nyuuzyou/wikis huggingface dataset:
Tangent of 150 Degrees

== Theorem ==
:$\tan 150 \degrees = \tan \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 15 Degrees

== Theorem ==
:$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$
where $\tan$ denotes tangent.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Category:Tangent of 15 Degrees
Tangent of 165 Degrees

== Theorem ==
:$\tan 165 \degrees = \tan \dfrac {11 \pi} {12} = -\paren {2 - \sqrt 3}$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 195 Degrees

== Theorem ==
:$\tan 195 \degrees = \tan \dfrac {13 \pi} {12} = 2 - \sqrt 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 210 Degrees

== Theorem ==
:$\tan 210 \degrees = \tan \dfrac {7 \pi} 6 = \dfrac {\sqrt 3} 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 22.5 Degrees

== Theorem ==
:$\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$
where $\tan$ denotes tangent.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Sine of $22.5 \degrees$
* Cosine of $22.5 \degrees$
Category:Tangent of 22.5 Degrees
Category:Tangent Function
Tangent of 240 Degrees

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

== Theorem ==
:$\tan 255 \degrees = \tan \dfrac {17 \pi} {12} = 2 + \sqrt 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 285 Degrees

== Theorem ==
:$\tan 285 \degrees = \tan \dfrac {19 \pi} {12} = -\paren {2 + \sqrt 3}$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 300 Degrees

== Theorem ==
:$\tan 300 \degrees = \tan \dfrac {5 \pi} 3 = -\sqrt 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 30 Degrees

== Theorem ==
:$\tan 30 \degrees = \tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$
where $\tan$ denotes tangent.
== Proof ==
== Also presented as ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
* 
* 
* : Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
* : Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles
Category:Tangent of 30 Degrees
Category:Tangent Function
Tangent of 30 Degrees/Also presented as

== Tangent of 30 Degrees: Also presented as == 
Some sources present the tangent of $30 \degrees$ as:
:$\tan 30 \degrees = \tan \dfrac \pi 6 = \dfrac 1 {\sqrt 3}$
== Sources ==
* 
* 
* : Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
* : Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles
Category:Tangent of 30 Degrees
Tangent of 330 Degrees

== Theorem ==
:$\tan 330^\circ = \tan \dfrac {11 \pi} 6 = -\dfrac {\sqrt 3} 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 345 Degrees

== Theorem ==
:$\tan 345^\circ = \tan \dfrac {23 \pi} {12} = -\left({2 - \sqrt 3}\right)$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of 45 Degrees

== Theorem ==
:$\tan 45 \degrees = \tan \dfrac \pi 4 = 1$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
* 
* 
* : Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
* : Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles
Category:Tangent Function
Tangent of 60 Degrees

== Theorem ==
:$\tan 60 \degrees = \tan \dfrac \pi 3 = \sqrt 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
* : Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
* : Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles
Category:Tangent Function
Tangent of 75 Degrees

== Theorem ==
:$\tan 75 \degrees = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$
where $\tan$ denotes tangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent 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:
:$\tan \theta = \dfrac y x$
where $\tan$ denotes the tangent of $\theta$.
== Proof ==
:
Let a unit circle $C$ be drawn with its center at the origin $O$.
Let a tangent line be drawn to $C$ parallel to $PS$ meeting $C$ at $R$.
Let $Q$ be the point on $OP$ which intersects this tangent line.
$\angle OSP = \angle ORQ$, as both are right angles.
Both $\triangle OSP$ and $\triangle ORQ$ share angle $\theta$.
By Triangles with Two Equal Angles are Similar it follows that $\triangle OSP$ and $\triangle ORQ$ are similar.
Thus:
Then:
When $\theta$ is obtuse, the same argument holds, but both $x$ and $\tan \theta$ are negative.
When $\theta = \dfrac \pi 2$ we have that $x = 0$.
Then $OP$ is parallel to the tangent line at $R$ which it therefore does not meet.
Thus when $\theta = \dfrac \pi 2$, it follows that $\tan \theta$ is not defined.
Likewise $\dfrac y x$ is not defined when $x = 0$.
Thus the relation holds for $\theta = \dfrac \pi 2$.
When $\pi < \theta < 2 \pi$ the diagram can be reflected in the $x$-axis.
In this case, $y$ is negative.
Thus the relation continues to hold.
When $\theta = 0$ and $\theta = \pi$ we have that $y = 0$ and $\tan \theta = 0 = \dfrac y x$.
Hence the result.
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.9$
Category:Tangent Function
Tangent of Angle minus Three Right Angles

== Theorem ==
:$\map \tan {x - \dfrac {3 \pi} 2} = \cot x$
== Proof ==
== Also see ==
* Sine of Angle plus Three Right Angles
* Cosine of Angle plus Three Right Angles
* Cotangent of Angle plus Three Right Angles
* Secant of Angle plus Three Right Angles
* Cosecant of Angle plus Three Right Angles
== Sources ==
* : $\text V$. Trigonometry: Angles larger than $90 \degrees$: Examples
Category:Tangent Function
Tangent of Angle plus Full Angle

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

== Theorem ==
:$\map \tan {x + \dfrac \pi 2} = -\cot x$
== Proof ==
== Also see ==
* Sine of Angle plus Right Angle
* Cosine of Angle plus Right Angle
* Cotangent of Angle plus Right Angle
* Secant of Angle plus Right Angle
* Cosecant of Angle plus Right Angle
== Sources ==
* : $\text V$. Trigonometry: Exercise $\text {XXXI}$: $6.$
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
* : Appendix $12$: Trigonometric formulae: Shifts and periodicity
* : Appendix $14$: Trigonometric formulae: Shifts and periodicity
Category:Tangent Function
Category:Reduction Formulae (Trigonometry)
Tangent of Angle plus Straight Angle

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

== Theorem ==
:$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$
== Proof ==
== Also see ==
* Sine of Angle plus Three Right Angles
* Cosine of Angle plus Three Right Angles
* Cotangent of Angle plus Three Right Angles
* Secant of Angle plus Three Right Angles
* Cosecant 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:Tangent Function
Tangent of Complement equals Cotangent

== Theorem ==
:$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ 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 $\sin \theta \ne 0$.
== Proof ==
The above is valid only where $\sin \theta \ne 0$, as otherwise $\dfrac {\cos \theta} {\sin \theta}$ is undefined.
From Sine of Multiple of Pi it follows that this happens when $\theta \ne n \pi$.
== Also see ==
* Sine of Complement equals Cosine
* Cosine of Complement equals Sine
* Cotangent of Complement equals Tangent
* Secant of Complement equals Cosecant
* Cosecant of Complement equals Secant
== Sources ==
* : $\text V$. Trigonometry: Formulae $(6)$
* : $\text V$. Trigonometry: Complementary angles
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
* : Appendix $12$: Trigonometric formulae: Symmetry
* : Appendix $14$: Trigonometric formulae: Symmetry
Category:Tangent Function
Category:Cotangent Function
Category:Complementary Angles
Tangent of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
=== Formulation 4 ===
where:
:$\tan$ denotes the tangent function (real and complex)
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
:$\tanh$ denotes the hyperbolic tangent function.
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
Category:Tangent Function
Category:Complex Numbers
Category:Tangent of Complex Number
Tangent of Complex Number/Formulation 1

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$
where:
:$\tan$ denotes the complex tangent 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
* Cosecant of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
Category:Tangent of Complex Number
Tangent of Complex Number/Formulation 2

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$
where:
:$\tan$ denotes the tangent function (real and complex)
:$\tanh$ denotes the hyperbolic tangent function.
== Proof ==
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
Category:Tangent of Complex Number
Tangent of Complex Number/Formulation 3

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$
where:
:$\tan$ denotes the tangent function (real and complex)
:$\tanh$ denotes the hyperbolic tangent function.
== Proof ==
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
Category:Tangent of Complex Number
Tangent of Complex Number/Formulation 4

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$
where:
:$\tan$ denotes the complex tangent 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
* Cosecant of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
Category:Tangent of Complex Number
Tangent of Conjugate Angle

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

== Corollary to Tangent of Sum ==
:$\map \tan {a - b} = \dfrac {\tan a - \tan b} {1 + \tan a \tan b}$
where $\tan$ is tangent.
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(18)$
* : $\S 5$: Trigonometric Functions: $5.36$
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
* 
* 
* 
* : Appendix $12$: Trigonometric formulae: Addition formulae
* : Appendix $14$: Trigonometric formulae: Addition formulae
Category:Tangent of Difference
Category:Tangent Function
Category:Trigonometric Subtraction Formulas
Tangent of Half Angle for Spherical Triangles

== Theorem ==
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$
where $s = \dfrac {a + b + c} 2$.
== Proof ==
The result follows.
== Also presented as ==
== Also see ==
* The other Half Angle Formulas for Spherical Triangles:
** Sine of Half Angle for Spherical Triangles
** Cosine of Half Angle for Spherical Triangles
* Half Side Formulas for Spherical Triangles:
** Sine of Half Side for Spherical Triangles
** Cosine of Half Side for Spherical Triangles
** Tangent of Half Side for Spherical Triangles
== Sources ==
* : Chapter $\text I$. Spherical Trigonometry: $5$. The cosine-formula.
Category:Half Angle Formulas for Spherical Triangles
Tangent of Half Angle in Triangle

== Theorem ==
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 $C$.
Let $s$ denote the semiperimeter of $\triangle ABC$:
:$s = \dfrac {a + b + c} 2$
Then:
:$\tan \dfrac C 2 = \sqrt {\dfrac {\paren {s - a} \paren {s - b} } {s \paren {s - c} } }$
== Proof ==
== Also presented as ==
== Also see ==
* Cosine of Half Angle in Triangle
* Sine of Half Angle in Triangle
== Historical Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(43)$
Category:Tangent of Half Angle in Triangle
Category:Triangles
Category:Tangent Function
Tangent of Half Side for Spherical Triangles

== Theorem ==
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$
where $S = \dfrac {a + b + c} 2$.
== Proof ==
Hence the result.
== Also presented as ==
== Also see ==
* The other Half Side Formulas for Spherical Triangles:
** Sine of Half Side for Spherical Triangles
** Cosine of Half Side for Spherical Triangles
* Half Angle Formulas for Spherical Triangles:
** Sine of Half Angle for Spherical Triangles
** Cosine of Half Angle for Spherical Triangles
** Tangent of Half Angle for Spherical Triangles
Category:Half Side Formulas for Spherical Triangles
Tangent of Right Angle

== Theorem ==
:$\tan 90 \degrees = \tan \dfrac \pi 2$ is undefined
where $\tan$ denotes tangent.
== Proof ==
From Tangent is Sine divided by Cosine:
:$\tan \theta = \dfrac {\sin \theta} {\cos \theta}$
When $\cos \theta = 0$, $\dfrac {\sin \theta} {\cos \theta}$ can be defined only if $\sin \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = \dfrac \pi 2$, $\cos \theta = 0$.
Thus $\tan \theta$ is undefined at this value.
== Also defined as ==
Some sources give that:
:$\tan 90 \degrees = \infty$
but this naïve approach is overly simplistic and cannot be backed up with mathematical rigour.
== Also see ==
* Sine of Right Angle
* Cosine of Right Angle
* Cotangent of Right Angle
* Secant of Right Angle
* Cosecant of Right Angle
== Sources ==
* : $\text V$. Trigonometry: Special angles
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
* : Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
* : Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles
Category:Tangent Function
Tangent of Straight Angle

== Theorem ==
:$\tan 180 \degrees = \tan \pi = 0$
where $\tan$ denotes tangent.
== Proof ==
== Also see ==
* Sine of Straight Angle
* Cosine of Straight Angle
* Cotangent of Straight Angle
* Secant of Straight Angle
* Cosecant of Straight Angle
== Sources ==
* : $\text V$. Trigonometry: Special angles
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
* 
* 
Category:Tangent Function
Tangent of Sum

== Theorem ==
:$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$
where $\tan$ is tangent.
=== Corollary ===
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(17)$
* : $\S 5$: Trigonometric Functions: $5.36$
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
* 
* 
* 
* : Appendix $12$: Trigonometric formulae: Addition formulae
* : Appendix $14$: Trigonometric formulae: Addition formulae
Category:Tangent of Sum
Category:Tangent Function
Category:Trigonometric Addition Formulas
Tangent of Sum of Series of Angles

== Theorem ==
Let $\theta_1, \theta_2, \ldots, \theta_n$ be angles.
For all $k \in \set {1, 2, 3, \ldots, n}$, let $s_k$ be defined as the sum of the product of $\map \tan {\theta_1}, \map \tan {\theta_2}, \ldots, \map \tan {\theta_n}$ taken $k$ at a time:
Then:
:$\map \tan {\theta_1 + \theta_2 + \theta_3 + \cdots + \theta_n} = \dfrac {s_1 - s_3 + s_5 - \cdots} {1 - s_2 + s_4 - \cdots}$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(21)$
* 
Category:Tangent Function
Category:Tangent of Sum of Series of Angles
Tangent of Sum of Three Angles

== Theorem ==
:$\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Cotangent of Sum of Three Angles
== Sources ==
* : $\text V$. Trigonometry: Formulae $(20)$
Category:Tangent of Sum of Three Angles
Category:Tangent Function
Category:Trigonometric Addition Formulas
Tangent of Supplementary Angle

== Theorem ==
:$\map \tan {\pi - \theta} = -\tan \theta$
where $\tan$ denotes tangent.
That is, the tangent of an angle is the negative of its supplement.
== Proof ==
== Also see ==
* Sine of Supplementary Angle
* Cosine of Supplementary Angle
* Cotangent of Supplementary Angle
* Secant of Supplementary Angle
* Cosecant of Supplementary Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
* : Appendix $12$: Trigonometric formulae: Symmetry
* : Appendix $14$: Trigonometric formulae: Symmetry
Category:Tangent Function
Category:Supplementary Angles
Tangent of Three Right Angles

== Theorem ==
:$\tan 270 \degrees = \tan \dfrac {3 \pi} 2$ is undefined
where $\tan$ denotes tangent.
== Proof ==
We have:
But from Tangent of Right Angle, $\tan 90 \degrees$ is undefined.
Hence so is $\tan 270 \degrees$.
== Also defined as ==
Some sources give that:
:$\tan 270 \degrees = \infty$
but this naïve approach is overly simplistic and cannot be backed up with mathematical rigour.
== Also see ==
* Sine of Three Right Angles
* Cosine of Three Right Angles
* Cotangent of Three Right Angles
* Secant of Three Right Angles
* Cosecant of Three Right Angles
== Sources ==
* : $\text V$. Trigonometry: Special angles
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Tangent of Three Right Angles less Angle

== Theorem ==
:$\tan \left({\dfrac {3 \pi} 2 - \theta}\right) = \cot \theta$
where $\tan$ and $\cot$ are tangent and cotangent respectively.
== Proof ==
== Also see ==
* Sine of Three Right Angles less Angle
* Cosine of Three Right Angles less Angle
* Cotangent of Three Right Angles less Angle
* Secant of Three Right Angles less Angle
* Cosecant 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:Tangent Function
Tangent of Three Right Angles minus Angle

== Theorem ==
:$\map \tan {x - \dfrac {3 \pi} 2} = \cot x$
== Proof ==
== Also see ==
* Sine of Angle minus Three Right Angles
* Cosine of Angle minus Three Right Angles
* Cotangent of Angle minus Three Right Angles
* Secant of Angle minus Three Right Angles
* Cosecant of Angle minus Three Right Angles
Category:Tangent Function
Tangent of Uniform Distribution has Standard Cauchy Distribution

== Theorem ==
Let $X$ be a continuous random variable with a uniform distribution on the closed real interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$:
:$X \sim \ContinuousUniform {-\dfrac \pi 2} {\dfrac \pi 2}$
Let $Y$ be a continuous random variable such that:
:$Y = \tan X$
where $\tan$ denotes the tangent function.
Then $Y$ has the standard Cauchy distribution:
:$Y \sim \Cauchy 0 1$
== Proof ==
== Examples ==
== Sources ==
* 
Category:Tangent of Uniform Distribution has Standard Cauchy Distribution
Category:Uniform Distribution
Category:Standard Cauchy Distribution
Tangent of Uniform Distribution has Standard Cauchy Distribution/Examples/Radiation Particles

== Example of Use of Tangent of Uniform Distribution has Standard Cauchy Distribution ==
Let particles be emitted from a source of radiation $S$ in a plane in random (equally likely) directions.
Let these particles travel in a straight line to a plane collector some distance from $A$.
Then the distribution of the points of impact of the particles on the collector has a standard Cauchy distribution.
== Proof ==
== Sources ==
* 
Category:Tangent of Uniform Distribution has Standard Cauchy Distribution