hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = \tan \frac \theta 2       | r = \pm \sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} }       | c = Half Angle Formula for Tangent }} {{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta}^2} {\paren {1 + \cos \theta} \paren {1 - \cos \theta} } }       | c = multiplying both numerator and denominator by $\sqrt {1 - \cos \theta}$ }} {{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta}^2} {1 - \cos^2 \theta} }       | c = Difference of Two Squares }} {{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta}^2} {\sin^2 \theta} }       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = \pm \frac {1 - \cos \theta} {\sin \theta} }} {{end-eqn}} When $\sin \theta = 0$, the above is undefined. This happens when $\theta = k \pi$ for $k \in \Z$. When $\theta = \paren {2 k + 1} \pi$, the value of $1 - \cos \theta$ is $2$. Thus at $\theta = \paren {2 k + 1} \pi$, the value of $\tan \dfrac \theta 2$ is undefined. {{qed|lemma}} When $\theta = 2 k \pi$, the value of $\cos \theta = 1$ and so $1 - \cos \theta$ is $0$. Then: {{begin-eqn}} {{eqn | l = \lim_{x \mathop \to 0^+} \frac {1 - \cos \theta} {\sin \theta}       | r = \lim_{x \mathop \to 0^+} \frac {\map {\dfrac \d {\d \theta} } {1 - \cos \theta} } {\dfrac \d {\d \theta} \sin \theta}       | c = L'Hôpital's Rule }} {{eqn | r = \lim_{x \mathop \to 0^+} \frac {\sin \theta} {\cos \theta}       | c =  }} {{eqn | r = \frac 0 1       | c =  }} {{eqn | r = 0       | c =  }} {{end-eqn}} Thus $\tan \dfrac \theta 2$ is defined at $\theta = 2 k \pi$, and equals $0$. {{qed|lemma}} At all other values of $\theta$, $1 - \cos \theta > 0$. Therefore the sign of $\dfrac {1 - \cos \theta} {\sin \theta}$ is equal to the sign of $\sin \theta$. We recall: :In quadrant $\text I$ and quadrant $\text {II}$: $\sin \theta > 0$ :In quadrant $\text {III}$ and quadrant $\text {IV}$: $\sin \theta < 0$ Thus it follows that the same applies to $\dfrac {1 - \cos \theta} {\sin \theta}$. Let $\dfrac \theta 2$ be in quadrant $\text I$ or quadrant $\text {III}$. Then from Bisection of Angle in Cartesian Plane: Corollary, $\theta$ is in quadrant $\text I$ or quadrant $\text {II}$. Therefore $\dfrac {1 - \cos \theta} {\sin \theta}$ is positive. Let $\dfrac \theta 2$ be in quadrant $\text {II}$ or quadrant $\text {IV}$. Then from Bisection of Angle in Cartesian Plane: Corollary, $\theta$ is in quadrant $\text {III}$ or quadrant $\text {IV}$. Therefore $\dfrac {1 - \cos \theta} {\sin \theta}$ is negative. {{qed}}
{{begin-eqn}} {{eqn | l = \tan \frac \theta 2       | r = \frac {1 - \cos \theta} {\sin \theta}       | c = Half Angle Formula for Tangent: Corollary 2 }} {{eqn | r = \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}       | c =  }} {{eqn | r = \csc \theta - \cot \theta       | c = Cosecant is Reciprocal of Sine and Cotangent is Cosine divided by Sine }} {{end-eqn}} When $\theta = k \pi$, both $\cot \theta$ and $\csc \theta$ are undefined. {{qed}}
:500px Let $A$ be a point on the polar axis. By definition of bisection, $\angle AOQ = \dfrac \theta 2$. This is the angular coordinate of $Q$. {{qed|lemma}} Consider the conjugate angle $\map \complement {\angle AOP}$ of $\angle AOP$. By definition of conjugate angle: :$\map \complement {\angle AOP} = -2 \pi - \theta$ where the negative sign arises from the fact that it is measured clockwise. Then the angle $\angle AOR$ is half of $\map \complement {\angle AOP}$: {{begin-eqn}} {{eqn | l = \angle AOR       | r = -\paren {\frac {2 \pi - \theta} 2}       | c =  }} {{eqn | r = -\pi + \frac \theta 2       | c =  }} {{end-eqn}} The angular coordinate of point $R$ is the conjugate angle $\map \complement {\angle AOR}$ of $\angle AOR$: {{begin-eqn}} {{eqn | l = \map \complement {\angle AOR}       | r = 2 \pi - \paren {-\pi + \frac \theta 2}       | c =  }} {{eqn | r = 2 \pi - \pi + \frac \theta 2       | c =  }} {{eqn | r = \pi + \frac \theta 2       | c =  }} {{end-eqn}} {{qed}} Category:Analytic Geometry 3i1zg2688efn9v7ec6z4nnkjzmv5mf7
{{begin-eqn}} {{eqn | l = \sin 15^\circ       | r = \sin \frac {30^\circ} 2       | c =  }} {{eqn | r = \sqrt {\frac {1 - \cos 30^\circ} 2}       | c = Half Angle Formula for Sine: $\theta$ is in Quadrant I }} {{eqn | r = \sqrt {\frac {1 - \frac {\sqrt 3} 2} 2}       | c = Cosine of $30^\circ$ }} {{eqn | r = \sqrt {\frac {2 - \sqrt 3} 4}       | c =  }} {{eqn | r = \sqrt {\frac {8 - 4 \sqrt 3} {16} }       | c =  }} {{eqn | r = \sqrt {\frac {6 + 2 - 2 \sqrt 2 \sqrt 6} {16} }       | c =  }} {{eqn | r = \sqrt {\frac {\left({\sqrt 6 - \sqrt 2}\right)^2} {16} }       | c =  }} {{eqn | r = \frac {\sqrt 6 - \sqrt 2} 4       | c = positive because $\theta$ is in the first quadrant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 15^\circ       | r = \sin \left({45^\circ - 30^\circ}\right)       | c =  }} {{eqn | r = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ       | c = Sine of Difference }} {{eqn | r = \left({\frac {\sqrt 2} 2}\right) \left({\frac {\sqrt 3} 2}\right) - \left({\frac {\sqrt 2} 2}\right) \left({\dfrac 1 2}\right)       | c = Sine of $45^\circ$, Cosine of $30^\circ$, Cosine of $45^\circ$, Sine of $30^\circ$ }} {{eqn | r = \frac {\sqrt 6} 4 - \frac {\sqrt 2} 4       | c =  }} {{eqn | r = \frac {\sqrt 6 - \sqrt 2} 4       | c =  }} {{end-eqn}} {{qed}}
:200px Let $\triangle ABC$ be an equilateral triangle of side $r$. By definition, each angle of $\triangle ABC$ is equal. From Sum of Angles of Triangle equals Two Right Angles it follows that each angle measures $60^\circ$. Let $CD$ be a perpendicular dropped from $C$ to $AB$ at $D$. Then $AD = \dfrac r 2$ while: : $\angle ACD = \dfrac {60 \degrees} 2 = 30 \degrees$ So by definition of sine function: : $\sin \paren {\angle ACD} = \dfrac {r / 2} r = \dfrac 1 2$ {{qed}}
:300px 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: :$\map \sin {\angle CAB} = \dfrac r {r \sqrt 2} = \dfrac {\sqrt 2} 2$ {{qed}}
{{begin-eqn}} {{eqn | l = \sin 45 \degrees       | r = \map \sin {30 \degrees + 15 \degrees}       | c =  }} {{eqn | r = \sin 30 \degrees \cos 15 \degrees + \cos 30 \degrees \sin 15 \degrees       | c = Sine of Sum }} {{eqn | r = \paren {\frac 1 2} \paren {\frac {\sqrt 6 + \sqrt 2} 4} + \paren {\frac {\sqrt 3} 2} \paren {\dfrac {\sqrt 6 - \sqrt 2} 4}       | c = Sine of $30 \degrees$, Cosine of $15 \degrees$, Cosine of $30 \degrees$, Sine of $15 \degrees$ }} {{eqn | r = \frac 1 8 \paren {\sqrt 6 + \sqrt 2 + \sqrt 3 \paren {\sqrt 6 - \sqrt 2} }       | c =  }} {{eqn | r = \frac 1 8 \paren {\sqrt 3 \sqrt 2 + \sqrt 2 + \sqrt 3 \sqrt 3 \sqrt 2 - \sqrt 3 \sqrt 2}       | c =  }} {{eqn | r = \frac 1 8 \paren {\sqrt 2 + 3 \sqrt 2}       | c =  }} {{eqn | r = \frac 1 8 \paren {4 \sqrt 2}       | c =  }} {{eqn | r = \frac {\sqrt 2} 2       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 45 \degrees       | r = \map \sin {3 \times 15 \degrees}       | c =  }} {{eqn | r = 3 \sin 15 \degrees - 4 \sin^3 15 \degrees       | c = Triple Angle Formula for Sine }} {{eqn | r = 3 \paren {\frac {\sqrt 6 - \sqrt 2} 4} - 4 \paren {\frac {\sqrt 6 - \sqrt 2} 4}^3       | c = Sine of $15 \degrees$ }} {{eqn | r = \frac {3 \sqrt 6 - 3 \sqrt 2} 4 - \frac {6 \sqrt 6 - 18 \sqrt 2 + 6 \sqrt 6 - 2 \sqrt 2} {16}       | c =  }} {{eqn | r = \frac {3 \sqrt 6 - 3 \sqrt 2} 4 - \frac {12 \sqrt 6 - 20\sqrt 2} {16}       | c =  }} {{eqn | r = \frac {3 \sqrt 6 - 3 \sqrt 2} 4 - \frac {3 \sqrt 6 - 5 \sqrt 2} 4       | c =  }} {{eqn | r = \frac {2 \sqrt 2} 4       | c =  }} {{eqn | r = \frac {\sqrt 2} 2       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 45 \degrees       | r = \map \sin {60 \degrees - 15 \degrees} }} {{eqn | r = \sin 60 \degrees \cos 15 \degrees - \cos 60 \degrees \sin 15 \degrees       | c = Sine of Difference }} {{eqn | r = \paren {\frac {\sqrt 3} 2} \paren {\frac {\sqrt 6 + \sqrt 2} 4} - \paren {\frac 1 2} \paren {\frac {\sqrt 6 - \sqrt 2} 4}       | c = Sine of $60 \degrees$, Cosine of $15 \degrees$, Cosine of $60 \degrees$, Sine of $15 \degrees$ }} {{eqn | r = \frac 1 8 \paren {\sqrt 3 \paren {\sqrt 6 + \sqrt 2} - \paren {\sqrt 6 - \sqrt 2} }       | c = simplifying }} {{eqn | r = \frac 1 8 \paren {3 \sqrt 2 + \sqrt 6 - \sqrt 6 + \sqrt 2}       | c = simplifying }} {{eqn | r = \frac 1 8 \paren {3 \sqrt 2 + \sqrt 2}       | c = simplifying }} {{eqn | r = \frac 1 8 \paren {4 \sqrt 2}       | c = simplifying }} {{eqn | r = \frac {\sqrt 2} 2       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 90 \degrees       | r = 1       | c = Sine of Right Angle }} {{eqn | ll= \leadsto       | l = \map \sin {2 \times 45 \degrees}       | r = 1       | c =  }} {{eqn | ll= \leadsto       | l = 2 \sin 45 \degrees \cos 45 \degrees       | r = 1       | c = Double Angle Formula for Sine }} {{eqn | ll= \leadsto       | l = 2 \sin 45 \degrees \map \sin {90 \degrees - 45 \degrees}       | r = 1       | c = Sine of Complement equals Cosine }} {{eqn | ll= \leadsto       | l = 2 \sin 45 \degrees \sin 45 \degrees       | r = 1       | c =  }} {{eqn | ll= \leadsto       | l = 2 \sin^2 45 \degrees       | r = 1       | c = by $(1)$ }} {{eqn | ll= \leadsto       | l = \sin 45 \degrees       | r = \pm \frac 1 {\sqrt 2}       | c =  }} {{end-eqn}} The negative solution is rejected because $45 \degrees$ is an acute angle and Sine of Acute Angle is Positive. Therefore: :$\sin 45 \degrees = \dfrac 1 {\sqrt 2} = \dfrac {\sqrt 2} 2$ {{qed}}
{{begin-eqn}} {{eqn | l = \sin 60 \degrees       | r = \map \cos {90 \degrees - 60 \degrees}       | c = Cosine of Complement equals Sine }} {{eqn | r = \cos 30 \degrees       | c =  }} {{eqn | r = \dfrac {\sqrt 3} 2       | c = Cosine of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 75 \degrees       | r = \map \sin {60 \degrees + 15 \degrees}       | c =  }} {{eqn | r = \sin 60 \degrees \cos 15 \degrees + \cos 60 \degrees \sin 15 \degrees       | c = Sine of Sum }} {{eqn | r = \paren {\frac {\sqrt 3} 2} \paren {\frac {\sqrt 6 + \sqrt 2} 4} + \paren {\frac 1 2} \paren {\dfrac {\sqrt 6 - \sqrt 2} 4}       | c = Sine of $60 \degrees$, Cosine of $15 \degrees$, Cosine of $60 \degrees$, Sine of $15 \degrees$ }} {{eqn | r = \frac 1 8 \paren {\sqrt 3 \paren {\sqrt 6 + \sqrt 2} + \sqrt 6 - \sqrt 2}       | c =  }} {{eqn | r = \frac 1 8 \paren {\sqrt 3 \sqrt 3 \sqrt 2 + \sqrt 3 \sqrt 2 + \sqrt 3 \sqrt 2 - \sqrt 2}       | c =  }} {{eqn | r = \frac 1 8 \paren {3 \sqrt 2 + 2 \sqrt 3 \sqrt 2 - \sqrt 2}       | c =  }} {{eqn | r = \frac 1 8 \paren {2 \sqrt 2 + 2 \sqrt 6}       | c =  }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 75 \degrees       | r = \map \cos {90 \degrees - 75 \degrees}       | c = Cosine of Complement equals Sine }} {{eqn | r = \cos 15^\circ       | c =  }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4       | c = Cosine of $15 \degrees$ }} {{end-eqn}} {{qed}}
A direct implementation of Sine of Half-Integer Multiple of Pi: :$\forall n \in \Z: \sin \left({n + \dfrac 1 2}\right) \pi = \left({-1}\right)^n$ In this case, $n = 0$ and so: :$\sin \dfrac 1 2 \pi = \left({-1}\right)^0 = 1$ {{qed}}
{{begin-eqn}} {{eqn | l = \sin \paren {x + \frac \pi 2}       | r = \sin x \cos \frac \pi 2 + \cos x \sin \frac \pi 2       | c = Sine of Sum }} {{eqn | r = \sin x \cdot 0 + \cos x \cdot 1       | c = Cosine of Right Angle and Sine of Right Angle }} {{eqn | r = \cos x       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 105^\circ       | r = \sin \left({90^\circ + 15^\circ}\right)       | c =  }} {{eqn | r = \cos 15^\circ       | c = Sine of Angle plus Right Angle }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4       | c = Cosine of 15 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 120 \degrees       | r = \map \sin {90 \degrees + 30 \degrees}       | c =  }} {{eqn | r = \cos 30 \degrees       | c = Sine of Angle plus Right Angle }} {{eqn | r = \frac {\sqrt 3} 2       | c = Cosine of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 135 \degrees       | r = \map \sin {90 \degrees + 45 \degrees}       | c =  }} {{eqn | r = \cos 45 \degrees       | c = Sine of Angle plus Right Angle }} {{eqn | r = \frac {\sqrt 2} 2       | c = Cosine of $45 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 150 \degrees       | r = \map \sin {90 \degrees + 60 \degrees}       | c =  }} {{eqn | r = \cos 60 \degrees       | c = Sine of Angle plus Right Angle }} {{eqn | r = \frac 1 2       | c = Cosine of $60 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 165 \degrees       | r = \map \sin {90 \degrees + 75 \degrees}       | c =  }} {{eqn | r = \cos 75 \degrees       | c = Sine of Angle plus Right Angle }} {{eqn | r = \frac {\sqrt 6 - \sqrt 2} 4       | c = Cosine of $75 \degrees$ }} {{end-eqn}} {{qed}}
A direct implementation of Sine of Multiple of Pi: :$\forall n \in \Z: \sin n \pi = 0$ In this case, $n = 1$ and so: :$\sin \pi = 0$ {{qed}}
{{begin-eqn}} {{eqn | l = \sin \paren {\pi - \theta}       | r = \sin \pi \cos \theta - \cos \pi \sin \theta       | c = Sine of Difference }} {{eqn | r = 0 \times \cos \theta - \paren {-1} \times \sin \theta       | c = Sine of Straight Angle and Cosine of Straight Angle }} {{eqn | r = \sin \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cos {\pi - \theta}       | r = \cos \pi \cos \theta + \sin \pi \sin \theta       | c = Cosine of Difference }} {{eqn | r = \paren {-1} \times \cos \theta + 0 \times \sin \theta       | c = Cosine of Straight Angle and Sine of Straight Angle }} {{eqn | r = -\cos \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \sin {2 \pi - \theta}       | r = \map \sin {2 \pi} \cos \theta - \map \cos {2 \pi} \sin \theta       | c = Sine of Difference }} {{eqn | r = 0 \times \cos \theta - 1 \times \sin \theta       | c = Sine of Full Angle and Cosine of Full Angle }} {{eqn | r = -\sin \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cos {2 \pi - \theta}       | r = \map \cos {2 \pi} \cos \theta + \map \sin {2 \pi} \sin \theta       | c = Cosine of Difference }} {{eqn | r = 1 \times \cos \theta + 0 \times \sin \theta       | c = Cosine of Full Angle and Sine of Full Angle }} {{eqn | r = \cos \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 195 \degrees       | r = \map \sin {360 \degrees - 165 \degrees}       | c =  }} {{eqn | r = -\sin 165 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\frac {\sqrt 6 - \sqrt 2} 4       | c = Sine of $165 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 210 \degrees       | r = \map \sin {360 \degrees - 150 \degrees}       | c =  }} {{eqn | r = -\sin 150 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\frac 1 2       | c = Sine of $150 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 225 \degrees       | r = \map \sin {360 \degrees - 135 \degrees}       | c =  }} {{eqn | r = -\sin 135 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\frac {\sqrt 2} 2       | c = Sine of $135 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 240 \degrees       | r = \map \sin {360 \degrees - 120 \degrees}       | c =  }} {{eqn | r = -\sin 120 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\frac {\sqrt 3} 2       | c = Sine of $120 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 255^\circ       | r = \sin \left({360^\circ - 105^\circ}\right)       | c =  }} {{eqn | r = - \sin 105^\circ       | c = Sine of Conjugate Angle }} {{eqn | r = - \frac {\sqrt 6 + \sqrt 2} 4       | c = Sine of 105 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 270 \degrees       | r = \map \sin {360 \degrees - 90 \degrees}       | c =  }} {{eqn | r = -\sin 90 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -1       | c = Sine of Right Angle }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 285^\circ       | r = \sin \left({360^\circ - 75^\circ}\right)       | c =  }} {{eqn | r = - \sin 75^\circ       | c = Sine of Conjugate Angle }} {{eqn | r = - \dfrac {\sqrt 6 + \sqrt 2} 4       | c = Sine of $75^\circ$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 300 \degrees       | r = \map \sin {360 \degrees - 60 \degrees}       | c =  }} {{eqn | r = -\sin 60 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\dfrac {\sqrt 3} 2       | c = Sine of $60 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 315 \degrees       | r = \map \sin {360 \degrees - 45 \degrees}       | c =  }} {{eqn | r = -\sin 45 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\dfrac {\sqrt 2} 2       | c = Sine of $45 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 330 \degrees       | r = \map \sin {360 \degrees - 30 \degrees}       | c =  }} {{eqn | r = -\sin 30 \degrees       | c = Sine of Conjugate Angle }} {{eqn | r = -\dfrac 1 2       | c = Sine of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sin 345^\circ       | r = \sin \left({360^\circ - 15^\circ}\right)       | c =  }} {{eqn | r = - \sin 15^\circ       | c = Sine of Conjugate Angle }} {{eqn | r = - \dfrac {\sqrt 6 - \sqrt 2} 4       | c = Sine of 15 Degrees }} {{end-eqn}} {{qed}}
A direct implementation of Sine of Multiple of Pi: :$\forall n \in \Z: \sin n \pi = 0$ In this case, $n = 2$ and so: :$\sin 2 \pi = 0$ {{qed}}
{{begin-eqn}} {{eqn | l = \cos 15^\circ       | r = \cos \frac {30 \degrees} 2       | c =  }} {{eqn | r = \sqrt {\frac {1 + \cos 30 \degrees} 2}       | c = Half Angle Formula for Cosine: $\theta$ is in Quadrant I }} {{eqn | r = \sqrt {\frac {1 + \frac {\sqrt 3} 2} 2}       | c = Cosine of $30 \degrees$ }} {{eqn | r = \sqrt {\frac {2 + \sqrt 3} 4}       | c =  }} {{eqn | r = \sqrt {\frac {8 + 4 \sqrt 3} {16} }       | c =  }} {{eqn | r = \sqrt {\frac {6 + 2 + 2 \sqrt 2 \sqrt 6} {16} }       | c =  }} {{eqn | r = \sqrt {\frac {\paren {\sqrt 6 + \sqrt 2}^2} {16} }       | c =  }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4       | c = positive because $\theta$ is in the first quadrant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 15 \degrees       | r = \map \cos {45 \degrees - 30 \degrees}       | c =  }} {{eqn | r = \cos 45 \degrees \cos 30 \degrees + \sin 45 \degrees \sin 30 \degrees       | c = Cosine of Difference }} {{eqn | r = \paren {\frac {\sqrt 2} 2} \paren {\frac {\sqrt 3} 2} + \paren {\frac {\sqrt 2} 2} \paren {\dfrac 1 2}       | c = Cosine of $45 \degrees$, Cosine of $30 \degrees$, Sine of $45 \degrees$, Sine of $30 \degrees$ }} {{eqn | r = \frac {\sqrt 6} 4 + \frac {\sqrt 2} 4       | c =  }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \paren {\cos 30 \degrees}^2       | r = 1 - \paren {\sin 30 \degrees}^2       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 1 - \paren {\frac 1 2}^2       | c = Sine of $30 \degrees$ }} {{eqn | r = \frac 3 4       | c =  }} {{eqn | ll= \leadsto       | l = \cos 30 \degrees       | r = \sqrt {\frac 3 4}       | c =  }} {{eqn | r = \dfrac {\sqrt 3} 2       | c = positive because $\cos 30 \degrees$ is in Quadrant I }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \paren {\cos 45 \degrees}^2       | r = 1 - \paren {\sin 45 \degrees}^2       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 1 - \paren {\frac {\sqrt 2} 2}^2       | c = Sine of $45 \degrees$ }} {{eqn | r = \frac 1 2       | c =  }} {{eqn | ll= \leadsto       | l = \cos 45 \degrees       | r = \sqrt {\frac 1 2}       | c = positive because $\cos 45 \degrees$ is in Quadrant I }} {{eqn | r = \dfrac {\sqrt 2} 2       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 60 \degrees       | r = \map \cos {90 \degrees - 30 \degrees}       | c =  }} {{eqn | r = \sin 30 \degrees       | c = Cosine of Complement equals Sine }} {{eqn | r = \dfrac 1 2       | c = Sine of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 75^\circ       | r = \cos \left({90^\circ - 15^\circ}\right)       | c =  }} {{eqn | r = \sin 15^\circ       | c = Cosine of Complement equals Sine }} {{eqn | r = \dfrac {\sqrt 6 - \sqrt 2} 4       | c = Sine of $15^\circ$ }} {{end-eqn}} {{qed}}
A direct implementation of Cosine of Half-Integer Multiple of Pi: :$\forall n \in \Z: \cos \left({n + \dfrac 1 2}\right) \pi = 0$ In this case, $n = 0$ and so: :$\cos \dfrac 1 2 \pi = 0$ {{qed}}
{{begin-eqn}} {{eqn | l = \cos 105 \degrees       | r = \map \cos {90 \degrees + 15 \degrees}       | c =  }} {{eqn | r = -\sin 15 \degrees       | c = Cosine of Angle plus Right Angle }} {{eqn | r = -\frac {\sqrt 6 - \sqrt 2} 4       | c = Sine of $15 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cos {x + \frac \pi 2}       | r = \cos x \cos \frac \pi 2 - \sin x \sin \frac \pi 2       | c = Cosine of Sum }} {{eqn | r = \cos x \cdot 0 - \sin x \cdot 1       | c = Cosine of Right Angle and Sine of Right Angle }} {{eqn | r = -\sin x       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 120 \degrees       | r = \map \cos {90 \degrees + 30 \degrees}       | c =  }} {{eqn | r = -\sin 30 \degrees       | c = Cosine of Angle plus Right Angle }} {{eqn | r = -\frac 1 2       | c = Sine of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 135 \degrees       | r = \map \cos {90 \degrees + 45 \degrees}       | c =  }} {{eqn | r = -\sin 45 \degrees       | c = Cosine of Angle plus Right Angle }} {{eqn | r = -\frac {\sqrt 2} 2       | c = Sine of $45 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 150 \degrees       | r = \map \cos {90 \degrees + 60 \degrees}       | c =  }} {{eqn | r = -\sin 60 \degrees       | c = Cosine of Angle plus Right Angle }} {{eqn | r = -\frac {\sqrt 3} 2       | c = Sine of $60 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 165 \degrees       | r = \map \cos {90 \degrees + 75 \degrees}       | c =  }} {{eqn | r = -\sin 75 \degrees       | c = Cosine of Angle plus Right Angle }} {{eqn | r = -\dfrac {\sqrt 6 + \sqrt 2} 4       | c = Sine of $75 \degrees$ }} {{end-eqn}} {{qed}}
A direct implementation of Cosine of Multiple of Pi: :$\forall n \in \Z: \cos n \pi = \paren {-1}^n$ In this case, $n = 1$ and so: :$\cos \pi = -1^1 = -1$ {{qed}}
{{begin-eqn}} {{eqn | l = \cos 195 \degrees       | r = \cos \paren {360 \degrees - 165 \degrees}       | c =  }} {{eqn | r = \cos 165 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = - \frac {\sqrt 6 + \sqrt 2} 4       | c = Cosine of $165 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 210 \degrees       | r = \map \cos {360 \degrees - 150 \degrees}       | c =  }} {{eqn | r = \cos 150 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = -\frac {\sqrt 3} 2       | c = Cosine of $150 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 225 \degrees       | r = \map \cos {360 \degrees - 135 \degrees}       | c =  }} {{eqn | r = \cos 135 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = -\frac {\sqrt 2} 2       | c = Cosine of $135 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 240 \degrees       | r = \map \cos {360 \degrees - 120 \degrees}       | c =  }} {{eqn | r = \cos 120 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = -\frac 1 2       | c = Cosine of $120 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 255^\circ       | r = \cos \left({360^\circ - 105^\circ}\right)       | c =  }} {{eqn | r = \cos 105^\circ       | c = Cosine of Conjugate Angle }} {{eqn | r = - \frac {\sqrt 6 - \sqrt 2} 4       | c = Cosine of 105 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 270^\circ       | r = \cos \left({360^\circ - 90^\circ}\right)       | c =  }} {{eqn | r = \cos 90^\circ       | c = Cosine of Conjugate Angle }} {{eqn | r = 0       | c = Cosine of Right Angle }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 285^\circ       | r = \cos \left({360^\circ - 75^\circ}\right)       | c =  }} {{eqn | r = \cos 75^\circ       | c = Cosine of Conjugate Angle }} {{eqn | r = \frac {\sqrt 6 - \sqrt 2} 4       | c = Cosine of 75 Degrees }} {{end-eqn}} {{qed}}