nyuuzyou/wikis huggingface dataset:
Sine of 330 Degrees

== Theorem ==
:$\sin 330 \degrees = \sin \dfrac {11 \pi} 6 = -\dfrac 1 2$
where $\sin$ denotes the sine function.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Sine Function
Sine of 345 Degrees

== Theorem ==
:$\sin 345^\circ = \sin \dfrac {23 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$
where $\sin$ denotes the sine function.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Sine Function
Sine of 36 Degrees

== Theorem ==
:$\sin 36^\circ = \sin \dfrac \pi 5 = \dfrac {\sqrt {\sqrt 5 / \phi} } 2 = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$
where $\phi$ denotes the golden mean.
== Proof ==
We also have:
== Sources ==
* : $\S 1.2.8$: Fibonacci Numbers: Exercise $19$
Category:Sine Function
Category:Golden Mean
Sine of 3 Degrees

== Theorem ==
:$\sin 3^\circ = \sin \dfrac \pi {60} = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$
where $\sin$ denotes the sine function.
== Proof ==
Category:Sine Function
Sine of 45 Degrees

== Theorem ==
:$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$
where $\sin$ denotes the sine function.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Proof 5 ==
== Also presented as ==
== 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:Sine of 45 Degrees
Category:Sine Function
Sine of 45 Degrees/Also presented as

== Sine of 45 Degrees: Also presented as == 
Some sources present the sine of $45 \degrees$ as:
:$\sin 45 \degrees = \dfrac 1 {\sqrt 2}$
== Sources ==
* : $\text V$. Trigonometry: Special angles
* 
* 
* : Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
* : Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles
Category:Sine of 45 Degrees
Sine of 45 Degrees/Proof 1

== Theorem ==
== Proof ==
:
Let $ABCD$ be a square of side $r$.
By definition, each angle of $\triangle ABCD$ is equal to $90 \degrees$.
Let $AC$ be a diagonal of $ABCD$.
As $\triangle ABC$ is a right angled triangle, it follows from Pythagoras's Theorem that $AC = \sqrt 2 A B$.
As $AC$ is a bisector of $\angle DAB$ it follows that $\angle CAB = 45 \degrees$.
So by definition of sine function:
:$\map \sin {\angle CAB} = \dfrac r {r \sqrt 2} = \dfrac {\sqrt 2} 2$
Category:Sine of 45 Degrees
Sine of 45 Degrees/Proof 5

== Theorem ==
== Proof ==
The negative solution is rejected because $45 \degrees$ is an acute angle and Sine of Acute Angle is Positive.
Therefore:
:$\sin 45 \degrees = \dfrac 1 {\sqrt 2} = \dfrac {\sqrt 2} 2$
Category:Sine of 45 Degrees
Sine of 60 Degrees

== Theorem ==
:$\sin 60 \degrees = \sin \dfrac \pi 3 = \dfrac {\sqrt 3} 2$
where $\sin$ denotes the sine function.
== Proof ==
== 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:Sine Function
Sine of 72 Degrees

== Theorem ==
:$\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$
where $\sin$ denotes the sine function.
== Proof ==
== Sources ==
* : $\S 1.2.8$: Fibonacci Numbers: Exercise $19$
Category:Sine Function
Sine of 75 Degrees

== Theorem ==
:$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
where $\sin$ denotes the sine function.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Sine Function
Category:Sine of 75 Degrees
Sine 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:
:$\sin \theta = \dfrac y r$
where $\sin$ denotes the sine of $\theta$.
== Proof ==
:
Let a unit circle $C$ be drawn with its center at the origin $O$.
Let $Q$ be the point on $C$ which intersects $OP$.
$\angle OSP = \angle ORQ$, 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.
By definition of similarity:
Then:
When $\theta$ is obtuse, the same argument holds.
When $\theta = \dfrac \pi 2$ we have that $x = 0$.
Thus $y = r$ and $\sin \theta = 1 \dfrac y r$.
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, both $\sin \theta$ and $y$ are negative.
Thus the relation continues to hold.
When $\theta = 0$ and $\theta = \pi$ we have that $y = 0$ and $\sin \theta = 0 = \dfrac y r$.
Hence the result.
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.7$
Category:Sine Function
Sine of Angle of Triangle by Semiperimeter

== 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 $C$.
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: $s = \dfrac {a + b + c} 2$.
== Proof ==
Let $Q$ be the area of $\triangle ABC$.
From Area of Triangle in Terms of Two Sides and Angle:
:$Q = \dfrac {b c \sin A} 2$
From Heron's Formula:
:$Q = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
Equating the two:
:$\dfrac {b c \sin A} 2 = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
from which follows the result.
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.95$
Category:Triangles
Category:Sine Function
Sine of Angle plus Full Angle

== Theorem ==
:$\map \sin {x + 2 \pi} = \sin x$
=== Corollary ===
== Proof ==
== Also see ==
* Cosine of Angle plus Full Angle
* Tangent 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:Sine Function
Sine of Angle plus Full Angle/Corollary

== Theorem ==
Let $n \in \Z$ be an integer.
Then:
:$\map \sin {x + 2 n \pi} = \sin x$
== Proof ==
From Sine of Angle plus Full Angle:
:$\map \sin {x + 2 \pi} = \sin x$
The result follows from the General Periodicity Property:
If:
:$\forall x \in X: \map f x = \map f {x + L}$
then:
:$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$
== Sources ==
* : Appendix $12$: Trigonometric formulae: Shifts and periodicity
* : Appendix $14$: Trigonometric formulae: Shifts and periodicity
Category:Sine Function
Sine of Angle plus Right Angle

== Theorem ==
: $\sin \paren {x + \dfrac \pi 2} = \cos x$
== Proof ==
== Also see ==
* Cosine of Angle plus Right Angle
* Tangent of Angle plus Right Angle
* Cotangent of Angle plus Right Angle
* Secant of Angle plus Right Angle
* Cosecant of Angle plus Right Angle
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.19)$
* : $\S 5$: 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:Sine Function
Category:Reduction Formulae (Trigonometry)
Sine of Angle plus Straight Angle

== Theorem ==
:$\map \sin {x + \pi} = -\sin x$
== Proof ==
== Also see ==
* Cosine of Angle plus Straight Angle
* Tangent of Angle plus Straight Angle
* Cotangent of Angle plus Straight Angle
* Secant of Angle plus Straight Angle
* Cosecant of Angle plus Straight Angle
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
* : $\S 5$: 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:Sine Function
Sine of Angle plus Three Right Angles

== Theorem ==
:$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$
== Proof ==
== Also see ==
* 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
* 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:Sine Function
Sine of Complement equals Cosine

== Theorem ==
:$\map \sin {\dfrac \pi 2 - \theta} = \cos \theta$
where $\sin$ and $\cos$ are sine and cosine respectively.
That is, the cosine of an angle is the sine of its complement.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Examples ==
== Also see ==
* Cosine of Complement equals Sine
* Tangent of Complement equals Cotangent
* Cotangent of Complement equals Tangent
* Secant of Complement equals Cosecant
* Cosecant of Complement equals Secant
== Sources ==
* : $\text V$. Trigonometry: Formulae $(5)$
* : $\S 5$: 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:Sine of Complement equals Cosine
Category:Sine Function
Category:Cosine Function
Category:Complementary Angles
Sine of Complement equals Cosine/Examples/2 Theta minus 90

== Example of Sine of Complement equals Cosine ==
:$\map \sin {2 \theta - 90 \degrees} = -\cos 2 \theta$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercise $\text {XXXI}$: $10.$
Category:Sine of Complement equals Cosine
Sine of Complement equals Cosine/Proof 4

== Theorem ==
== Proof ==
:
Let $\angle xOP$ and $\angle QOy$ be complementary.
Then:
:$\angle xOP = \angle QOy$
Hence:
:the projection of $OP$ on the $x$-axis
equals:
:the projection of $OQ$ on the $y$-axis.
Hence the result.
== Sources ==
* : $\text V$. Trigonometry: Complementary angles
Category:Sine of Complement equals Cosine
Sine of Complex Number

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

== Theorem ==
== Proof ==
== Also see ==
* Cosine of Complex Number
* Tangent of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
* Cotangent of Complex Number
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
Category:Sine of Complex Number
Sine of Conjugate Angle

== Theorem ==
:$\map \sin {2 \pi - \theta} = -\sin \theta$
where $\sin$ denotes sine.
That is, the sine of an angle is the negative of its conjugate.
== Proof ==
== Also see ==
* Cosine of Conjugate Angle
* Tangent 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:Sine Function
Category:Conjugate Angles
Sine of Difference

== Corollary to Sine of Sum ==
:$\map \sin {a - b} = \sin a \cos b - \cos a \sin b$
where $\sin$ denotes the sine and $\cos$ denotes the cosine.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Cosine of Difference
== Historical Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(10)$
* : $\S 5$: Trigonometric Functions: $5.34$
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
* 
* 
* 
* : Appendix $12$: Trigonometric formulae: Addition formulae
* : Appendix $14$: Trigonometric formulae: Addition formulae
Category:Sine of Difference
Category:Sine Function
Category:Trigonometric Subtraction Formulas
Sine of Full Angle

== Theorem ==
:$\sin 360^\circ = \sin 2 \pi = 0$
where $\sin$ denotes the sine function and $360^\circ = 2 \pi$ is the full angle.
== Proof ==
A direct implementation of Sine of Multiple of Pi:
:$\forall n \in \Z: \sin n \pi = 0$
In this case, $n = 2$ and so:
:$\sin 2 \pi = 0$
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Sine Function
Sine of Half-Integer Multiple of Pi

== Theorem ==
Let $x \in \R$ be a real number.
Let $\sin x$ be the sine of $x$.
Then:
:$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$
or:
== Proof ==
From the discussion of Sine and Cosine are Periodic on Reals:
:$\map \sin {x + \dfrac \pi 2} = \cos x$
The result then follows directly from the Cosine of Multiple of Pi.
Category:Sine Function
Sine 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:
:$\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$.
== Proof ==
The result follows.
== Also see ==
* The other Half Angle Formulas for Spherical Triangles:
** Cosine of Half Angle for Spherical Triangles
** Tangent 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
Sine 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 $\triangle ABC$.
Then:
:$\sin \dfrac C 2 = \sqrt {\dfrac {\paren {s - a} \paren {s - b} } {a b} }$
== Proof ==
== Also see ==
* Cosine of Half Angle in Triangle
* Tangent of Half Angle in Triangle
== Sources ==
* : $\text V$. Trigonometry: Formulae $(42)$
Category:Sine of Half Angle in Triangle
Category:Triangles
Category:Sine Function
Sine 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:
:$\sin \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} }$
where $S = \dfrac {A + B + C} 2$.
== Proof ==
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
:not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
:but also $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.
Let $s' = \dfrac {a' + b' + c'} 2$.
We have:
Then:
and:
The result follows.
== Also see ==
* The other Half Side Formulas for Spherical Triangles:
** Cosine of Half Side for Spherical Triangles
** Tangent 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
Sine of Integer Multiple of Argument

== Theorem ==
For $n \in \Z_{>0}$:
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
=== Formulation 4 ===
=== Formulation 5 ===
=== Formulation 6 ===
=== Formulation 7 ===
=== Formulation 8 ===
=== Formulation 9 ===
== Also see ==
* Cosine of Integer Multiple of Argument
Category:Sine Function
Category:Sine of Integer Multiple of Argument
Sine of Integer Multiple of Argument/Formulation 1

== Theorem ==
For $n \in \Z_{>0}$:
== Proof ==
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$
=== Basis for the Induction ===
$\map P 1$ is the case:
So $\map P 1$ is seen to hold.
$\map P 2$ is the case:
So $\map P 2$ is also seen to hold.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P n$ is true, where $n > 2$, then it logically follows that $\map P {n + 1}$ is true.
So this is our induction hypothesis:
:$\ds \map \sin {n \theta} = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$
from which we are to show:
:$\ds \map \sin {\paren {n + 1} \theta} = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n + 1 - \paren {k + 1} } k \paren {2 \cos \theta}^{n + 1 - \paren {2 k + 1} } }$
=== Induction Step ===
This is our induction step:
To proceed, we will require the following lemma:
=== Lemma ===
The result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \forall n \in \Z_{>0}: \sin n \theta = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$
== Examples ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.68$
Category:Sine of Integer Multiple of Argument
Sine of Integer Multiple of Argument/Formulation 1/Examples/Sine of Quintuple Angle

== Example of Use of ==
:$\sin 5 \theta = \sin \theta \paren { \paren {2 \cos \theta}^4 - 3 \paren {2 \cos \theta}^2 + 1 }$
== Proof ==
Category:Quintuple Angle Formula for Sine
Category:Quintuple Angle Formulas
Sine of Integer Multiple of Argument/Formulation 1/Examples/Sine of Sextuple Angle

== Example of Use of ==
:$\sin 6 \theta = \sin \theta \paren { \paren {2 \cos \theta}^5 - 4 \paren {2 \cos \theta}^3 + 3 \paren {2 \cos \theta} }$
== Proof ==
Category:Sextuple Angle Formula for Sine
Category:Sextuple Angle Formulas
Sine of Integer Multiple of Argument/Formulation 2

== Theorem ==
For $n \in \Z_{>0}$:
== Proof ==
By De Moivre's Formula:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \in \Z_{>0}$, we use the Binomial Theorem on the , resulting in:
:$\ds \cos n \theta + i \sin n \theta = \sum_{k \mathop \ge 0} \binom n k \paren {\cos^{n - k} \theta} \paren {i \sin \theta}^k$
When $k$ is odd, the expression being summed is imaginary.
Equating the imaginary parts of both sides of the equation, replacing $k$ with $2 k + 1$ to make $k$ odd, gives:
== Examples ==
Category:Sine of Integer Multiple of Argument
Sine of Integer Multiple of Argument/Formulation 2/Examples/Sine of Quintuple Angle

== Example of Use of ==
:$\sin 5 \theta = \cos^5 \theta \paren {5 \tan \theta - 10 \tan^3 \theta + \tan^5 \theta}$
== Proof ==
Category:Quintuple Angle Formula for Sine
Category:Quintuple Angle Formulas
Sine of Integer Multiple of Argument/Formulation 2/Examples/Sine of Sextuple Angle

== Example of Use of ==
:$\map \sin {6 \theta } = \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta}$
== Proof ==
Category:Sextuple Angle Formula for Sine
Category:Sextuple Angle Formulas
Sine of Integer Multiple of Argument/Formulation 3

== Theorem ==
For $n \in \Z_{>0}$:
== Proof ==
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop \ge 0} \frac {\cos k \theta} {\cos^k \theta}$
=== Basis for the Induction ===
$\map P 1$ is the case:
So $\map P 1$ is seen to hold.
$\map P 2$ is the case:
So $\map P 2$ is also seen to hold.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P n$ is true, where $n > 2$, then it logically follows that $\map P {n + 1}$ is true.
So this is our induction hypothesis:
:$\ds \map \sin {n \theta} = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop = 0}^{n - 1} \frac {\cos k \theta } {\cos^k \theta}$
from which we are to show:
:$\ds \map \sin {\paren {n + 1} \theta} = \sin \theta \cos^n \theta \sum_{k \mathop = 0}^n \frac {\cos k \theta} {\cos^k \theta}$
=== Induction Step ===
This is our induction step:
For the first part:
The result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \forall n \in \Z_{>0}: \sin n \theta = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop = 0}^{n - 1 } \frac {\cos k \theta} {\cos^k \theta}$
Category:Sine of Integer Multiple of Argument
Sine of Integer Multiple of Argument/Formulation 5

== Theorem ==
For $n \in \Z_{>0}$:
== Proof ==
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$
=== Basis for the Induction ===
$\map P 1$ is the case:
So $\map P 1$ is seen to hold.
$\map P 2$ is the case:
So $\map P 2$ is also seen to hold.
$\map P 3$ is the case:
So $\map P 3$ is also seen to hold.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P n$ is true, where $n > 3$ and $n$ even, then it logically follows that $\map P {n + 2}$ is true.
We also need to show that, if $\map P n$ is true, where $n > 3$ and $n$ odd, then it logically follows that $\map P {n + 2}$ is true.
So this is our induction hypothesis:
:$\ds \map \sin {n \theta} = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$
from which we are to show:
:$\ds \map \sin {\paren {n + 2} \theta} = \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n + 2} \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }$
=== Induction Step ===
This is our induction step:
For $n$ even or for $n$ odd (the argument is identical):
The result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \forall n \in \Z_{>0}: \sin n \theta = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$
== Examples ==
Category:Sine of Integer Multiple of Argument
Sine of Integer Multiple of Argument/Formulation 5/Examples/Sine of Quintuple Angle

== Example of Use of Sine of Integer Multiple of Argument/Formulation 5 ==
:$\sin 5 \theta = \sin \theta + 2 \cos \theta \paren {\sin 4 \theta - \sin 2 \theta }$
== Proof ==
Follows directly from the :
Explicit derivation illustrated below:
Category:Examples of Sine of Integer Multiple of Argument
Category:Quintuple Angle Formula for Sine
Category:Quintuple Angle Formulas
Sine of Integer Multiple of Argument/Formulation 5/Examples/Sine of Sextuple Angle

== Example of Use of Sine of Integer Multiple of Argument/Formulation 5 ==
:$\sin 6 \theta = 2 \cos \theta \paren {\sin 5 \theta - \sin 3 \theta + \sin \theta}$
== Proof ==
Follows directly from the :
Explicit derivation illustrated below:
Category:Examples of Sine of Integer Multiple of Argument
Category:Sextuple Angle Formula for Sine
Category:Sextuple Angle Formulas