = \tan \dfrac {7 \pi} 4 = -1$


Tangent of 330 Degrees

:$\tan 330^\circ = \tan \dfrac {11 \pi} 6 = -\dfrac {\sqrt 3} 3$


Tangent of 345 Degrees

:$\tan 345^\circ = \tan \dfrac {23 \pi} {12} = -\left({2 - \sqrt 3}\right)$


Tangent of Full Angle

:$\tan 360^\circ = \tan 2 \pi = 0$


Cotangent of Zero

:$\cot 0$ is undefined


Cotangent of 30 Degrees

:$\cot 30^\circ = \cot \dfrac {\pi} 6 = \sqrt 3$


Cotangent of 15 Degrees

:$\cot 15^\circ = \cot \dfrac {\pi} {12} = 2 + \sqrt 3$


Cotangent of 45 Degrees

:$\cot 45 \degrees = \cot \dfrac \pi 4 = 1$


Cotangent of 60 Degrees

:$\cot 60^\circ = \cot \dfrac {\pi} 3 = \dfrac {\sqrt 3} 3$


Cotangent of 75 Degrees

:$\cot 75 \degrees = \cot \dfrac {5 \pi} {12} = 2 - \sqrt 3$


Cotangent of Right Angle

:$\cot 90^\circ = \cot \dfrac \pi 2 = 0$


Cotangent of 105 Degrees

:$\cot 105^\circ = \cot \dfrac {7 \pi} {12} = -\left({2 - \sqrt 3}\right)$


Cotangent of 120 Degrees

:$\cot 120 \degrees = \cot \dfrac {2 \pi} 3 = -\dfrac {\sqrt 3} 3$


Cotangent of 135 Degrees

:$\cot 135 \degrees = \cot \dfrac {3 \pi} 4 = -1$


Cotangent of 150 Degrees

:$\cot 150 \degrees = \cot \dfrac {5 \pi} 6 = -\sqrt 3$


Cotangent of 165 Degrees

:$\cot 165 \degrees = \cot \dfrac {11 \pi} {12} = -\paren {2 + \sqrt 3}$


Cotangent of Straight Angle

:$\cot 180^\circ = \cot \pi$ is undefined


Cotangent of 195 Degrees

:$\cot 195 \degrees = \cot \dfrac {13 \pi} {12} = 2 + \sqrt 3$


Cotangent of 210 Degrees

:$\cot 210^\circ = \cot \dfrac {7 \pi} 6 = \sqrt 3$


Cotangent of 225 Degrees

:$\cot 225^\circ = \cot \dfrac {5 \pi} 4 = 1$


Cotangent of 240 Degrees

:$\cot 240^\circ = \cot \dfrac {4 \pi} 3 = \dfrac {\sqrt 3} 3$


Cotangent of 255 Degrees

:$\cot 255 \degrees = \cot \dfrac {17 \pi} {12} = 2 - \sqrt 3$


Cotangent of Three Right Angles

:$\cot 270^\circ = \cot \dfrac {3 \pi} 2 = 0$


Cotangent of 285 Degrees

:$\cot 285 \degrees = \cot \dfrac {19 \pi} {12} = -\paren {2 - \sqrt 3}$


Cotangent of 300 Degrees

:$\cot 300 \degrees = \cot \dfrac {5 \pi} 3 = - \dfrac {\sqrt 3} 3$


Cotangent of 315 Degrees

:$\cot 315^\circ = \cot \dfrac {7 \pi} 4 = -1$


Cotangent of 330 Degrees

:$\cot 330^\circ = \cot \dfrac {11 \pi} 6 = -\sqrt 3$


Cotangent of 345 Degrees

:$\cot 345 \degrees = \cot \dfrac {23 \pi} {12} = -\paren {2 + \sqrt 3}$


Cotangent of Full Angle

:$\cot 360^\circ = \cot 2 \pi$ is undefined


Secant of Zero

:$\sec 0 = 0$


Secant of 15 Degrees

:$\sec 15 \degrees = \sec \dfrac \pi {12} = \sqrt 6 - \sqrt 2$


Secant of 30 Degrees

:$\sec 30 \degrees = \sec \dfrac \pi 6 = \dfrac {2 \sqrt 3} 3$


Secant of 45 Degrees

:$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$


Secant of 60 Degrees

:$\sec 60 \degrees = \sec \dfrac \pi 3 = 2$


Secant of 75 Degrees

:$\sec 75 \degrees = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$


Secant of Right Angle

:$\sec 90 \degrees = \sec \dfrac \pi 2$ is undefined


Secant of Angle plus Right Angle

: $\sec \left({x + \dfrac \pi 2}\right) = -\csc x$


Cosecant of Angle plus Right Angle

: $\csc \left({x + \dfrac \pi 2}\right) = \sec x$


Cosecant of Supplementary Angle

:$\csc \left({\pi - \theta}\right) = \csc \theta$ where $\csc$ denotes cosecant. That is, the cosecant of an angle equals its supplement.


Secant of Conjugate Angle

:$\map \sec {2 \pi - \theta} = \sec \theta$ where $\sec$ denotes secant. That is, the secant of an angle equals its conjugate.


Cosecant of Conjugate Angle

:$\map \csc {2 \pi - \theta} = -\csc \theta$ where $\csc$ denotes cosecant. That is, the cosecant of an angle is the negative of its conjugate.


Secant of 105 Degrees

:$\sec 105 \degrees = \sec \dfrac {7 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$


Secant of 120 Degrees

:$\sec 120 \degrees = \sec \dfrac {2 \pi} 3 = -2$


Secant of 135 Degrees

:$\sec 135 \degrees = \sec \dfrac {3 \pi} 4 = -\sqrt 2$


Secant of 150 Degrees

:$\sec 150 \degrees = \sec \dfrac {5 \pi} 6 = -\dfrac {2 \sqrt 3} 3$


Secant of 165 Degrees

:$\sec 165 \degrees = \sec \dfrac {11 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$


Secant of Straight Angle

:$\sec 180 \degrees = \sec \pi = -1$


Secant of 195 Degrees

:$\sec 195 \degrees = \sec \dfrac {13 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$


Secant of 210 Degrees

:$\sec 210 \degrees = \sec \dfrac {7 \pi} 6 = -2 \dfrac {\sqrt 3} 3$


Secant of 225 Degrees

:$\sec 225 \degrees = \sec \dfrac {5 \pi} 4 = -\sqrt 2$


Secant of 240 Degrees

:$\sec 240 \degrees = \sec \dfrac {4 \pi} 3 = - 2$


Secant of 255 Degrees

:$\sec 255 \degrees = \sec \dfrac {17 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$


Secant of Three Right Angles

:$\sec 270 \degrees = \sec \dfrac {3 \pi} 2$ is undefined


Secant of 285 Degrees

:$\sec 285 \degrees = \sec \dfrac {19 \pi} {12} = \sqrt 6 + \sqrt 2$


Secant of 300 Degrees

:$\sec 300 \degrees = \sec \dfrac {5 \pi} 3 = 2$


Secant of 315 Degrees

:$\sec 315 \degrees = \sec \dfrac {7 \pi} 4 = \sqrt 2$


Secant of 330 Degrees

:$\sec 330 \degrees = \sec \dfrac {11 \pi} 6 = 2 \dfrac {\sqrt 3} 3$


Secant of 345 Degrees

:$\sec 345 \degrees = \sec \dfrac {23 \pi} {12} = \sqrt 6 - \sqrt 2$


Secant of Full Angle

:$\sec 360 \degrees = \sec 2 \pi = 1$


Cosecant of Zero

:$\csc 0$ is undefined


Cosecant of 15 Degrees

:$\csc 15^\circ = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$


Cosecant of 30 Degrees

:$\csc 30^\circ = \csc \dfrac \pi 6 = 2$


Cosecant of 45 Degrees

:$\csc 45^\circ = \csc \dfrac \pi 4 = \sqrt 2$


Cosecant of 60 Degrees

:$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$


Cosecant of 75 Degrees

:$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$


Cosecant of Right Angle

:$\csc 90^\circ = \csc \dfrac \pi 2 = 1$


Cosecant of 105 Degrees

:$\csc 105 \degrees = \csc \dfrac {7 \pi} {12} = \sqrt 6 - \sqrt 2$


Cosecant of 120 Degrees

:$\csc 120 \degrees = \csc \dfrac {2 \pi} 3 = \dfrac {2 \sqrt 3} 3$


Cosecant of 135 Degrees

:$\csc 135 \degrees = \csc \dfrac {3 \pi} 4 = \sqrt 2$


Cosecant of 150 Degrees

:$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$


Cosecant of 165 Degrees

:$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$


Cosecant of Straight Angle

:$\csc 180^\circ = \csc \pi$ is undefined


Cosecant of 195 Degrees

:$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$


Cosecant of 210 Degrees

:$\csc 210^\circ = \csc \dfrac {7 \pi} 6 = -2$


Cosecant of 225 Degrees

:$\csc 225 \degrees = \csc \dfrac {5 \pi} 4 = -\sqrt 2$


Cosecant of 240 Degrees

:$\csc 240 \degrees = \csc \dfrac {4 \pi} 3 = -\dfrac {2 \sqrt 3} 3$


Cosecant of 255 Degrees

:$\csc 255^\circ = \csc \dfrac {17 \pi} {12} = -\left({\sqrt 6 - \sqrt 2}\right)$


Cosecant of Three Right Angles

:$\csc 270^\circ = \csc \dfrac {3 \pi} 2 = -1$


Cosecant of 285 Degrees

:$\csc 285^\circ = \csc \dfrac {19 \pi} {12} = -\left({\sqrt 6 - \sqrt 2}\right)$


Cosecant of 300 Degrees

:$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$


Cosecant of 315 Degrees

:$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$


Cosecant of 330 Degrees

:$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$


Cosecant of 345 Degrees

:$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$


Cosecant of Full Angle

:$\csc 360 \degrees = \csc 2 \pi$ is undefined


Shape of Sine Function/Graph


Shape of Cosine Function/Graph


Shape of Tangent Function/Graph


Shape of Cotangent Function/Graph


Shape of Secant Function/Graph


Cosecant Function is Odd

:$\map \csc {-x} = -\csc x$ That is, the cosecant function is odd.


Secant Function is Even

:$\map \sec {-x} = \sec x$ That is, the secant function is even.


Cotangent Function is Odd

:$\map \cot {-x} = -\cot x$ That is, the cotangent function is odd.


Cotangent of Sum

:$\cot \left({a + b}\right) = \dfrac {\cot a \cot b - 1} {\cot b + \cot a}$ where $\cot $ is cotangent. === Corollary === {{:Cotangent of Sum/Corollary}}


Half-Integer is Half Odd Integer

Let $r$ be a number. Then $r$ is a half-integer {{iff}} $r = \dfrac n 2$ where $n$ is an odd integer.


Sine of Angle plus Straight Angle

:$\map \sin {x + \pi} = -\sin x$


Cosine of Angle plus Straight Angle

:$\map \cos {x + \pi} = -\cos x$


Sine of Angle plus Full Angle

:$\map \sin {x + 2 \pi} = \sin x$


Cosine of Angle plus Full Angle

: $\cos \paren {x + 2 \pi} = \cos x$


Tangent of Angle plus Full Angle

: $\tan \left({x + 2 \pi}\right) = \tan x$


Tangent of Angle plus Straight Angle

: $\tan \left({x + \pi}\right) = \tan x$


Cotangent of Angle plus Straight Angle

:$\map \cot {x + \pi} = \cot x$


Secant of Angle plus Straight Angle

:$\map \sec {x + \pi} = -\sec x$


Cosecant of Angle plus Straight Angle

:$\map \csc {x + \pi} = -\csc x$


Cotangent of Angle plus Full Angle

: $\cot \left({x + 2 \pi}\right) = \cot x$


Secant of Angle plus Full Angle

: $\sec \left({x + 2 \pi}\right) = \sec x$


Cosecant of Angle plus Full Angle

: $\csc \left({x + 2 \pi}\right) = \csc x$


Sine of Angle plus Three Right Angles

:$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$


Cosine of Angle plus Three Right Angles

:$\map \cos {x + \dfrac {3 \pi} 2} = \sin x$


Tangent of Angle plus Three Right Angles

:$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$


Cotangent of Angle plus Three Right Angles

:$\map \cot {x + \dfrac {3 \pi} 2} = -\tan x$


Sine of Three Right Angles less Angle

:$\map \sin {\dfrac {3 \pi} 2 - \theta} = -\cos \theta$ where $\sin$ and $\cos$ are sine and cosine respectively.


Cosine of Three Right Angles less Angle

:$\cos \left({\dfrac {3 \pi} 2 - \theta}\right) = - \sin \theta$ where $\cos$ and $\sin$ are cosine and sine respectively.


Cosecant of Three Right Angles less Angle

:$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$ where $\csc$ and $\sec$ are cosecant and secant respectively.


Double Angle Formulas/Sine

:$\sin 2 \theta = 2 \sin \theta \cos \theta$


Double Angle Formulas/Cosine

:$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$


Double Angle Formulas/Tangent

:$\map \tan {2 \theta} = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$


Triple Angle Formulas

=== Triple Angle Formula for Sine === {{:Triple Angle Formulas/Sine}} === Triple Angle Formula for Cosine === {{:Triple Angle Formulas/Cosine}} === Triple Angle Formula for Tangent === {{:Triple Angle Formulas/Tangent}}


Triple Angle Formulas/Sine

:$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$


Triple Angle Formulas/Cosine

:$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$


Triple Angle Formulas/Tangent

:$\map \tan {3 \theta} = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$


Quadruple Angle Formulas

=== Quadruple Angle Formula for Sine === {{:Quadruple Angle Formulas/Sine}} === Quadruple Angle Formula for Cosine === {{:Quadruple Angle Formulas/Cosine}} === Quadruple Angle Formula for Tangent === {{:Quadruple Angle Formulas/Tangent}}


Quadruple Angle Formulas/Sine

:$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$


Quadruple Angle Formulas/Cosine

:$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$


Quadruple Angle Formulas/Tangent

:$\tan 4 \theta = \dfrac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}$


Quintuple Angle Formulas/Sine

:$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$


Quintuple Angle Formulas/Cosine

:$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$


Quintuple Angle Formulas/Tangent

:$\tan 5 \theta = \dfrac {\tan^5 \theta - 10 \tan^3 \theta + 5 \tan \theta} {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$


Power Reduction Formulas/Sine Squared

:$\sin^2 x = \dfrac {1 - \cos 2 x} 2$


Power Reduction Formulas/Cosine Squared

:$\cos^2 x = \dfrac {1 + \cos 2 x} 2$


Power Reduction Formulas/Tangent Squared

:$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$


Power Reduction Formulas/Sine Cubed

:$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$


Power Reduction Formulas/Cosine Cubed

:$\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$


Power Reduction Formulas/Sine to 4th

:$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$


Power Reduction Formulas/Cosine to 4th

:$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$


Power Reduction Formulas/Sine to 5th

:$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$


Power Reduction Formulas/Cosine to 5th

:$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$


Orthocomplement of Subset of Orthocomplement is Superset

Let $H$ be a Hilbert space. Let $A, B \subseteq H$ be subsets of $H$ such that $B \subseteq A^\perp$, where $A^\perp$ is the orthocomplement of $A$. Then $A \subseteq B^\perp$.


Simpson's Formulas/Cosine by Cosine

:$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$


Simpson's Formulas/Sine by Sine

:$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$


Simpson's Formulas/Cosine by Sine

:$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$


Simpson's Formulas/Sine by Cosine

:$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$


Prosthaphaeresis Formulas/Sine plus Sine

:$\sin \alpha + \sin \beta = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$


Prosthaphaeresis Formulas/Sine minus Sine

:$\sin \alpha - \sin \beta = 2 \map \cos {\dfrac {\alpha + \beta} 2} \map \sin {\dfrac {\alpha - \beta} 2}$


Prosthaphaeresis Formulas/Cosine plus Cosine

:$\cos \alpha + \cos \beta = 2 \, \map \cos {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$


Prosthaphaeresis Formulas/Cosine minus Cosine

:$\cos \alpha - \cos \beta = -2 \, \map \sin {\dfrac {\alpha + \beta} 2} \, \map \sin {\dfrac {\alpha - \beta} 2}$


Sine of Integer Multiple of Argument/Formulation 1

{{begin-eqn}} {{eqn | l = \sin n \theta       | r = \sin \theta \paren {\paren {2 \cos \theta}^{n - 1} - \dbinom {n - 2} 1 \paren {2 \cos \theta}^{n - 3} + \dbinom {n - 3} 2 \paren {2 \cos \theta}^{n - 5} - \cdots}       | c =  }} {{eqn | r = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }       | c =  }} {{end-eqn}}


Cosine to Power of Odd Integer

:$\displaystyle \cos^{2n+1} \theta = \frac 1 {2^{2n}} \sum_{k \mathop = 0}^n \binom {2n+1} k \cos \left({2n - 2k + 1}\right) \theta$


Sum of Arctangent and Arccotangent

Let $x \in \R$ be a real number. Then: : $\arctan x + \operatorname{arccot} x = \dfrac \pi 2$ where $\arctan$ and $\operatorname{arccot}$ denote arctangent and arccotangent respectively.


Arcsine of Reciprocal equals Arccosecant

:$\map \arcsin {\dfrac 1 x} = \arccsc x$


Arccosine of Reciprocal equals Arcsecant

:$\map \arccos {\dfrac 1 x} = \arcsec x$


Arctangent of Reciprocal equals Arccotangent

:$\map \arctan {\dfrac 1 x} = \arccot x$


Inverse Sine is Odd Function

Everywhere that the function is defined: :$\map \arcsin {-x} = -\arcsin x$


Arccosine of Negative Argument

Everywhere that the function is defined: :$\map \arccos {-x} = \pi - \arccos x$


Inverse Tangent is Odd Function

Everywhere that the function is defined: :$\map \arctan {-x} = -\arctan x$


Arccotangent of Negative Argument

Everywhere that the function is defined: :$\map \arccot {-x} = \pi - \arccot x$


Arcsecant of Negative Argument

Everywhere that the function is defined: :$\map \arcsec {-x} = \pi - \arcsec x$


Inverse Cosecant is Odd Function

Everywhere that the function is defined: :$\map \arccsc {-x} = -\arccsc x$


Sine of Angle of Triangle by Semiperimeter

Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: : $\sin A = \dfrac 2 {b c} \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$ where $\sin$ denotes sine and $s$ is the semiperimeter: $s = \dfrac {a + b + c} 2$.


Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Implication

: $\left\{{\neg, \land, \lor, \implies}\right\}$: Not, And, Or and Implies


Functionally Complete Logical Connectives/Negation and Disjunction

:$\set {\neg, \lor}$: Not and Or


Functionally Complete Logical Connectives/Negation and Conjunction

:$\set {\neg, \land}$: Not and And


Functionally Complete Logical Connectives/Negation and Conditional

:$\set {\neg, \implies}$: Not and Implies


Functionally Complete Logical Connectives/Conjunction, Negation and Disjunction

:$\set {\neg, \land, \lor}$: Not, And and Or


Functionally Complete Logical Connectives/NAND

:$\set {\uparrow}$: NAND


Functionally Complete Logical Connectives/NOR

:$\set {\downarrow}$: NOR


NOR with Equal Arguments/Proof 1

Let $\downarrow$ signify the NOR operation. Then for any proposition $p$: :$p \downarrow p \dashv \vdash \neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator.


NOR with Equal Arguments/Proof 2


NAND with Equal Arguments/Proof 1

Let $\uparrow$ signify the NAND operation. Then, for any proposition $p$: :$p \uparrow p \dashv \vdash \neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation.


NAND with Equal Arguments/Proof 2


Conditional and Converse are not Equivalent

A conditional statement: :$p \implies q$ is not logically equivalent to its converse: :$q \implies p$


NOR is Commutative/Proof 1

Let $\downarrow$ signify the NOR operation. Then, for any two propositions $p$ and $q$: :$p \downarrow q \dashv \vdash q \downarrow p$ That is, NOR is commutative.


NOR is Commutative/Proof 2


NOR is not Associative/Proof 1

Let $\downarrow$ signify the NOR operation. Then there exist propositions $p,q,r$ such that: :$p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$ That is, NOR is not associative.


NOR is not Associative/Proof 2

Let $\downarrow$ signify the NOR operation. Then there exist propositions $p, q, r$ such that: :$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$ That is, NOR is not associative.


De Morgan's Laws (Predicate Logic)/Assertion of Universality

:$\forall x: \map P x \dashv \vdash \neg \exists x: \neg \map P x$ ::''If everything '''is''', there exists nothing that '''is not'''.''


De Morgan's Laws (Predicate Logic)/Denial of Existence

:$\forall x: \neg \map P x \dashv \vdash \neg \exists x: \map P x$ ::''If everything '''is not''', there exists nothing that '''is'''.''


De Morgan's Laws (Predicate Logic)/Denial of Universality

:$\neg \forall x: \map P x \dashv \vdash \exists x: \neg \map P x$ ::''If not everything '''is''', there exists something that '''is not'''.''


De Morgan's Laws (Predicate Logic)/Assertion of Existence

:$\neg \forall x: \neg \map P x \dashv \vdash \exists x: \map P x$ ::''If not everything '''is not''', there exists something that '''is'''.''


Conjunction in terms of NAND

:$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \uparrow q}$ where $\land$ denotes logical conjunction and $\uparrow$ denotes logical NAND.


Disjunction in terms of NOR

:$p \lor q \dashv \vdash \paren {p \downarrow q} \downarrow \paren {p \downarrow q}$ where $\lor$ denotes logical disjunction and $\downarrow$ denotes logical NOR.


Disjunction in terms of NAND

:$p \lor q \dashv \vdash \paren {p \uparrow p} \uparrow \paren {q \uparrow q}$ where $\lor$ denotes logical disjunction and $\uparrow$ denotes logical NAND.


Implication in terms of NAND

:$p \implies q \dashv \vdash p \uparrow \paren {q \uparrow q}$


Mapping is Injection and Surjection iff Inverse is Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then: : $f: S \to T$ can be defined as a bijection in the sense that: ::$(1): \quad f$ is an injection ::$(2): \quad f$ is a surjection {{iff}}: :the inverse $f^{-1}$ of $f$ is itself a mapping.


Condition for Composite Mapping to be Identity

Let $S$ and $T$ be sets. Let $f: S \to T$ and $g: T \to S$ be mappings such that: : $g \circ f = I_S$ where $I_S$ is the identity mapping on $S$. Then $f$ is an injection and $g$ is a surjection.


Eigenvalues of G-Representation are Roots of Unity

Let $G$ be a finite group. Let $\left({K, +, \cdot}\right)$ be a field. Let $V$ be a $G$-module over $K$ (i.e. $V$ is a $K \left[{G}\right]$-module). Then $\forall g \in G$, the eigenvalues of the action by the vector $g \in K \left[{G}\right]$ on $V$ are roots of unity.


Character of Representations over C are Algebraic Integers

Let $G$ be a finite group. Let $\chi$ be the character of any [[Definition:G-Module|$\C \left[{G}\right]$-module]] $\left({V, \rho}\right)$. Then for all $g \in G$, it follows that $\chi \left({g}\right)$ is an algebraic integer.


Injection from Finite Set to Itself is Surjection

Let $S$ be a finite set. Let $f: S \to S$ be an injection. Then $f$ is also a surjection.


Codomain of Bijection is Domain of Inverse

Let $S$ and $T$ be sets. Let $f: S \to T$ be a bijection. Let $f^{-1}: T \to S$ be the inverse of $f$. Then the domain of $f^{-1}$ equals the codomain of $f$.


Domain of Bijection is Codomain of Inverse

Let $S$ and $T$ be sets. Let $f: S \to T$ be a bijection. Let $f^{-1}: T \to S$ be the inverse of $f$. Then the codomain of $f^{-1}$ equals the domain of $f$.


Conditions for Uniqueness of Left Inverse Mapping

Let $S$ and $T$ be sets such that $S \ne \O$. Let $f: S \to T$ be an injection. Then a left inverse mapping of $f$ is in general not unique. Uniqueness occurs under either of two circumstances: :$(1): \quad S$ is a singleton :$(2): \quad f$ is a bijection.


Surjection iff Right Inverse/Non-Uniqueness

A right inverse of $f$ is in general not unique. Uniqueness occurs {{iff}} $f$ is a bijection.


Mapping reflects Preordering

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let ${\precsim} \subseteq T \times T$ be a preordering on $T$. Let $\RR$ be the relation defined on $S$ by the rule: :$x \mathrel \RR y \iff \map f x \precsim \map f y$ Then $\RR$ is a preordering on $S$.


Bernoulli's Inequality

Let $x \in \R$ be a real number such that $x > -1$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\paren {1 + x}^n \ge 1 + n x$


Multiple of Ring Product

Let $\left({R, +, \circ}\right)$ be a ring. Let $x, y \in \left({R, +, \circ}\right)$. Then: :$\forall n \in \Z_{> 0}: \left({n \cdot x} \right) \circ y = n \cdot \left({x \circ y}\right) = x \circ \left({n \cdot y}\right)$ where $n \cdot x$ denotes the $n$th multiple of $x$.


Ordering is Preserved on Integers by Addition

The usual ordering on the integers is preserved by the operation of addition: :$\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$


Power Set of Finite Set is Finite

Let $S$ be a finite set. Then the power set of $S$ is likewise finite.


Common Divisor Divides Difference

Let $c$ be a common divisor of two integers $a$ and $b$. That is: :$a, b, c \in \Z: c \divides a \land c \divides b$ Then: :$c \divides \paren {a - b}$


Divisor Divides Multiple

Let $a, b$ be integers. Let: :$a \divides b$ where $\divides$ denotes divisibility. Then: :$\forall c \in \Z: a \divides b c$


One is not Prime

The integer $1$ (one) is not a prime number.


Initial Part of String is Substring

Let $S$ be a string, and let $T$ be an initial part of $S$. Then $T$ is a substring of $S$.


Ring Zero is Idempotent

Let $\left({R, +, \circ}\right)$ be a ring whose ring zero is $0_R$. Then $0_R$ is an idempotent element of $R$ under the ring product $\circ$: :$0_R \circ 0_R = 0_R$


Prime iff Coprime to all Smaller Positive Integers

Let $p$ be a prime number Then: :$\forall x \in \Z, 0 < x < p: x \perp p$ That is, $p$ is relatively prime to all smaller (strictly) positive integers.


Division Theorem/Positive Divisor/Positive Dividend

<onlyinclude> For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, a \ge 0, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$ <onlyinclude> In the above equation: * $a$ is the '''dividend''' * $b$ is the '''divisor''' * $q$ is the '''quotient''' * $r$ is the '''principal remainder''', or, more usually, just the '''remainder'''.


Division Theorem/Positive Divisor

<onlyinclude> For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$ <onlyinclude> In the above equation: :$a$ is the '''dividend''' :$b$ is the '''divisor''' :$q$ is the '''quotient''' :$r$ is the '''principal remainder''', or, more usually, just the '''remainder'''.


Division Theorem/Positive Divisor/Positive Dividend/Existence

For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, a \ge 0, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$


Division Theorem/Positive Divisor/Positive Dividend/Uniqueness

For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, a \ge 0, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$


Division Theorem/Positive Divisor/Existence/Proof 2

For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$


Division Theorem/Positive Divisor/Existence


Division Theorem/Positive Divisor/Uniqueness/Proof 1

For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$


Division Theorem/Positive Divisor/Uniqueness/Proof 2


Division Theorem/Positive Divisor/Uniqueness


Division Theorem/Positive Divisor/Existence/Proof 3


Division Theorem/Positive Divisor/Uniqueness/Proof 3


Integer Coprime to all Factors is Coprime to Whole

Let $a, b \in \Z$. Let $\displaystyle b = \prod_{j \mathop = 1}^r b_j$ Let $a$ be coprime to each of $b_1, \ldots, b_r$. Then $a$ is coprime to $b$.


All Factors Divide Integer then Whole Divides Integer

Let $S = \set {a_1, a_2, \ldots, a_r} \subseteq \Z$ be a finite subset of the integers. Let $S$ be pairwise coprime. Let: :$\forall j \in \set {1, 2, \ldots, r}: a_r \divides b$ where $\divides$ denotes divisibility. Then: :$\displaystyle \prod_{j \mathop = 1}^r a_j \divides b$


Piecewise Continuous Function with One-Sided Limits is Bounded

Let $f$ be a real function defined on a closed interval $\closedint a b$. Let $f$ be a piecewise continuous function with one-sided limits. Then $f$ is a bounded piecewise continuous function.


Upper Bounds for Prime Numbers/Result 1

:$\forall n \in \N: \map p n \le 2^{2^{n - 1} }$


Upper Bounds for Prime Numbers/Result 2

:$\forall n \in \N: \map p n \le \paren {p \paren {n - 1} }^{n - 1} + 1$


Product of Integers of form 4n + 1

Let $m, n \in \Z$ such that both $m$ and $n$ are of the form $4 k + 1$ where $k \in \Z$. Then $m n$ is also of the form $4 k + 1$.


Infinite Number of Primes of form 4n - 1

There are infinitely many prime numbers of the form $4 n - 1$.


Integer Coprime to Modulus iff Linear Congruence to 1 exists

Let $a, m \in \Z$. The linear congruence: :$a x \equiv 1 \pmod m$ has a solution $x$ {{iff}} $a$ and $m$ are coprime.


Integer Coprime to Modulus iff Linear Congruence to 1 exists/Corollary

Let $p$ be a prime number. The linear congruence: :$a x \equiv 1 \pmod p$ has a solution $x$ {{iff}} $a \not \equiv 0 \pmod p$.


Solution of Linear Congruence/Existence

$a x \equiv b \pmod n$ has at least one solution {{iff}}: : $\gcd \set {a, n} \divides b$ that is, {{iff}} $\gcd \set {a, n}$ is a divisor of $b$.


Solution of Linear Congruence/Number of Solutions

Let $\gcd \set {a, n} = d$. Then $a x \equiv b \pmod n$ has $d$ solutions which are given by the unique solution modulo $\dfrac n d$ of the congruence: : $\dfrac a d x \equiv \dfrac b d \paren {\bmod \dfrac n d}$


Solution of Linear Congruence/Unique iff Coprime to Modulus

If $\gcd \set {a, n} = 1$, then $a x \equiv b \pmod n$ has a unique solution.


Ring Direct Product of Modulo Integers is Isomorphic to Ring Modulo Product iff Coprime

Let $m, n \in \Z_{>1}$. Let $\struct {\Z_m, +_m, \times_m}$ and $\struct {\Z_n, +_n, \times_n}$ be the rings of integers modulo $m$ and $n$ respectively. Let $\struct {\Z_m \times \Z_n}$ be the direct product of $\Z_m$ and $\Z_n$. Let $\struct {\Z_{m n}, +_{m n}, \times_{m n} }$ be the ring of integers modulo $mn$. Then $\struct {\Z_m \times \Z_n}$ is isomorphic to $\struct {\Z_{m n}, +_{m n}, \times_{m n} }$ {{iff}} $m$ and $n$ are coprime.


Trivial Norm on Division Ring is Non-Archimedean

Let $\struct {R, +, \circ}$ be a division ring whose ring zero is $0_R$. Then the trivial norm $\norm {\, \cdot \,}: R \to \R_{\ge 0}$, which is given by: :$\norm x = \begin{cases}   0 & : x = 0_R \\   1 & : \text{ otherwise} \end{cases}$ is non-archimedean: :$\norm {x + y} \le \max \set {\norm x, \norm y}$


Ring of Integers Modulo m is Ring

For all $m \in \N: m \ge 2$, the ring of integers modulo $m$: :$\struct {\Z_m, +_m, \times_m}$ is a commutative ring with unity $\eqclass 1 m$. The zero of $\struct {\Z_m, +_m, \times_m}$ is $\eqclass 0 m$.


Piecewise Continuous Function with One-Sided Limits is Uniformly Continuous on Each Piece

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ be piecewise continuous with one-sided limits: {{:Definition:Piecewise Continuous Function with One-Sided Limits}} Then: :for all $i \in \left\{ {1, 2, \ldots, n}\right\}$, $f$ is uniformly continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$ .


Principle of Definition by Structural Induction

Let $\LL$ be a formal language. Let the formal grammar of $\LL$ be a bottom-up grammar with unique parsability. A definition $\map D \phi$ (in the metalanguage of $\LL$) for all well-formed formulas $\phi$ of $\LL$ is uniquely specified by: :$(1): \quad$ A definition $\map D a$ for each letter $a$ of $\LL$ :$(2): \quad$ For each rule of formation for $\LL$, a definition $\map D \phi$ of the resultant WFF $\phi$ in terms of the constituent WFFs' definitions $\map D {\phi_1}, \ldots, \map D {\phi_n}$.


Principle of Structural Induction

Let $\LL$ be a formal language. Let the formal grammar of $\LL$ be a bottom-up grammar. Let $\map P \phi$ be a statement (in the metalanguage of $\LL$) about well-formed formulas $\phi$ of $\LL$. Then $P$ is true for all WFFs of $\LL$ {{iff}} both: :$\map P a$ is true for all letters $a$ of $\LL$, and, for each rule of formation of $\LL$, if $\phi$ is a WFF resulting from WFFs $\phi_1, \ldots, \phi_n$ by applying that rule, then: :$\map P \phi$ is true only if $\map P {\phi_1}, \ldots, \map P {\phi_n}$ are all true.


Integers under Multiplication form Monoid

The set of integers under multiplication $\struct {\Z, \times}$ is a monoid.


Integers under Multiplication form Semigroup

The set of integers under multiplication $\struct {\Z, \times}$ is a semigroup.


Canonical Form of Rational Number is Unique

The canonical form of a rational number is unique.


Existence of Canonical Form of Rational Number

Let $r \in \Q$. Then: :$\exists p \in \Z, q \in \Z_{>0}: r = \dfrac p q, p \perp q$ That is, every rational number can be expressed in its canonical form.


Existence of Dyadic Rational between two Rationals

Let $a$ and $b$ be rational numbers such that $a < b$. Then there exist integers $m$ and $r$ such that: :$a < \dfrac m {2^r} < b$ That is, there exists a dyadic rational between any pair of rational numbers.


Field with 4 Elements has only Order 2 Elements

Let $\struct {\GF, +, \times}$ be a field which has exactly $4$ elements. Then: :$\forall a \in \GF: a + a = 0_\GF$ where $0_\GF$ is the zero of $\GF$.


Count of Truth Functions

There are $2^{\paren {2^k} }$ distinct truth functions on $k$ variables.


Unary Truth Functions

There are $4$ distinct unary truth functions: :$(1): \quad$ The constant function $\map f p = \F$ :$(2): \quad$ The constant function $\map f p = \T$ :$(3): \quad$ The identity function $\map f p = p$ :$(4): \quad$ The logical not function $\map f p = \neg p$


Binary Truth Functions

There are $16$ distinct binary truth functions: * Two constant functions: ** $\map {f_\F} {p, q} = \F$ ** $\map {f_\T} {p, q} = \T$ * Two projections: ** $\map {\pr_1} {p, q} = p$ ** $\map {\pr_2} {p, q} = q$ * Two negated projections: ** $\map {\overline {\pr_1} } {p, q} = \neg p$ ** $\map {\overline {\pr_2} } {p, q} = \neg q$ * The conjunction: $p \land q$ * The disjunction: $p \lor q$ * Two conditionals: ** $p \implies q$ ** $q \implies p$ * The biconditional: $p \iff q$ * The exclusive or: $\map \neg {p \iff q}$ * Two negated conditionals: ** $\map \neg {p \implies q}$ ** $\map \neg {q \implies p}$ * The NAND: $p \uparrow q$ * The NOR: $p \downarrow q$


Inverse Hyperbolic Sine Logarithmic Formulation

{{finish|The true story is more complicated than this. The inverse hyperbolic sine is a multifunction. So is $\ln$ in the complex plane. And $\sqrt {z^2 + 1}$ has two values, pos and neg, so you need to justify which one is taken. The best context to put this page is probably into Definition:Inverse Hyperbolic Sine/Arcsine, where an explanation needs to be made as to why the principal value is taken thus.}} For any complex number $z \in \C$: :$\operatorname {arsinh} z = \map \ln {z + \sqrt {z^2 + 1} }$ where $\operatorname {arsinh} z$ is the inverse hyperbolic sine.


Value of Cauchy Determinant

Let $D_n$ be a Cauchy determinant of order $n$: :$\begin{vmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{vmatrix}$ Then the value of $D_n$ is given by: :$D_n = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_j - y_i}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i + y_j}\right)}$ If $D_n$ is given by: :$\begin{vmatrix} \dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\ \end{vmatrix}$ then its determinant is given by: :$D_n = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_i - y_j}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i - y_j}\right)}$


Integers under Addition form Abelian Group

The set of integers under addition $\struct {\Z, +}$ forms an abelian group.


Piecewise Continuous Function with One-Sided Limits is Darboux Integrable

Let $f$ be a real function defined on a closed interval $\closedint a b$. Let $f$ be piecewise continuous with one-sided limits on $\closedint a b$. Then $f$ is Darboux integrable on $\closedint a b$.


Schröder Rule/Proof 1

Let $A$, $B$ and $C$ be relations on a set $S$. Then the following are equivalent statements: :$(1): \quad A \circ B \subseteq C$ :$(2): \quad A^{-1} \circ \overline{C} \subseteq \overline{B}$ :$(3): \quad \overline{C} \circ B^{-1} \subseteq \overline{A}$ where: : $\circ$ denotes relation composition : $A^{-1}$ denotes the inverse of $A$ : $\overline{A}$ denotes the complement of $A$.


Condition for Well-Foundedness/Forward Implication

Let $\struct {S, \preceq}$ be an ordered set. Let $\struct {S, \preceq}$ be well-founded. Then there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that $\forall n \in \N: a_{n + 1} \prec a_n$.


Extendability Theorem for Function Continuous on Open Interval

Let $f$ be a continuous real function that is defined on an open interval $\openint a b$. Let $g$ be a real function that satisfies: :$g$ is defined on $\closedint a b$ :$g$ is continuous on $\closedint a b$ :$g = f$ on $\openint a b$. Then $g$ exists {{iff}} $\displaystyle \lim_{x \mathop \to a^+} \map f x$ and $\displaystyle \lim_{x \mathop \to b^-} \map f x$ exist.


Definition:Trivial Ordering

The '''trivial ordering''' is an ordering $\mathcal R$ defined on a set $S$ by: :$\forall a, b \in S: a \mathrel{\mathcal R} b \iff a = b$ That is, there is no ordering defined on any two distinct elements of the set $S$.


Integrability Theorem for Functions Continuous on Open Intervals

Let $f$ be a real function defined on a closed interval $\closedint a b$ such that $a < b$. Let $f$ be continuous on $\openint a b$. Let the one-sided limits $\displaystyle \lim_{x \mathop \to a^+} \map f x$ and $\displaystyle \lim_{x \mathop \to b^-} \map f x$ exist. Then $f$ is Darboux integrable on $\closedint a b$.


Set of Rational Numbers whose Numerator Divisible by p is Closed under Multiplication

Let $p$ be a prime number. Let $A_p$ be the set of all rational numbers which, when expressed in canonical form has a numerator which is divisible by $p$. Then $A_p$ is closed under rational multiplication.


Integer Less One divides Power Less One

Let $q, n \in \Z_{>0}$. Then: :$\paren {q - 1} \divides \paren {q^n - 1}$ where $\divides$ denotes divisibility.


Number Plus One divides Power Plus One iff Odd

Let $q, n \in \Z_{>0}$. Then: :$\paren {q + 1} \divides \paren {q^n + 1}$ {{iff}} $n$ is odd. In the above, $\divides$ denotes divisibility.


Mapping Bounded on Union iff Bounded on Each Component/Real-Valued Function

Let $f$ be a real-valued function. Then: : $f$ is bounded on the union of a finite number of sets within the domain of $f$ {{iff}}: :$f$ is bounded on each of the sets.


Parity of Integer equals Parity of Positive Power

Let $p \in \Z$ be an integer. Let $n \in \Z_{>0}$ be a strictly positive integer. Then $p$ is even {{iff}} $p^n$ is even. That is, the parity of an integer equals the parity of all its (strictly) positive powers.


Two divides Power Plus One iff Odd

Let $q, n \in \Z_{>0}$. Then: :$2 \divides \paren {q^n + 1}$ {{iff}} $q$ is odd. In the above, $\divides$ denotes divisibility.


Primes of form Power plus One

Let $q, n \in \Z_{>0}$ such that $q > 1$. Then $q^n + 1$ is prime only if: :$(1): \quad q$ is even and :$(2): \quad n$ is of the form $2^m$ for some positive integer $m$.


Fermat Number whose Index is Sum of Integers

Let $F_n = 2^{\left({2^n}\right)} + 1$ be the $n$th Fermat number. Let $k \in \Z_{>0}$. Then: :$F_{n + k} - 1 = \left({F_n - 1}\right)^{2^k}$


Goldbach's Theorem

Let $F_m$ and $F_n$ be Fermat numbers such that $m \ne n$. Then $F_m$ and $F_n$ are coprime.


Number of Boolean Interpretations for Finite Set of Variables

Let $\mathcal P_0$ be the vocabulary of language of propositional logic. Let $S \subseteq \mathcal P_0$ be a finite set of $n$ letters from $\mathcal P_0$. Then there are $2^n$ different partial boolean interpretations for $S$.


Rational Numbers under Addition form Monoid

The set of rational numbers under addition $\struct {\Q, +}$ forms a monoid.


Real Numbers form Field

The set of real numbers $\R$ forms a field under addition and multiplication: $\struct {\R, +, \times}$.


Rational Numbers form Field

Consider the algebraic structure $\struct {\Q, +, \times}$, where: :$\Q$ is the set of all rational numbers :$+$ is the operation of rational addition :$\times$ is the operation of rational multiplication. Then $\struct {\Q, +, \times}$ forms a field.


Integers Modulo m under Addition form Abelian Group

Let $\Z_m$ is the set of integers modulo $m$ Let $+_m$ be the operation of addition modulo $m$. Then the structure $\struct {\Z_m, +_m}$ is an abelian group.


Rational Numbers under Multiplication form Monoid

The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a monoid.


Real Numbers under Multiplication form Monoid

The set of real numbers under multiplication $\struct {\R, \times}$ forms a monoid.


Complex Numbers under Multiplication form Monoid

The set of complex numbers under multiplication $\struct {\C, \times}$ forms a monoid.


Non-Zero Natural Numbers under Addition do not form Monoid

Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \left\{{0}\right\}$. The structure $\left({\N_{>0}, +}\right)$ does ''not'' form a monoid.


Natural Numbers Bounded Below under Addition form Commutative Semigroup

Let $m \in \N$ where $\N$ is the set of natural numbers. Let $M \subseteq \N$ be defined as: :$M := \set {x \in \N: x \ge m}$ That is, $M$ is the set of all natural numbers greater than or equal to $m$. Then the algebraic structure $\struct {M, +}$ is a commutative semigroup.


Positive Real Numbers under Max Operation form Monoid

Let $\R_{\ge 0}$ be the set of positive (that is, non-negative) real numbers. Let $\max: \R_{\ge 0}^2 \to \R_{\ge 0}$ be the max operation on $\R_{\ge 0}$. Then $\left({\R_{\ge 0}, \max}\right)$ is a monoid whose identity is $0$.


Free Commutative Monoid on One Element is Isomorphic to Positive Integers under Addition

Let $X = \left\{{x}\right\}$ be a singleton. Let $M$ be the free commutative monoid on $X$. Then $M$ is isomorphic to the additive monoid of natural numbers.


Finite Monoid with Right Cancellable Operation is Group

Let $\struct {S, \circ}$ be a finite monoid. Let $\circ$ be a right cancellable operation. Then $\struct {S, \circ}$ is a group.


Multiplicative Group of Field is Abelian Group

Let $\struct {F, +, \times}$ be a field. Let $F^* := F \setminus \set 0$ be the set $F$ less its zero. The algebraic structure $\struct {F^*, \times}$ is an abelian group.


One and Minus One form Subgroup of Multiplicative Group of Rational Numbers

Let $\left({\Q_{\ne 0}, \times}\right)$ be the multiplicative group of rational numbers. Let $S \subseteq \Q$ where $S = \left\{{1, -1}\right\}$. Then $\left({S, \times}\right)$ is a subgroup of $\left({\Q_{\ne 0}, \times}\right)$.


Symmetric Group is not Abelian

Let $S_n$ be the symmetric group of order $n$ where $n \ge 3$. Then $S_n$ is not abelian.


Möbius Function is Multiplicative/Corollary

Let $\gcd \set {m, n} > 1$. Then: :$\map \mu {m n} = 0$


Identity of Affine Group of One Dimension

The $1$-dimensional affine group on $\R$ $\operatorname{Af}_1 \left({\R}\right)$ has $f_{1, 0}$ as an identity element.


Inverse in Affine Group of One Dimension

Let $\map {\operatorname {Af}_1} \R$ denote the $1$-dimensional affine group on $\R$. Let $f_{a b} \in \map {\operatorname {Af}_1} \R$. Let $c = \dfrac 1 a$ and $d = \dfrac {-b} a$. Then $f_{c d} \in \map {\operatorname {Af}_1} \R$ is the inverse of $f_{a b}$.


Commutation with Group Elements implies Commuation with Product with Inverse

Let $G$ be a group. Let $a, b, c \in G$ such that $a$ commutes with both $b$ and $c$. Then $a$ commutes with $b c^{-1}$.


Centralizer of Subset is Intersection of Centralizers of Elements

Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. Let $\map {C_G} S$ be the centralizer of $S$ in $G$. Then: :$\ds \map {C_G} S = \bigcap_{x \mathop \in S} \map {C_G} x$ where $\map {C_G} z$ is the centralizer of $x$ in $G$.


Group Action of Symmetric Group

Let $\N_n$ denote the set $\set {1, 2, \ldots, n}$. Let $\struct {S_n, \circ}$ denote the symmetric group on $\N_n$. The mapping $*: S_n \times \N_n \to \N_n$ defined as: :$\forall \pi \in S_n, \forall n \in \N_n: \pi * n = \map \pi n$ is a group action.


Trivial Group Action is Group Action

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $S$ be a set. Let $*: G \times S \to S$ be the trivial group action: :$\forall \tuple {g, s} \in G \times S: g * s = s$ Then $*$ is indeed a group action.


Right Regular Representation by Inverse is Group Action

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*: G \times G \to G$ be the operation: :$\forall g, h \in G: g * h = \map {\rho_{g^{-1} } } h$ where $\rho_g$ is the right regular representation of $G$ with respect to $g$. Then $*$ is a group action.


Left Regular Representation is Group Action

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*: G \times G \to G$ be the operation: :$\forall g, h \in G: g * h = \map {\lambda_g} h$ where $\lambda_g$ is the left regular representation of $G$ with respect to $g$. Then $*$ is a group action.


Right Regular Representation by Inverse is Transitive Group Action

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*: G \times G \to G$ be the group action: :$\forall g, h \in G: g * h = \map {\rho_{g^{-1} } } h$ where $\rho_g$ is the right regular representation of $G$ with respect to $g$. Then $*$ is a transitive group action.


Conjugacy Action is not Transitive

Let $\struct {G, \circ}$ be a non-trivial group whose identity is $e$. Let $*: G \times G \to G$ be the conjugacy group action: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Then $*$ is ''not'' a transitive group action.


Group Action on Subgroup by Right Regular Representation

Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $*: H \times G \to G$ be the operation defined as: :$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$ where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$. Then $*$ is a group action.


Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset

Let $G$ be a group. Let $H$ be a proper subgroup of $G$. Let $*: H \times G \to G$ be the group action defined as: :$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$ where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$. Let $x \in G$. Then the orbit of $x$ under $*$ is given by: :$\forall x \in G: \Orb x = H x$ where $H x$ is the right coset of $H$ by $x$.


Left Coset by Identity

Then: : $e H = H$ where $e H$ is the left coset of $H$ by $e$.


Right Coset by Identity

Then: : $H = H e$ where $H e$ is the right coset of $H$ by $e$.