nyuuzyou/wikis huggingface dataset:
Cosecant Function is Odd

== Theorem ==
Let $x \in \R$ be a real number.
Let $\csc x$ be the cosecant of $x$.
Then, whenever $\csc x$ is defined:
:$\map \csc {-x} = -\csc x$
That is, the cosecant function is odd.
== Proof ==
== Also see ==
* Sine Function is Odd
* Cosine Function is Even
* Tangent Function is Odd
* Cotangent Function is Odd
* Secant Function is Even
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.31$
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Category:Examples of Odd Functions
Cosecant in terms of Hyperbolic Cosecant

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$i \csc = -\csch \paren {i z}$
where:
: $\csc$ denotes the cosecant function
: $\csch$ denotes the hyperbolic cosecant
: $i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
 
== Also see ==
* Sine in terms of Hyperbolic Sine
* Cosine in terms of Hyperbolic Cosine
* Tangent in terms of Hyperbolic Tangent
* Cotangent in terms of Hyperbolic Cotangent
* Secant in terms of Hyperbolic Secant
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.83$: Relationship between Hyperbolic and Trigonometric Functions
Category:Cosecant Function
Category:Hyperbolic Cosecant Function
Cosecant is Reciprocal of Sine

== Theorem ==
Let $\theta$ be an angle such that $\sin \theta \ne 0$.
Then:
:$\csc \theta = \dfrac 1 {\sin \theta}$
where $\csc$ and $\sin$ mean cosecant and sine respectively.
== Proof ==
Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.
Then:
When $\sin \theta = 0$, $\dfrac 1 {\sin \theta}$ is not defined.
== Sources ==
* : $\text V$. Trigonometry: Definitions of the ratios
* : $\S 5$: Trigonometric Functions: $5.18$
* 
* 
* : Appendix $12$: Trigonometric formulae
* : Appendix $14$: Trigonometric formulae
Category:Sine Function
Category:Cosecant Function
Cosecant of 105 Degrees

== Theorem ==
:$\csc 105 \degrees = \csc \dfrac {7 \pi} {12} = \sqrt 6 - \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 120 Degrees

== Theorem ==
:$\csc 120 \degrees = \csc \dfrac {2 \pi} 3 = \dfrac {2 \sqrt 3} 3$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 135 Degrees

== Theorem ==
:$\csc 135 \degrees = \csc \dfrac {3 \pi} 4 = \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 150 Degrees

== Theorem ==
:$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 15 Degrees

== Theorem ==
:$\csc 15 \degrees = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 165 Degrees

== Theorem ==
:$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 195 Degrees

== Theorem ==
:$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 210 Degrees

== Theorem ==
:$\csc 210 \degrees = \csc \dfrac {7 \pi} 6 = -2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 225 Degrees

== Theorem ==
:$\csc 225 \degrees = \csc \dfrac {5 \pi} 4 = -\sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 240 Degrees

== Theorem ==
:$\csc 240 \degrees = \csc \dfrac {4 \pi} 3 = -\dfrac {2 \sqrt 3} 3$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 255 Degrees

== Theorem ==
:$\csc 255 \degrees = \csc \dfrac {17 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 285 Degrees

== Theorem ==
:$\csc 285 \degrees = \csc \dfrac {19 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 300 Degrees

== Theorem ==
:$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 30 Degrees

== Theorem ==
:$\csc 30^\circ = \csc \dfrac \pi 6 = 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 315 Degrees

== Theorem ==
:$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 330 Degrees

== Theorem ==
:$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 345 Degrees

== Theorem ==
:$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 45 Degrees

== Theorem ==
:$\csc 45^\circ = \csc \dfrac \pi 4 = \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 60 Degrees

== Theorem ==
:$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of 75 Degrees

== Theorem ==
:$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$
where $\csc$ denotes cosecant.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of Angle in Cartesian Plane

== Theorem ==
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 $\theta$.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.12$
Category:Cosecant Function
Cosecant of Angle plus Full Angle

== Theorem ==
: $\csc \left({x + 2 \pi}\right) = \csc x$
== Proof ==
== Also see ==
* Sine of Angle plus Full Angle
* Cosine of Angle plus Full Angle
* Tangent of Angle plus Full Angle
* Cotangent of Angle plus Full Angle
* Secant of Angle plus Full Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Cosecant of Angle plus Right Angle

== Theorem ==
: $\csc \left({x + \dfrac \pi 2}\right) = \sec x$
== Proof ==
== Also see ==
* Sine of Angle plus Right Angle
* Cosine of Angle plus Right Angle
* Tangent of Angle plus Right Angle
* Cotangent of Angle plus Right Angle
* Secant of Angle plus Right Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Category:Reduction Formulae (Trigonometry)
Cosecant of Angle plus Straight Angle

== Theorem ==
:$\map \csc {x + \pi} = -\csc x$
== Proof ==
== Also see ==
* Sine of Angle plus Straight Angle
* Cosine of Angle plus Straight Angle
* Tangent of Angle plus Straight Angle
* Cotangent of Angle plus Straight Angle
* Secant of Angle plus Straight Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Cosecant of Angle plus Three Right Angles

== Theorem ==
:$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$
== Proof ==
== Also see ==
* Sine of Angle plus Three Right Angles
* Cosine of Angle plus Three Right Angles
* Tangent of Angle plus Three Right Angles
* Cotangent of Angle plus Three Right Angles
* Secant of Angle plus Three Right Angles
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Cosecant of Complement equals Secant

== Theorem ==
:$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
where $\csc$ and $\sec$ are cosecant and secant respectively.
That is, the secant of an angle is the cosecant of its complement.
This relation is defined wherever $\cos \theta \ne 0$.
== Proof ==
The above is valid only where $\cos \theta \ne 0$, as otherwise $\dfrac 1 {\cos \theta}$ is undefined.
From Cosine of Half-Integer Multiple of Pi it follows that this happens when $\theta \ne \paren {2 n + 1} \dfrac \pi 2$.
== Also see ==
* Sine of Complement equals Cosine
* Cosine of Complement equals Sine
* Tangent of Complement equals Cotangent
* Cotangent of Complement equals Tangent
* Secant of Complement equals Cosecant
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Secant Function
Category:Cosecant Function
Category:Complementary Angles
Cosecant of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$
where:
:$\csc$ denotes the complex cosecant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
== Proof ==
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Tangent of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
Category:Cosecant Function
Category:Complex Numbers
Cosecant of Conjugate Angle

== Theorem ==
:$\map \csc {2 \pi - \theta} = -\csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle is the negative of its conjugate.
== Proof ==
== Also see ==
* Sine of Conjugate Angle
* Cosine of Conjugate Angle
* Tangent of Conjugate Angle
* Cotangent of Conjugate Angle
* Secant of Conjugate Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Category:Conjugate Angles
Cosecant of Full Angle

== Theorem ==
:$\csc 360 \degrees = \csc 2 \pi$ is undefined
where $\csc$ denotes cosecant.
== Proof ==
From Cosecant is Reciprocal of Sine:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Full Angle:
:$\sin 360 \degrees = 0$
Thus $\csc \theta$ is undefined at this value.
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of Right Angle

== Theorem ==
:$\csc 90 \degrees = \csc \dfrac \pi 2 = 1$
where $\csc$ denotes cosecant.
== Proof ==
== Also see ==
* Sine of Right Angle
* Cosine of Right Angle
* Tangent of Right Angle
* Cotangent of Right Angle
* Secant of Right Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of Straight Angle

== Theorem ==
:$\csc 180 \degrees = \csc \pi$ is undefined
where $\csc$ denotes cosecant.
== Proof ==
From Cosecant is Reciprocal of Sine:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Straight Angle:
:$\sin \pi = 0$
Thus $\csc \theta$ is undefined at this value.
== Also see ==
* Sine of Straight Angle
* Cosine of Straight Angle
* Tangent of Straight Angle
* Cotangent of Straight Angle
* Secant of Straight Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of Supplementary Angle

== Theorem ==
:$\map \csc {\pi - \theta} = \csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle equals its supplement.
== Proof ==
== Examples ==
== Also see ==
* Sine of Supplementary Angle
* Cosine of Supplementary Angle
* Tangent of Supplementary Angle
* Cotangent of Supplementary Angle
* Secant of Supplementary Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant of Supplementary Angle
Category:Cosecant Function
Category:Supplementary Angles
Cosecant of Supplementary Angle/Examples/4 Theta minus 180

== Example of Cosecant of Supplementary Angle ==
:$\map \csc {4 \theta - 180 \degrees} = -\map \csc {4 \theta}$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercise $\text {XXXI}$: $8.$
Category:Cosecant of Supplementary Angle
Cosecant of Three Right Angles

== Theorem ==
:$\csc 270 \degrees = \csc \dfrac {3 \pi} 2 = -1$
where $\csc$ denotes cosecant.
== Proof ==
== Also see ==
* Sine of Three Right Angles
* Cosine of Three Right Angles
* Tangent of Three Right Angles
* Cotangent of Three Right Angles
* Secant of Three Right Angles
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of Three Right Angles less Angle

== Theorem ==
:$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$
where $\csc$ and $\sec$ are cosecant and secant respectively.
== Proof ==
== Also see ==
* Sine of Three Right Angles less Angle
* Cosine of Three Right Angles less Angle
* Tangent of Three Right Angles less Angle
* Cotangent of Three Right Angles less Angle
* Secant of Three Right Angles less Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cosecant Function
Cosecant of Zero

== Theorem ==
:$\csc 0$ is undefined
where $\csc$ denotes cosecant.
== Proof ==
From Cosecant is Reciprocal of Sine:
: $\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Zero is Zero:
: $\sin 0 = 0$
Thus $\csc \theta$ is undefined at this value.
== Also see ==
* Sine of Zero is Zero
* Cosine of Zero is One
* Tangent of Zero
* Cotangent of Zero
* Secant of Zero
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cosecant Function
Cosecant of i

== Theorem ==
:$\csc i = \paren {\dfrac {2 e} {1 - e^2} } i$
where $\csc$ denotes the complex cosecant function and $i$ is the imaginary unit.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Sine of $i$
* Cosine of $i$
* Tangent of $i$
* Cotangent of $i$
* Secant of $i$
Category:Cosecant of i
Category:Cosecant Function
Category:Imaginary Unit
Coset/Examples

== Examples of Cosets ==
=== Symmetry Group of Equilateral Triangle: Cosets of Reflection Subgroup ===
=== Symmetric Group on 3 Letters: Cosets of Alternating Subgroup ===
=== Dihedral Group $D_3$: Cosets of $\gen b$ ===
=== Subgroup of Infinite Cyclic Group ===
Category:Examples of Cosets
Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b

== Examples of Cosets ==
Consider the dihedral group $D_3$.
Let $H \subseteq D_3$ be defined as:
:$H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
As $b$ has order $2$, it follows that:
:$\gen b = \set {e, b}$
=== Left Cosets ===
=== Right Cosets ===
Category:Examples of Cosets
Category:Dihedral Group D3
Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b/Left Cosets

== Examples of Left Cosets ==
Consider the dihedral group $D_3$.
Let $H \subseteq D_3$ be defined as:
:$H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
As $b$ has order $2$, it follows that:
:$\gen b = \set {e, b}$
The left cosets of $H$ are:
== Proof ==
The Cayley table of $D_3$ is presented as:
Thus:
== Sources ==
* : Chapter $5$: Cosets and Lagrange's Theorem: Example $5.2$
Category:Examples of Cosets
Category:Dihedral Group D3
Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b/Right Cosets

== Examples of Right Cosets ==
Consider the dihedral group $D_3$.
Let $H \subseteq D_3$ be defined as:
:$H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
As $b$ has order $2$, it follows that:
:$\gen b = \set {e, b}$
The right cosets of $H$ are:
== Proof ==
The Cayley table of $D_3$ is presented as:
Thus:
== Sources ==
* : Chapter $5$: Cosets and Lagrange's Theorem: Proposition $5.15$
Category:Examples of Cosets
Category:Dihedral Group D3
Coset/Examples/Subgroup of Infinite Cyclic Group

== Examples of Cosets ==
Let $G = \gen a$ be an infinite cyclic group.
Let $s \in \Z_{>0}$ be a (strictly) positive integer.
Let $H$ be the subgroup of $G$ defined as:
:$H := \gen {a^s}$
Then a complete repetition-free list of the cosets of $H$ in $G$ is:
:$S = \set {H, aH, a^2 H, \ldots, a^{s - 1} H}$
== Proof ==
First it is demonstrated that $S$ is complete.
Let $x \in G$ be arbitrary.
Then:
:$\exists n \in \Z: x = a^n$
By the Division Theorem:
:$n = q s + r$
for $q, r \in \Z$ such that $0 \le r \le s - 1$.
Thus:
Thus:
:$x \in a^r H$
and as $0 \le r \le s - 1$ it follows that $a^r H \in S$.
Thus every $x \in G$ belongs to at least one of the cosets on $S$.
:
:$a^i H = a^j H$
where $0 \le i < j \le s - 1$.
Then from Sundry Coset Results:
:$\paren {a^i}^{-1} a^j \in H$
that is:
:$a^{j - i} \in H$
and so:
:$a^{j - i} = a^{k s}$
for some $k \in \Z$.
But $a$ is of infinite order.
So it follows from Subgroup Generated by Infinite Order Element is Infinite that:
:$j - i = k s$
But this contradicts our statement that $0 < j - i < s$.
Hence $S$ must be repetition-free.
== Sources ==
* : $\S 42$. Another approach to cosets: Worked Example $3$
Category:Examples of Cosets
Category:Infinite Cyclic Group
Coset/Examples/Symmetric Group on 3 Letters/Cosets of Alternating Subgroup

== Examples of Cosets ==
Consider the symmetric group on 3 Letters.
Let $H \subseteq S_3$ be defined as:
:$H = \set {e, \tuple {1 2 3}, \tuple {1 3 2} }$
The cosets of $H$ are:
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.5 \ \text{(b)}$
Category:Examples of Cosets
Category:Symmetric Group on 3 Letters
Coset/Examples/Symmetry Group of Equilateral Triangle/Cosets of Reflection Subgroup

== Examples of Cosets ==
Consider the symmetry group of the equilateral triangle $D_3$.
:
Let $H \subseteq D_3$ be defined as:
:$H = \set {e, r}$
where:
:$e$ denotes the identity mapping
:$r$ denotes reflection in the line $r$.
The left cosets of $H$ are:
The right cosets of $H$ are:
== Transversals ==
== Sources ==
* : $\S 6.1$. The quotient sets of a subgroup: Example $112$
* : $\S 42$. Another approach to cosets: Worked Example $1$
Category:Examples of Cosets
Category:Symmetry Group of Equilateral Triangle
Coset/Examples/Symmetry Group of Equilateral Triangle/Cosets of Reflection Subgroup/Transversals

== Examples of Cosets ==
Consider the symmetry group of the equilateral triangle $D_3$.
:
Let $H \subseteq D_3$ be defined as:
:$H = \set {e, r}$
where:
:$e$ denotes the identity mapping
:$r$ denotes reflection in the line $r$.
Some of the left transversals of $H$ are given by:
:$\set {e, s, t}$
:$\set {e, q, p}$
:$\set {r, s, p}$
and so on.
== Sources ==
* : $\S 6.3$. Index. Transversals: Example $116$
Category:Symmetry Group of Equilateral Triangle
Category:Examples of Transversals
Coset Decoding Table/Examples/(4, 2) code in Z3

== Example of Coset Decoding Table ==
Let $C$ be the linear code:
:$C = \set {0000, 0112, 0221, 1011, 1120, 1202, 2022, 2101, 2210}$
Then the coset decoding table $T$ for $C$ is:
:$\begin{array} {cccccccc}
0000 & 0112 & 0221 & 1011 & 1120 & 1202 & 2022 & 2101 & 2210 \\
1000 & 1112 & 1221 & 2011 & 2120 & 2202 & 0022 & 0101 & 0210 \\
0100 & 0212 & 0021 & 1111 & 1220 & 1002 & 2122 & 2201 & 2010 \\
0010 & 0122 & 0201 & 1021 & 1100 & 1212 & 2002 & 2111 & 2220 \\
0001 & 0110 & 0222 & 1012 & 1121 & 1200 & 2020 & 2102 & 2211 \\
2000 & 2112 & 2221 & 0011 & 0120 & 0202 & 1022 & 1101 & 1210 \\
0200 & 0012 & 0121 & 1211 & 1020 & 1102 & 2222 & 2001 & 2110 \\
0020 & 0102 & 0211 & 1001 & 1110 & 1222 & 2012 & 2121 & 2200 \\
0002 & 0111 & 0220 & 1010 & 1122 & 1201 & 2021 & 2100 & 2212 \\
\end{array}$
== Sources ==
* : Chapter $6$: Error-correcting codes: Exercise $4$
Category:Examples of Linear Codes
Coset Decoding Table/Examples/(6, 3) code in Z2

== Example of Coset Decoding Table ==
Let $C$ be the linear code:
:$C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$
Then the coset decoding table $T$ for $C$ is:
:$\begin{array} {cccccccc}
000000 & 100110 & 010101 & 110011 & 001011 & 101101 & 011110 & 111000 \\
100000 & 000110 & 110101 & 010011 & 101011 & 001101 & 111110 & 011000 \\
010000 & 110110 & 000101 & 100011 & 011011 & 111101 & 001110 & 101000 \\
001000 & 101110 & 011101 & 111011 & 000011 & 100101 & 010110 & 110000 \\
000100 & 100010 & 010001 & 110111 & 001111 & 101001 & 011010 & 111100 \\
000010 & 100100 & 010111 & 110001 & 001001 & 101111 & 011100 & 111010 \\
000001 & 100111 & 010100 & 110010 & 001010 & 101100 & 011111 & 111001 \\
100001 & 000111 & 110100 & 010010 & 101010 & 001100 & 111111 & 011001 \\
\end{array}$
Note that it is not easy to find the $8$th coset leader.
It has to be an example of an element $v$ of $\map V {6, 2}$ of weight $2$ which does not exist anywhere in the other $7$ rows.
Hence we must try each one in turn, and scan the coset decoding table as it has been constructed so far to see whether it is already there.
This may take some time to do.
Having chosen such a coset leader, it is noted that it is not unique: $010010$ or $001100$ could equally well have been chosen.
 
== Example of Use ==
To decode an element $v$ of $\map V {6, 2}$, we locate $v$ in $T$, and replace it with the element found in the first row of the same column that $v$ is in.
Thus, for example, let $v := 011010$.
$v$ is in the $5$th row and the $7$th column.
Hence its corresponding transmitted codeword $u$ is found in the $1$st row of the $7$th column:
:$u = 011110$
== Sources ==
* : Chapter $6$: Error-correcting codes: Example $6.16$
Category:Examples of Linear Codes
Coset Equals Subgroup iff Element in Subgroup

== Theorem ==
Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Let $x \in G$.
Let:
: $x H$ denote the left coset of $H$ by $x$
: $H x$ denote the right coset of $H$ by $x$.
==== Left Coset Equals Subgroup iff Element in Subgroup ====
==== Right Coset Equals Subgroup iff Element in Subgroup ====
Category:Cosets