id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-7900 | Bisection of Angle in Cartesian Plane | Let $\theta$ be the angular coordinate of a point $P$ in a polar coordinate plane.
Let $QOR$ be a straight line that bisects the angle $\theta$.
Then the angular coordinates of $Q$ and $R$ are $\dfrac \theta 2$ and $\pi + \dfrac \theta 2$. | :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:
:$\map \complement {\angle AOP} = -2 \pi - \theta$... | Let $\theta$ be the [[Definition:Angular Coordinate|angular coordinate]] of a [[Definition:Point|point]] $P$ in a [[Definition:Polar Coordinate Plane|polar coordinate plane]].
Let $QOR$ be a [[Definition:Straight Line|straight line]] that [[Definition:Bisection|bisects]] the [[Definition:Angle|angle]] $\theta$.
Then... | :[[File:BisectedAngle.png|500px]]
Let $A$ be a [[Definition:Point|point]] on the [[Definition:Polar Axis (Polar Coordinates)|polar axis]].
By definition of [[Definition:Bisection|bisection]], $\angle AOQ = \dfrac \theta 2$.
This is the [[Definition:Angular Coordinate|angular coordinate]] of $Q$.
{{qed|lemma}}
Con... | Bisection of Angle in Cartesian Plane | https://proofwiki.org/wiki/Bisection_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Bisection_of_Angle_in_Cartesian_Plane | [
"Analytic Geometry"
] | [
"Definition:Polar Coordinates/Angular Coordinate",
"Definition:Point",
"Definition:Polar Coordinates/Polar Plane",
"Definition:Line/Straight Line",
"Definition:Bisection",
"Definition:Angle",
"Definition:Polar Coordinates/Angular Coordinate"
] | [
"File:BisectedAngle.png",
"Definition:Point",
"Definition:Polar Coordinates/Polar Axis",
"Definition:Bisection",
"Definition:Polar Coordinates/Angular Coordinate",
"Definition:Conjugate Angles",
"Definition:Conjugate Angles",
"Definition:Clockwise",
"Definition:Angle",
"Definition:Polar Coordinate... |
proofwiki-7901 | Sine of 15 Degrees | :$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \sin 15 \degrees
| r = \sin \frac {30 \degrees} 2
| c =
}}
{{eqn | r = \sqrt {\frac {1 - \cos 30 \degrees} 2}
| c = Half Angle Formula for Sine: $\theta$ is in Quadrant I
}}
{{eqn | r = \sqrt {\frac {1 - \frac {\sqrt 3} 2} 2}
| c = {{cos|30}}
}}
{{eqn | r = \sqrt {\fra... | :$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \sin 15 \degrees
| r = \sin \frac {30 \degrees} 2
| c =
}}
{{eqn | r = \sqrt {\frac {1 - \cos 30 \degrees} 2}
| c = [[Half Angle Formula for Sine]]: $\theta$ is in [[Definition:Sine/Definition from Circle/First Quadrant|Quadrant I]]
}}
{{eqn | r = \sqrt {\frac {1 - \frac {\s... | Sine of 15 Degrees/Proof 1 | https://proofwiki.org/wiki/Sine_of_15_Degrees | https://proofwiki.org/wiki/Sine_of_15_Degrees/Proof_1 | [
"Sine Function",
"Sine of 15 Degrees"
] | [] | [
"Half Angle Formulas/Sine",
"Definition:Sine/Definition from Circle/First Quadrant",
"Definition:Sine/Definition from Circle/First Quadrant"
] |
proofwiki-7902 | Sine of 15 Degrees | :$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \sin 15 \degrees
| r = \map \sin {45 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \sin 45 \degrees \cos 30 \degrees - \cos 45 \degrees \sin 30 \degrees
| c = Sine of Difference
}}
{{eqn | r = \paren {\frac {\sqrt 2} 2} \paren {\frac {\sqrt 3} 2} - \paren {\frac {\sqrt 2} 2} ... | :$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \sin 15 \degrees
| r = \map \sin {45 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \sin 45 \degrees \cos 30 \degrees - \cos 45 \degrees \sin 30 \degrees
| c = [[Sine of Difference]]
}}
{{eqn | r = \paren {\frac {\sqrt 2} 2} \paren {\frac {\sqrt 3} 2} - \paren {\frac {\sqrt 2}... | Sine of 15 Degrees/Proof 2 | https://proofwiki.org/wiki/Sine_of_15_Degrees | https://proofwiki.org/wiki/Sine_of_15_Degrees/Proof_2 | [
"Sine Function",
"Sine of 15 Degrees"
] | [] | [
"Sine of Difference",
"Sine of 45 Degrees",
"Cosine of 30 Degrees",
"Cosine of 45 Degrees",
"Sine of 30 Degrees"
] |
proofwiki-7903 | Sine of 30 Degrees | :$\sin 30 \degrees = \sin \dfrac \pi 6 = \dfrac 1 2$ | :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:
:$\a... | :$\sin 30 \degrees = \sin \dfrac \pi 6 = \dfrac 1 2$ | :[[File:Sine30.png|200px]]
Let $\triangle ABC$ be an [[Definition:Equilateral Triangle|equilateral triangle]] of [[Definition:Side of Polygon|side]] $r$.
By definition, each [[Definition:Angle|angle]] of $\triangle ABC$ is equal.
From [[Sum of Angles of Triangle equals Two Right Angles]] it follows that each [[Defin... | Sine of 30 Degrees | https://proofwiki.org/wiki/Sine_of_30_Degrees | https://proofwiki.org/wiki/Sine_of_30_Degrees | [
"Sine Function"
] | [] | [
"File:Sine30.png",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Side",
"Definition:Angle",
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Angle",
"Definition:Right Angle/Perpendicular",
"Definition:Sine"
] |
proofwiki-7904 | Sine of 45 Degrees | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | :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 $\angle... | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | :[[File:Sine45.png|300px]]
Let $ABCD$ be a [[Definition:Square (Geometry)|square]] of [[Definition:Side of Polygon|side]] $r$.
By definition, each [[Definition:Angle|angle]] of $\triangle ABCD$ is equal to $90 \degrees$.
Let $AC$ be a [[Definition:Diagonal of Quadrilateral|diagonal]] of $ABCD$.
As $\triangle ABC$ i... | Sine of 45 Degrees/Proof 1 | https://proofwiki.org/wiki/Sine_of_45_Degrees | https://proofwiki.org/wiki/Sine_of_45_Degrees/Proof_1 | [
"Sine of 45 Degrees",
"Sine Function"
] | [] | [
"File:Sine45.png",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Diameter of Quadrilateral",
"Definition:Triangle (Geometry)/Right-Angled",
"Pythagoras's Theorem",
"Definition:Bisection",
"Definition:Sine"
] |
proofwiki-7905 | Sine of 45 Degrees | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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} \pare... | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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} \... | Sine of 45 Degrees/Proof 2 | https://proofwiki.org/wiki/Sine_of_45_Degrees | https://proofwiki.org/wiki/Sine_of_45_Degrees/Proof_2 | [
"Sine of 45 Degrees",
"Sine Function"
] | [] | [
"Sine of Sum"
] |
proofwiki-7906 | Sine of 45 Degrees | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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 = {{sin|15}}
... | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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 = {{sin|1... | Sine of 45 Degrees/Proof 3 | https://proofwiki.org/wiki/Sine_of_45_Degrees | https://proofwiki.org/wiki/Sine_of_45_Degrees/Proof_3 | [
"Sine of 45 Degrees",
"Sine Function"
] | [] | [
"Triple Angle Formulas/Sine"
] |
proofwiki-7907 | Sine of 45 Degrees | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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 {\fr... | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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 ... | Sine of 45 Degrees/Proof 4 | https://proofwiki.org/wiki/Sine_of_45_Degrees | https://proofwiki.org/wiki/Sine_of_45_Degrees/Proof_4 | [
"Sine of 45 Degrees",
"Sine Function"
] | [] | [
"Sine of Difference",
"Sine of 60 Degrees",
"Cosine of 15 Degrees",
"Cosine of 60 Degrees",
"Sine of 15 Degrees"
] |
proofwiki-7908 | Sine of 45 Degrees | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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
}}
{{eq... | :$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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]]... | Sine of 45 Degrees/Proof 5 | https://proofwiki.org/wiki/Sine_of_45_Degrees | https://proofwiki.org/wiki/Sine_of_45_Degrees/Proof_5 | [
"Sine of 45 Degrees",
"Sine Function"
] | [] | [
"Sine of Right Angle",
"Double Angle Formulas/Sine",
"Sine of Complement equals Cosine",
"Definition:Acute Angle",
"Definition:Sine/Definition from Circle/First Quadrant"
] |
proofwiki-7909 | Sine of 60 Degrees | :$\sin 60 \degrees = \sin \dfrac \pi 3 = \dfrac {\sqrt 3} 2$ | {{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}} | :$\sin 60 \degrees = \sin \dfrac \pi 3 = \dfrac {\sqrt 3} 2$ | {{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|Cosine of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 60 Degrees | https://proofwiki.org/wiki/Sine_of_60_Degrees | https://proofwiki.org/wiki/Sine_of_60_Degrees | [
"Sine Function"
] | [] | [
"Cosine of Complement equals Sine",
"Cosine of 30 Degrees"
] |
proofwiki-7910 | Sine of 75 Degrees | :$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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} \pare... | :$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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} \... | Sine of 75 Degrees/Proof 1 | https://proofwiki.org/wiki/Sine_of_75_Degrees | https://proofwiki.org/wiki/Sine_of_75_Degrees/Proof_1 | [
"Sine Function",
"Sine of 75 Degrees"
] | [] | [
"Sine of Sum",
"Sine of 60 Degrees",
"Cosine of 15 Degrees",
"Cosine of 60 Degrees",
"Sine of 15 Degrees"
] |
proofwiki-7911 | Sine of 75 Degrees | :$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | :$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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|Cosine of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 75 Degrees/Proof 2 | https://proofwiki.org/wiki/Sine_of_75_Degrees | https://proofwiki.org/wiki/Sine_of_75_Degrees/Proof_2 | [
"Sine Function",
"Sine of 75 Degrees"
] | [] | [
"Cosine of Complement equals Sine",
"Cosine of 15 Degrees"
] |
proofwiki-7912 | Sine of Right Angle | :$\sin 90 \degrees = \sin \dfrac \pi 2 = 1$ | A direct implementation of Sine of Half-Integer Multiple of Pi:
:$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$
In this case, $n = 0$ and so:
:$\sin \dfrac 1 2 \pi = \paren {-1}^0 = 1$
{{qed}} | :$\sin 90 \degrees = \sin \dfrac \pi 2 = 1$ | A direct implementation of [[Sine of Half-Integer Multiple of Pi]]:
:$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$
In this case, $n = 0$ and so:
:$\sin \dfrac 1 2 \pi = \paren {-1}^0 = 1$
{{qed}} | Sine of Right Angle | https://proofwiki.org/wiki/Sine_of_Right_Angle | https://proofwiki.org/wiki/Sine_of_Right_Angle | [
"Sine Function"
] | [] | [
"Sine of Half-Integer Multiple of Pi"
] |
proofwiki-7913 | Sine of Angle plus Right Angle | : $\sin \paren {x + \dfrac \pi 2} = \cos x$ | {{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}} | : $\sin \paren {x + \dfrac \pi 2} = \cos x$ | {{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}} | Sine of Angle plus Right Angle | https://proofwiki.org/wiki/Sine_of_Angle_plus_Right_Angle | https://proofwiki.org/wiki/Sine_of_Angle_plus_Right_Angle | [
"Sine Function",
"Reduction Formulae (Trigonometry)"
] | [] | [
"Sine of Sum",
"Cosine of Right Angle",
"Sine of Right Angle"
] |
proofwiki-7914 | Sine of 105 Degrees | :$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | :$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | Sine of 105 Degrees | https://proofwiki.org/wiki/Sine_of_105_Degrees | https://proofwiki.org/wiki/Sine_of_105_Degrees | [
"Sine Function"
] | [] | [
"Sine of Angle plus Right Angle",
"Cosine of 15 Degrees"
] |
proofwiki-7915 | Sine of 120 Degrees | :$\sin 120 \degrees = \sin \dfrac {2 \pi} 3 = \dfrac {\sqrt 3} 2$ | {{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}} | :$\sin 120 \degrees = \sin \dfrac {2 \pi} 3 = \dfrac {\sqrt 3} 2$ | {{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|Cosine of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 120 Degrees | https://proofwiki.org/wiki/Sine_of_120_Degrees | https://proofwiki.org/wiki/Sine_of_120_Degrees | [
"Sine Function"
] | [] | [
"Sine of Angle plus Right Angle",
"Cosine of 30 Degrees"
] |
proofwiki-7916 | Sine of 135 Degrees | :$\sin 135 \degrees = \sin \dfrac {3 \pi} 4 = \dfrac {\sqrt 2} 2$ | {{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}} | :$\sin 135 \degrees = \sin \dfrac {3 \pi} 4 = \dfrac {\sqrt 2} 2$ | {{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|Cosine of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 135 Degrees | https://proofwiki.org/wiki/Sine_of_135_Degrees | https://proofwiki.org/wiki/Sine_of_135_Degrees | [
"Sine Function"
] | [] | [
"Sine of Angle plus Right Angle",
"Cosine of 45 Degrees"
] |
proofwiki-7917 | Sine of 150 Degrees | :$\sin 150 \degrees = \sin \dfrac {5 \pi} 6 = \dfrac 1 2$ | {{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}} | :$\sin 150 \degrees = \sin \dfrac {5 \pi} 6 = \dfrac 1 2$ | {{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|Cosine of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 150 Degrees | https://proofwiki.org/wiki/Sine_of_150_Degrees | https://proofwiki.org/wiki/Sine_of_150_Degrees | [
"Sine Function"
] | [] | [
"Sine of Angle plus Right Angle",
"Cosine of 60 Degrees"
] |
proofwiki-7918 | Sine of 165 Degrees | :$\sin 165 \degrees = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | :$\sin 165 \degrees = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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|Cosine of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 165 Degrees | https://proofwiki.org/wiki/Sine_of_165_Degrees | https://proofwiki.org/wiki/Sine_of_165_Degrees | [
"Sine Function"
] | [] | [
"Sine of Angle plus Right Angle",
"Cosine of 75 Degrees"
] |
proofwiki-7919 | Sine of Straight Angle | :$\sin 180 \degrees = \sin \pi = 0$ | 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}} | :$\sin 180 \degrees = \sin \pi = 0$ | 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}} | Sine of Straight Angle | https://proofwiki.org/wiki/Sine_of_Straight_Angle | https://proofwiki.org/wiki/Sine_of_Straight_Angle | [
"Sine Function"
] | [] | [
"Sine of Integer Multiple of Pi"
] |
proofwiki-7920 | Sine of Supplementary Angle | :$\map \sin {\pi - \theta} = \sin \theta$
where $\sin$ denotes sine.
That is, the sine of an angle equals its supplement. | {{begin-eqn}}
{{eqn | l = \map \sin {\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}} | :$\map \sin {\pi - \theta} = \sin \theta$
where $\sin$ denotes [[Definition:Sine|sine]].
That is, the [[Definition:Sine|sine]] of an [[Definition:Angle|angle]] equals its [[Definition:Supplement of Angle|supplement]]. | {{begin-eqn}}
{{eqn | l = \map \sin {\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-e... | Sine of Supplementary Angle | https://proofwiki.org/wiki/Sine_of_Supplementary_Angle | https://proofwiki.org/wiki/Sine_of_Supplementary_Angle | [
"Sine Function",
"Supplementary Angles"
] | [
"Definition:Sine",
"Definition:Sine",
"Definition:Angle",
"Definition:Supplementary Angles"
] | [
"Sine of Difference",
"Sine of Straight Angle",
"Cosine of Straight Angle"
] |
proofwiki-7921 | Cosine of Supplementary Angle | :$\map \cos {\pi - \theta} = -\cos \theta$
where $\cos$ denotes cosine.
That is, the cosine of an angle is the negative of its supplement. | {{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}}
{{qe... | :$\map \cos {\pi - \theta} = -\cos \theta$
where $\cos$ denotes [[Definition:Cosine|cosine]].
That is, the [[Definition:Cosine|cosine]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Supplement of Angle|supplement]]. | {{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
}}
{{en... | Cosine of Supplementary Angle | https://proofwiki.org/wiki/Cosine_of_Supplementary_Angle | https://proofwiki.org/wiki/Cosine_of_Supplementary_Angle | [
"Cosine of Supplementary Angle",
"Cosine Function",
"Supplementary Angles"
] | [
"Definition:Cosine",
"Definition:Cosine",
"Definition:Angle",
"Definition:Supplementary Angles"
] | [
"Cosine of Difference",
"Cosine of Straight Angle",
"Sine of Straight Angle"
] |
proofwiki-7922 | Sine of Conjugate Angle | :$\map \sin {2 \pi - \theta} = -\sin \theta$
where $\sin$ denotes sine.
That is, the sine of an angle is the negative of its conjugate. | {{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}}
{{qe... | :$\map \sin {2 \pi - \theta} = -\sin \theta$
where $\sin$ denotes [[Definition:Sine|sine]].
That is, the [[Definition:Sine|sine]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Conjugate Angle|conjugate]]. | {{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
}}
{{en... | Sine of Conjugate Angle | https://proofwiki.org/wiki/Sine_of_Conjugate_Angle | https://proofwiki.org/wiki/Sine_of_Conjugate_Angle | [
"Sine Function",
"Conjugate Angles"
] | [
"Definition:Sine",
"Definition:Sine",
"Definition:Angle",
"Definition:Conjugate Angles"
] | [
"Sine of Difference",
"Sine of Full Angle",
"Cosine of Full Angle"
] |
proofwiki-7923 | Cosine of Conjugate Angle | :$\map \cos {2 \pi - \theta} = \cos \theta$
where $\cos$ denotes cosine.
That is, the cosine of an angle equals its conjugate. | {{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}}
{{q... | :$\map \cos {2 \pi - \theta} = \cos \theta$
where $\cos$ denotes [[Definition:Cosine|cosine]].
That is, the [[Definition:Cosine|cosine]] of an [[Definition:Angle|angle]] equals its [[Definition:Conjugate Angle|conjugate]]. | {{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
}}
{{e... | Cosine of Conjugate Angle | https://proofwiki.org/wiki/Cosine_of_Conjugate_Angle | https://proofwiki.org/wiki/Cosine_of_Conjugate_Angle | [
"Cosine Function",
"Conjugate Angles"
] | [
"Definition:Cosine",
"Definition:Cosine",
"Definition:Angle",
"Definition:Conjugate Angles"
] | [
"Cosine of Difference",
"Cosine of Full Angle",
"Sine of Full Angle"
] |
proofwiki-7924 | Sine of 195 Degrees | :$\sin 195 \degrees = \sin \dfrac {13 \pi} {12} = -\dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | :$\sin 195 \degrees = \sin \dfrac {13 \pi} {12} = -\dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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|Sine of $165 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 195 Degrees | https://proofwiki.org/wiki/Sine_of_195_Degrees | https://proofwiki.org/wiki/Sine_of_195_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 165 Degrees"
] |
proofwiki-7925 | Sine of 210 Degrees | :$\sin 210 \degrees = \sin \dfrac {7 \pi} 6 = -\dfrac 1 2$ | {{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}} | :$\sin 210 \degrees = \sin \dfrac {7 \pi} 6 = -\dfrac 1 2$ | {{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|Sine of $150 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 210 Degrees | https://proofwiki.org/wiki/Sine_of_210_Degrees | https://proofwiki.org/wiki/Sine_of_210_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 150 Degrees"
] |
proofwiki-7926 | Sine of 225 Degrees | :$\sin 225 \degrees = \sin \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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}} | :$\sin 225 \degrees = \sin \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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|Sine of $135 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 225 Degrees | https://proofwiki.org/wiki/Sine_of_225_Degrees | https://proofwiki.org/wiki/Sine_of_225_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 135 Degrees"
] |
proofwiki-7927 | Sine of 240 Degrees | :$\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$ | {{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}} | :$\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$ | {{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|Sine of $120 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 240 Degrees/Proof 1 | https://proofwiki.org/wiki/Sine_of_240_Degrees | https://proofwiki.org/wiki/Sine_of_240_Degrees/Proof_1 | [
"Sine of 240 Degrees",
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 120 Degrees"
] |
proofwiki-7928 | Sine of 240 Degrees | :$\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$ | When $240 \degrees$ is embedded in a Cartesian plane, it makes an angle of $60 \degrees$ with the $x$-axis.
$240 \degrees$ can be found in the third quadrant.
Hence by definition of sine function in the third quadrant, $\sin 240 \degrees$ is negative.
Thus:
:$\sin 240 \degrees = -\sin 60 \degrees = -\dfrac {\sqrt 3} 2$... | :$\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$ | When $240 \degrees$ is embedded in a [[Definition:Cartesian Plane|Cartesian plane]], it makes an [[Definition:Angle|angle]] of $60 \degrees$ with the [[Definition:X-Axis|$x$-axis]].
$240 \degrees$ can be found in the [[Definition:Third Quadrant|third quadrant]].
Hence by definition of [[Definition:Sine/Definition fro... | Sine of 240 Degrees/Proof 2 | https://proofwiki.org/wiki/Sine_of_240_Degrees | https://proofwiki.org/wiki/Sine_of_240_Degrees/Proof_2 | [
"Sine of 240 Degrees",
"Sine Function"
] | [] | [
"Definition:Cartesian Plane",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane/Quadrants/Third",
"Definition:Sine/Definition from Circle/Third Quadrant",
"Definition:Negative Real Function"
] |
proofwiki-7929 | Sine of 255 Degrees | :$\sin 255^\circ = \sin \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | :$\sin 255^\circ = \sin \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | Sine of 255 Degrees | https://proofwiki.org/wiki/Sine_of_255_Degrees | https://proofwiki.org/wiki/Sine_of_255_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 105 Degrees"
] |
proofwiki-7930 | Sine of Three Right Angles | :$\sin 270 \degrees = \sin \dfrac {3 \pi} 2 = -1$ | {{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}} | :$\sin 270 \degrees = \sin \dfrac {3 \pi} 2 = -1$ | {{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}} | Sine of Three Right Angles | https://proofwiki.org/wiki/Sine_of_Three_Right_Angles | https://proofwiki.org/wiki/Sine_of_Three_Right_Angles | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of Right Angle"
] |
proofwiki-7931 | Sine of 285 Degrees | :$\sin 285^\circ = \sin \dfrac {19 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | :$\sin 285^\circ = \sin \dfrac {19 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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 Degrees|Sine of $75^\circ$]]
}}
{{end-eqn}}
{{qed}} | Sine of 285 Degrees | https://proofwiki.org/wiki/Sine_of_285_Degrees | https://proofwiki.org/wiki/Sine_of_285_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 75 Degrees"
] |
proofwiki-7932 | Sine of 300 Degrees | :$\sin 300 \degrees = \sin \dfrac {5 \pi} 3 = -\dfrac {\sqrt 3} 2$ | {{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}} | :$\sin 300 \degrees = \sin \dfrac {5 \pi} 3 = -\dfrac {\sqrt 3} 2$ | {{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|Sine of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 300 Degrees | https://proofwiki.org/wiki/Sine_of_300_Degrees | https://proofwiki.org/wiki/Sine_of_300_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 60 Degrees"
] |
proofwiki-7933 | Sine of 315 Degrees | :$\sin 315 \degrees = \sin \dfrac {7 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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}} | :$\sin 315 \degrees = \sin \dfrac {7 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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|Sine of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 315 Degrees | https://proofwiki.org/wiki/Sine_of_315_Degrees | https://proofwiki.org/wiki/Sine_of_315_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 45 Degrees"
] |
proofwiki-7934 | Sine of 330 Degrees | :$\sin 330 \degrees = \sin \dfrac {11 \pi} 6 = -\dfrac 1 2$ | {{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}} | :$\sin 330 \degrees = \sin \dfrac {11 \pi} 6 = -\dfrac 1 2$ | {{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|Sine of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Sine of 330 Degrees | https://proofwiki.org/wiki/Sine_of_330_Degrees | https://proofwiki.org/wiki/Sine_of_330_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle",
"Sine of 30 Degrees"
] |
proofwiki-7935 | Sine of 345 Degrees | :$\sin 345 \degrees = \sin \dfrac {23 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \sin 345 \degrees
| r = \map \sin {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = -\sin 15 \degrees
| c = Sine of Conjugate Angle
}}
{{eqn | r = -\dfrac {\sqrt 6 - \sqrt 2} 4
| c = {{sin|15}}
}}
{{end-eqn}}
{{qed}} | :$\sin 345 \degrees = \sin \dfrac {23 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \sin 345 \degrees
| r = \map \sin {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = -\sin 15 \degrees
| c = [[Sine of Conjugate Angle]]
}}
{{eqn | r = -\dfrac {\sqrt 6 - \sqrt 2} 4
| c = {{sin|15}}
}}
{{end-eqn}}
{{qed}} | Sine of 345 Degrees | https://proofwiki.org/wiki/Sine_of_345_Degrees | https://proofwiki.org/wiki/Sine_of_345_Degrees | [
"Sine Function"
] | [] | [
"Sine of Conjugate Angle"
] |
proofwiki-7936 | Sine of Full Angle | :$\sin 360^\circ = \sin 2 \pi = 0$ | 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}} | :$\sin 360^\circ = \sin 2 \pi = 0$ | 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}} | Sine of Full Angle | https://proofwiki.org/wiki/Sine_of_Full_Angle | https://proofwiki.org/wiki/Sine_of_Full_Angle | [
"Sine Function"
] | [] | [
"Sine of Integer Multiple of Pi"
] |
proofwiki-7937 | Cosine of 15 Degrees | :$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \cos 15 \degrees
| 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 | ... | :$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \cos 15 \degrees
| 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 [[Definition:Cosine/Definition from Circle/First Quadrant|Quadrant I]]
}}
{{eqn | r = \sqrt {\frac {1 + \frac... | Cosine of 15 Degrees/Proof 1 | https://proofwiki.org/wiki/Cosine_of_15_Degrees | https://proofwiki.org/wiki/Cosine_of_15_Degrees/Proof_1 | [
"Cosine Function",
"Cosine of 15 Degrees"
] | [] | [
"Half Angle Formulas/Cosine",
"Definition:Cosine/Definition from Circle/First Quadrant",
"Cosine of 30 Degrees",
"Definition:Cosine/Definition from Circle/First Quadrant"
] |
proofwiki-7938 | Cosine of 15 Degrees | :$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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... | :$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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 ... | Cosine of 15 Degrees/Proof 2 | https://proofwiki.org/wiki/Cosine_of_15_Degrees | https://proofwiki.org/wiki/Cosine_of_15_Degrees/Proof_2 | [
"Cosine Function",
"Cosine of 15 Degrees"
] | [] | [
"Cosine of Difference",
"Cosine of 45 Degrees",
"Cosine of 30 Degrees",
"Sine of 45 Degrees",
"Sine of 30 Degrees"
] |
proofwiki-7939 | Cosine of 30 Degrees | :$\cos 30 \degrees = \cos \dfrac \pi 6 = \dfrac {\sqrt 3} 2$ | {{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 = {{sin|30}}
}}
{{eqn | r = \frac 3 4
| c =
}}
{{eqn | ll= \leadsto
| l = \cos 30 \degrees
| r = \sqrt {\fr... | :$\cos 30 \degrees = \cos \dfrac \pi 6 = \dfrac {\sqrt 3} 2$ | {{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 = {{sin|30}}
}}
{{eqn | r = \frac 3 4
| c =
}}
{{eqn | ll= \leadsto
| l = \cos 30 \degrees
| r = \sqrt ... | Cosine of 30 Degrees | https://proofwiki.org/wiki/Cosine_of_30_Degrees | https://proofwiki.org/wiki/Cosine_of_30_Degrees | [
"Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Definition:Positive/Real Number",
"Definition:Cosine/Definition from Circle/First Quadrant"
] |
proofwiki-7940 | Cosine of 45 Degrees | :$\cos 45 \degrees = \cos \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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
... | :$\cos 45 \degrees = \cos \dfrac \pi 4 = \dfrac {\sqrt 2} 2$ | {{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|Sine of $45 \degrees$]]
}}
{{eqn | r = \frac 1 2
| c =
}}
{{eqn | ll= \leadsto
... | Cosine of 45 Degrees | https://proofwiki.org/wiki/Cosine_of_45_Degrees | https://proofwiki.org/wiki/Cosine_of_45_Degrees | [
"Cosine of 45 Degrees",
"Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Sine of 45 Degrees",
"Definition:Positive/Real Number",
"Definition:Cosine/Definition from Circle/First Quadrant"
] |
proofwiki-7941 | Cosine of 60 Degrees | :$\cos 60 \degrees = \cos \dfrac \pi 3 = \dfrac 1 2$ | {{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}} | :$\cos 60 \degrees = \cos \dfrac \pi 3 = \dfrac 1 2$ | {{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|Sine of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 60 Degrees | https://proofwiki.org/wiki/Cosine_of_60_Degrees | https://proofwiki.org/wiki/Cosine_of_60_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Complement equals Sine",
"Sine of 30 Degrees"
] |
proofwiki-7942 | Cosine of 75 Degrees | :$\cos 75^\circ = \cos \dfrac {5 \pi}{12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | :$\cos 75^\circ = \cos \dfrac {5 \pi}{12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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 Degrees|Sine of $15^\circ$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 75 Degrees | https://proofwiki.org/wiki/Cosine_of_75_Degrees | https://proofwiki.org/wiki/Cosine_of_75_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Complement equals Sine",
"Sine of 15 Degrees"
] |
proofwiki-7943 | Cosine of Right Angle | :$\cos 90 \degrees = \cos \dfrac \pi 2 = 0$ | A direct implementation of Cosine of Half-Integer Multiple of Pi:
:$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$
In this case, $n = 0$ and so:
:$\cos \dfrac 1 2 \pi = 0$
{{qed}} | :$\cos 90 \degrees = \cos \dfrac \pi 2 = 0$ | A direct implementation of [[Cosine of Half-Integer Multiple of Pi]]:
:$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$
In this case, $n = 0$ and so:
:$\cos \dfrac 1 2 \pi = 0$
{{qed}} | Cosine of Right Angle | https://proofwiki.org/wiki/Cosine_of_Right_Angle | https://proofwiki.org/wiki/Cosine_of_Right_Angle | [
"Cosine Function"
] | [] | [
"Cosine of Half-Integer Multiple of Pi"
] |
proofwiki-7944 | Cosine of 105 Degrees | :$\cos 105 \degrees = \cos \dfrac {7 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | :$\cos 105 \degrees = \cos \dfrac {7 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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|Sine of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 105 Degrees | https://proofwiki.org/wiki/Cosine_of_105_Degrees | https://proofwiki.org/wiki/Cosine_of_105_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Angle plus Right Angle",
"Sine of 15 Degrees"
] |
proofwiki-7945 | Cosine of Angle plus Right Angle | :$\map \cos {x + \dfrac \pi 2} = -\sin x$ | {{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}} | :$\map \cos {x + \dfrac \pi 2} = -\sin x$ | {{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}} | Cosine of Angle plus Right Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Right_Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Right_Angle | [
"Cosine Function",
"Reduction Formulae (Trigonometry)"
] | [] | [
"Cosine of Sum",
"Cosine of Right Angle",
"Sine of Right Angle"
] |
proofwiki-7946 | Cosine of 120 Degrees | :$\cos 120 \degrees = \cos \dfrac {2 \pi} 3 = -\dfrac 1 2$ | {{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 = {{sin|30}}
}}
{{end-eqn}}
{{qed}} | :$\cos 120 \degrees = \cos \dfrac {2 \pi} 3 = -\dfrac 1 2$ | {{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 = {{sin|30}}
}}
{{end-eqn}}
{{qed}} | Cosine of 120 Degrees | https://proofwiki.org/wiki/Cosine_of_120_Degrees | https://proofwiki.org/wiki/Cosine_of_120_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Angle plus Right Angle"
] |
proofwiki-7947 | Cosine of 135 Degrees | :$\cos 135 \degrees = \cos \dfrac {3 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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}} | :$\cos 135 \degrees = \cos \dfrac {3 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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|Sine of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 135 Degrees | https://proofwiki.org/wiki/Cosine_of_135_Degrees | https://proofwiki.org/wiki/Cosine_of_135_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Angle plus Right Angle",
"Sine of 45 Degrees"
] |
proofwiki-7948 | Cosine of 150 Degrees | :$\cos 150 \degrees = \cos \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 2$ | {{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}} | :$\cos 150 \degrees = \cos \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 2$ | {{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|Sine of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 150 Degrees | https://proofwiki.org/wiki/Cosine_of_150_Degrees | https://proofwiki.org/wiki/Cosine_of_150_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Angle plus Right Angle",
"Sine of 60 Degrees"
] |
proofwiki-7949 | Cosine of 165 Degrees | :$\cos 165 \degrees = \cos \dfrac {11 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | :$\cos 165 \degrees = \cos \dfrac {11 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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|Sine of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 165 Degrees | https://proofwiki.org/wiki/Cosine_of_165_Degrees | https://proofwiki.org/wiki/Cosine_of_165_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Angle plus Right Angle",
"Sine of 75 Degrees"
] |
proofwiki-7950 | Cosine of Straight Angle | :$\cos 180 \degrees = \cos \pi = -1$ | 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}} | :$\cos 180 \degrees = \cos \pi = -1$ | 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}} | Cosine of Straight Angle | https://proofwiki.org/wiki/Cosine_of_Straight_Angle | https://proofwiki.org/wiki/Cosine_of_Straight_Angle | [
"Cosine Function"
] | [] | [
"Cosine of Integer Multiple of Pi"
] |
proofwiki-7951 | Cosine of 195 Degrees | :$\cos 195 \degrees = \cos \dfrac {13 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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}} | :$\cos 195 \degrees = \cos \dfrac {13 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{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|Cosine of $165 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 195 Degrees | https://proofwiki.org/wiki/Cosine_of_195_Degrees | https://proofwiki.org/wiki/Cosine_of_195_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 165 Degrees"
] |
proofwiki-7952 | Cosine of 210 Degrees | :$\cos 210 \degrees = \cos \dfrac {7 \pi} 6 = -\dfrac {\sqrt 3} 2$ | {{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}} | :$\cos 210 \degrees = \cos \dfrac {7 \pi} 6 = -\dfrac {\sqrt 3} 2$ | {{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|Cosine of $150 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 210 Degrees | https://proofwiki.org/wiki/Cosine_of_210_Degrees | https://proofwiki.org/wiki/Cosine_of_210_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 150 Degrees"
] |
proofwiki-7953 | Cosine of 225 Degrees | :$\cos 225 \degrees = \cos \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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}} | :$\cos 225 \degrees = \cos \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$ | {{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|Cosine of $135 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 225 Degrees | https://proofwiki.org/wiki/Cosine_of_225_Degrees | https://proofwiki.org/wiki/Cosine_of_225_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 135 Degrees"
] |
proofwiki-7954 | Cosine of 240 Degrees | :$\cos 240 \degrees = \cos \dfrac {4 \pi} 3 = -\dfrac 1 2$ | {{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}} | :$\cos 240 \degrees = \cos \dfrac {4 \pi} 3 = -\dfrac 1 2$ | {{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|Cosine of $120 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 240 Degrees | https://proofwiki.org/wiki/Cosine_of_240_Degrees | https://proofwiki.org/wiki/Cosine_of_240_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 120 Degrees"
] |
proofwiki-7955 | Cosine of 255 Degrees | :$\cos 255^\circ = \cos \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | :$\cos 255^\circ = \cos \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | Cosine of 255 Degrees | https://proofwiki.org/wiki/Cosine_of_255_Degrees | https://proofwiki.org/wiki/Cosine_of_255_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 105 Degrees"
] |
proofwiki-7956 | Cosine of Three Right Angles | :$\cos 270 \degrees = \cos \dfrac {3 \pi} 2 = 0$ | {{begin-eqn}}
{{eqn | l = \cos 270 \degrees
| r = \map \cos {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = \cos 90 \degrees
| c = Cosine of Conjugate Angle
}}
{{eqn | r = 0
| c = Cosine of Right Angle
}}
{{end-eqn}}
{{qed}} | :$\cos 270 \degrees = \cos \dfrac {3 \pi} 2 = 0$ | {{begin-eqn}}
{{eqn | l = \cos 270 \degrees
| r = \map \cos {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = \cos 90 \degrees
| c = [[Cosine of Conjugate Angle]]
}}
{{eqn | r = 0
| c = [[Cosine of Right Angle]]
}}
{{end-eqn}}
{{qed}} | Cosine of Three Right Angles | https://proofwiki.org/wiki/Cosine_of_Three_Right_Angles | https://proofwiki.org/wiki/Cosine_of_Three_Right_Angles | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of Right Angle"
] |
proofwiki-7957 | Cosine of 285 Degrees | :$\cos 285^\circ = \cos \dfrac {19 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | :$\cos 285^\circ = \cos \dfrac {19 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$ | {{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}} | Cosine of 285 Degrees | https://proofwiki.org/wiki/Cosine_of_285_Degrees | https://proofwiki.org/wiki/Cosine_of_285_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 75 Degrees"
] |
proofwiki-7958 | Cosine of 300 Degrees | :$\cos 300 \degrees = \cos \dfrac {5 \pi} 3 = \dfrac 1 2$ | {{begin-eqn}}
{{eqn | l = \cos 300 \degrees
| r = \map \cos {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = \cos 60 \degrees
| c = Cosine of Conjugate Angle
}}
{{eqn | r = \frac 1 2
| c = {{cos|60}}
}}
{{end-eqn}}
{{qed}} | :$\cos 300 \degrees = \cos \dfrac {5 \pi} 3 = \dfrac 1 2$ | {{begin-eqn}}
{{eqn | l = \cos 300 \degrees
| r = \map \cos {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = \cos 60 \degrees
| c = [[Cosine of Conjugate Angle]]
}}
{{eqn | r = \frac 1 2
| c = {{cos|60}}
}}
{{end-eqn}}
{{qed}} | Cosine of 300 Degrees | https://proofwiki.org/wiki/Cosine_of_300_Degrees | https://proofwiki.org/wiki/Cosine_of_300_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle"
] |
proofwiki-7959 | Cosine of 315 Degrees | :$\cos 315 \degrees = \cos \dfrac {7 \pi} 4 = \dfrac {\sqrt 2} 2$ | {{begin-eqn}}
{{eqn | l = \cos 315 \degrees
| r = \map \cos {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = \cos 45 \degrees
| c = Cosine of Conjugate Angle
}}
{{eqn | r = \frac {\sqrt 2} 2
| c = Cosine of $45 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cos 315 \degrees = \cos \dfrac {7 \pi} 4 = \dfrac {\sqrt 2} 2$ | {{begin-eqn}}
{{eqn | l = \cos 315 \degrees
| r = \map \cos {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = \cos 45 \degrees
| c = [[Cosine of Conjugate Angle]]
}}
{{eqn | r = \frac {\sqrt 2} 2
| c = [[Cosine of 45 Degrees|Cosine of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 315 Degrees | https://proofwiki.org/wiki/Cosine_of_315_Degrees | https://proofwiki.org/wiki/Cosine_of_315_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 45 Degrees"
] |
proofwiki-7960 | Cosine of 330 Degrees | :$\cos 330 \degrees = \cos \dfrac {11 \pi} 6 = \dfrac {\sqrt 3} 2$ | {{begin-eqn}}
{{eqn | l = \cos 330 \degrees
| r = \map \cos {360 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \cos 30 \degrees
| c = Cosine of Conjugate Angle
}}
{{eqn | r = \frac {\sqrt 3} 2
| c = Cosine of $30 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cos 330 \degrees = \cos \dfrac {11 \pi} 6 = \dfrac {\sqrt 3} 2$ | {{begin-eqn}}
{{eqn | l = \cos 330 \degrees
| r = \map \cos {360 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \cos 30 \degrees
| c = [[Cosine of Conjugate Angle]]
}}
{{eqn | r = \frac {\sqrt 3} 2
| c = [[Cosine of 30 Degrees|Cosine of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 330 Degrees | https://proofwiki.org/wiki/Cosine_of_330_Degrees | https://proofwiki.org/wiki/Cosine_of_330_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 30 Degrees"
] |
proofwiki-7961 | Cosine of 345 Degrees | :$\cos 345 \degrees = \cos \dfrac {23 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \cos 345 \degrees
| r = \map \cos {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = \cos 15 \degrees
| c = Cosine of Conjugate Angle
}}
{{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4
| c = Cosine of $15 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cos 345 \degrees = \cos \dfrac {23 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$ | {{begin-eqn}}
{{eqn | l = \cos 345 \degrees
| r = \map \cos {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = \cos 15 \degrees
| c = [[Cosine of Conjugate Angle]]
}}
{{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4
| c = [[Cosine of 15 Degrees|Cosine of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosine of 345 Degrees | https://proofwiki.org/wiki/Cosine_of_345_Degrees | https://proofwiki.org/wiki/Cosine_of_345_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Conjugate Angle",
"Cosine of 15 Degrees"
] |
proofwiki-7962 | Cosine of Full Angle | :$\cos 360 \degrees = \cos 2 \pi = 1$ | A direct implementation of Cosine of Multiple of Pi:
:$\forall n \in \Z: \cos n \pi = \paren {-1}^n$
In this case, $n = 2$ and so:
:$\cos 2 \pi = \paren {-1}^2 = 1$
{{qed}} | :$\cos 360 \degrees = \cos 2 \pi = 1$ | A direct implementation of [[Cosine of Multiple of Pi]]:
:$\forall n \in \Z: \cos n \pi = \paren {-1}^n$
In this case, $n = 2$ and so:
:$\cos 2 \pi = \paren {-1}^2 = 1$
{{qed}} | Cosine of Full Angle | https://proofwiki.org/wiki/Cosine_of_Full_Angle | https://proofwiki.org/wiki/Cosine_of_Full_Angle | [
"Cosine Function"
] | [] | [
"Cosine of Integer Multiple of Pi"
] |
proofwiki-7963 | Tangent of Zero | :$\tan 0 = 0$ | {{begin-eqn}}
{{eqn | l = \tan 0
| r = \frac {\sin 0} {\cos 0}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac 0 1
| c = Sine of Zero is Zero and Cosine of Zero is One
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | :$\tan 0 = 0$ | {{begin-eqn}}
{{eqn | l = \tan 0
| r = \frac {\sin 0} {\cos 0}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac 0 1
| c = [[Sine of Zero is Zero]] and [[Cosine of Zero is One]]
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | Tangent of Zero | https://proofwiki.org/wiki/Tangent_of_Zero | https://proofwiki.org/wiki/Tangent_of_Zero | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Zero is Zero",
"Cosine of Zero is One"
] |
proofwiki-7964 | Tangent of 15 Degrees | :$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 15 \degrees
| r = \tan \frac {30 \degrees} 2
| c =
}}
{{eqn | r = \frac {\sin 30 \degrees} {1 + \cos 30 \degrees}
| c = {{Corollary|Half Angle Formula for Tangent|1}}
}}
{{eqn | r = \frac {\frac 1 2} {1 + \frac {\sqrt 3} 2}
| c = {{sin|30}} and {{cos|30}}
}}
{{eqn... | :$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 15 \degrees
| r = \tan \frac {30 \degrees} 2
| c =
}}
{{eqn | r = \frac {\sin 30 \degrees} {1 + \cos 30 \degrees}
| c = {{Corollary|Half Angle Formula for Tangent|1}}
}}
{{eqn | r = \frac {\frac 1 2} {1 + \frac {\sqrt 3} 2}
| c = {{sin|30}} and {{cos|30}}
}}
{{eqn... | Tangent of 15 Degrees/Proof 1 | https://proofwiki.org/wiki/Tangent_of_15_Degrees | https://proofwiki.org/wiki/Tangent_of_15_Degrees/Proof_1 | [
"Tangent of 15 Degrees",
"Tangent Function"
] | [] | [
"Difference of Two Squares"
] |
proofwiki-7965 | Tangent of 15 Degrees | :$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 15 \degrees
| r = \tan \frac {30 \degrees} 2
| c =
}}
{{eqn | r = \frac {1 - \cos 30 \degrees} {\sin 30 \degrees}
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \frac {1 - \frac {\sqrt 3} 2} {\frac 1 2}
| c = {{cos|30}} and {{sin|30}}
}}
{{eq... | :$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 15 \degrees
| r = \tan \frac {30 \degrees} 2
| c =
}}
{{eqn | r = \frac {1 - \cos 30 \degrees} {\sin 30 \degrees}
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \frac {1 - \frac {\sqrt 3} 2} {\frac 1 2}
| c = {{cos|30}} and {{sin|30}}
}}
{{eq... | Tangent of 15 Degrees/Proof 2 | https://proofwiki.org/wiki/Tangent_of_15_Degrees | https://proofwiki.org/wiki/Tangent_of_15_Degrees/Proof_2 | [
"Tangent of 15 Degrees",
"Tangent Function"
] | [] | [] |
proofwiki-7966 | Tangent of 15 Degrees | :$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 15 \degrees
| r = \frac {\sin 15 \degrees} {\cos 15 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}
| c = {{sin|15}} and {{cos|15}}
}}
{{eqn | r = \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + \sqr... | :$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 15 \degrees
| r = \frac {\sin 15 \degrees} {\cos 15 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}
| c = {{sin|15}} and {{cos|15}}
}}
{{eqn | r = \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + ... | Tangent of 15 Degrees/Proof 3 | https://proofwiki.org/wiki/Tangent_of_15_Degrees | https://proofwiki.org/wiki/Tangent_of_15_Degrees/Proof_3 | [
"Tangent of 15 Degrees",
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-7967 | Tangent of 30 Degrees | :$\tan 30 \degrees = \tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 30 \degrees
| r = \frac {\sin 30 \degrees} {\cos 30 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\frac 1 2} {\frac {\sqrt 3} 2}
| c = Sine of $30 \degrees$ and Cosine of $30 \degrees$
}}
{{eqn | r = \frac 1 {\sqrt 3}
| c =
}}
{{eqn | r =... | :$\tan 30 \degrees = \tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 30 \degrees
| r = \frac {\sin 30 \degrees} {\cos 30 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\frac 1 2} {\frac {\sqrt 3} 2}
| c = [[Sine of 30 Degrees|Sine of $30 \degrees$]] and [[Cosine of 30 Degrees|Cosine of $30 \degrees$]]
}}
{{eqn... | Tangent of 30 Degrees | https://proofwiki.org/wiki/Tangent_of_30_Degrees | https://proofwiki.org/wiki/Tangent_of_30_Degrees | [
"Tangent of 30 Degrees",
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of 30 Degrees",
"Cosine of 30 Degrees"
] |
proofwiki-7968 | Tangent of 45 Degrees | :$\tan 45 \degrees = \tan \dfrac \pi 4 = 1$ | {{begin-eqn}}
{{eqn | l = \tan 45 \degrees
| r = \frac {\sin 45 \degrees} {\cos 45 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
| c = Sine of $45 \degrees$ and Cosine of $45 \degrees$
}}
{{eqn | r = 1
| c = dividing top and bottom... | :$\tan 45 \degrees = \tan \dfrac \pi 4 = 1$ | {{begin-eqn}}
{{eqn | l = \tan 45 \degrees
| r = \frac {\sin 45 \degrees} {\cos 45 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
| c = [[Sine of 45 Degrees|Sine of $45 \degrees$]] and [[Cosine of 45 Degrees|Cosine of $45 \degrees$]]
... | Tangent of 45 Degrees | https://proofwiki.org/wiki/Tangent_of_45_Degrees | https://proofwiki.org/wiki/Tangent_of_45_Degrees | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of 45 Degrees",
"Cosine of 45 Degrees"
] |
proofwiki-7969 | Tangent of 60 Degrees | :$\tan 60 \degrees = \tan \dfrac \pi 3 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 60 \degrees
| r = \frac {\sin 60 \degrees} {\cos 60 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\frac {\sqrt 3} 2} {\frac 1 2}
| c = Sine of $60 \degrees$ and Cosine of $60 \degrees$
}}
{{eqn | r = \sqrt 3
| c = multiplying top and botto... | :$\tan 60 \degrees = \tan \dfrac \pi 3 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 60 \degrees
| r = \frac {\sin 60 \degrees} {\cos 60 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\frac {\sqrt 3} 2} {\frac 1 2}
| c = [[Sine of 60 Degrees|Sine of $60 \degrees$]] and [[Cosine of 60 Degrees|Cosine of $60 \degrees$]]
}}
{{eqn... | Tangent of 60 Degrees | https://proofwiki.org/wiki/Tangent_of_60_Degrees | https://proofwiki.org/wiki/Tangent_of_60_Degrees | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of 60 Degrees",
"Cosine of 60 Degrees"
] |
proofwiki-7970 | Tangent of 75 Degrees | :$\tan 75 \degrees = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 75 \degrees
| r = \frac {\sin 75 \degrees} {\cos 75 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{sin|75}} and {{cos|75}}
}}
{{eqn | r = \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - \sqr... | :$\tan 75 \degrees = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 75 \degrees
| r = \frac {\sin 75 \degrees} {\cos 75 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{sin|75}} and {{cos|75}}
}}
{{eqn | r = \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - ... | Tangent of 75 Degrees | https://proofwiki.org/wiki/Tangent_of_75_Degrees | https://proofwiki.org/wiki/Tangent_of_75_Degrees | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-7971 | Tangent of Complement equals Cotangent | :$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$
where $\tan$ and $\cot$ are tangent and cotangent respectively.
That is, the cotangent of an angle is the tangent of its complement.
This relation is defined wherever $\sin \theta \ne 0$. | {{begin-eqn}}
{{eqn | l = \map \tan {\frac \pi 2 - \theta}
| r = \frac {\map \sin {\frac \pi 2 - \theta} } {\map \cos {\frac \pi 2 - \theta} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\cos \theta} {\sin \theta}
| c = Sine and Cosine of Complementary Angles
}}
{{eqn | r = \cot \the... | :$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$
where $\tan$ and $\cot$ are [[Definition:Tangent Function|tangent]] and [[Definition:Cotangent|cotangent]] respectively.
That is, the [[Definition:Cotangent|cotangent]] of an [[Definition:Angle|angle]] is the [[Definition:Tangent Function|tange... | {{begin-eqn}}
{{eqn | l = \map \tan {\frac \pi 2 - \theta}
| r = \frac {\map \sin {\frac \pi 2 - \theta} } {\map \cos {\frac \pi 2 - \theta} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\cos \theta} {\sin \theta}
| c = [[Sine and Cosine of Complementary Angles]]
}}
{{eqn | r = \... | Tangent of Complement equals Cotangent | https://proofwiki.org/wiki/Tangent_of_Complement_equals_Cotangent | https://proofwiki.org/wiki/Tangent_of_Complement_equals_Cotangent | [
"Tangent Function",
"Cotangent Function",
"Complementary Angles"
] | [
"Definition:Tangent Function",
"Definition:Cotangent",
"Definition:Cotangent",
"Definition:Angle",
"Definition:Tangent Function",
"Definition:Complementary Angles"
] | [
"Tangent is Sine divided by Cosine",
"Sine and Cosine of Complementary Angles",
"Cotangent is Cosine divided by Sine",
"Sine of Integer Multiple of Pi"
] |
proofwiki-7972 | Cotangent of Complement equals Tangent | Let $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
Then:
:$\map \cot {\dfrac \pi 2 - \theta} = \tan \theta$ | {{begin-eqn}}
{{eqn | l = \map \cot {\frac \pi 2 - \theta}
| r = \frac {\map \cos {\frac \pi 2 - \theta} } {\map \sin {\frac \pi 2 - \theta} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\sin \theta} {\cos \theta}
| c = Sine and Cosine of Complementary Angles
}}
{{eqn | r = \tan \t... | Let $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
Then:
:$\map \cot {\dfrac \pi 2 - \theta} = \tan \theta$ | {{begin-eqn}}
{{eqn | l = \map \cot {\frac \pi 2 - \theta}
| r = \frac {\map \cos {\frac \pi 2 - \theta} } {\map \sin {\frac \pi 2 - \theta} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\sin \theta} {\cos \theta}
| c = [[Sine and Cosine of Complementary Angles]]
}}
{{eqn | r =... | Cotangent of Complement equals Tangent | https://proofwiki.org/wiki/Cotangent_of_Complement_equals_Tangent | https://proofwiki.org/wiki/Cotangent_of_Complement_equals_Tangent | [
"Cotangent of Complement equals Tangent",
"Tangent Function",
"Cotangent Function",
"Complementary Angles"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Sine and Cosine of Complementary Angles",
"Tangent is Sine divided by Cosine",
"Cosine of Half-Integer Multiple of Pi"
] |
proofwiki-7973 | Tangent of Angle plus Right Angle | :$\map \tan {x + \dfrac \pi 2} = -\cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x + \frac \pi 2}
| r = \frac {\map \sin {x + \frac \pi 2} } {\map \cos {x + \frac \pi 2} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\cos x} {- \sin x}
| c = Sine of Angle plus Right Angle and Cosine of Angle plus Right Angle
}}
{{eqn | r = -\c... | :$\map \tan {x + \dfrac \pi 2} = -\cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x + \frac \pi 2}
| r = \frac {\map \sin {x + \frac \pi 2} } {\map \cos {x + \frac \pi 2} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\cos x} {- \sin x}
| c = [[Sine of Angle plus Right Angle]] and [[Cosine of Angle plus Right Angle]]
}}
{{e... | Tangent of Angle plus Right Angle | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Right_Angle | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Right_Angle | [
"Tangent Function",
"Reduction Formulae (Trigonometry)"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Angle plus Right Angle",
"Cosine of Angle plus Right Angle",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-7974 | Cotangent of Angle plus Right Angle | :$\map \cot {x + \dfrac \pi 2} = -\tan x$ | {{begin-eqn}}
{{eqn | l = \map \cot {x + \frac \pi 2}
| r = \frac {\map \cos {x + \frac \pi 2} } {\map \sin {x + \frac \pi 2} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {-\sin x} {\cos x}
| c = Cosine of Angle plus Right Angle and Sine of Angle plus Right Angle
}}
{{eqn | r = -\... | :$\map \cot {x + \dfrac \pi 2} = -\tan x$ | {{begin-eqn}}
{{eqn | l = \map \cot {x + \frac \pi 2}
| r = \frac {\map \cos {x + \frac \pi 2} } {\map \sin {x + \frac \pi 2} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {-\sin x} {\cos x}
| c = [[Cosine of Angle plus Right Angle]] and [[Sine of Angle plus Right Angle]]
}}
{{... | Cotangent of Angle plus Right Angle | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Right_Angle | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Right_Angle | [
"Cotangent Function",
"Reduction Formulae (Trigonometry)"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Angle plus Right Angle",
"Sine of Angle plus Right Angle",
"Tangent is Sine divided by Cosine"
] |
proofwiki-7975 | Tangent of Supplementary Angle | :$\map \tan {\pi - \theta} = -\tan \theta$
where $\tan$ denotes tangent.
That is, the tangent of an angle is the negative of its supplement. | {{begin-eqn}}
{{eqn | l = \map \tan {\pi - \theta}
| r = \frac {\map \sin {\pi - \theta} } {\map \cos {\pi - \theta} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\sin \theta} {-\cos \theta}
| c = Sine of Supplementary Angle and Cosine of Supplementary Angle
}}
{{eqn | r = -\tan \the... | :$\map \tan {\pi - \theta} = -\tan \theta$
where $\tan$ denotes [[Definition:Tangent Function|tangent]].
That is, the [[Definition:Tangent Function|tangent]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Supplement of Angle|supplement]]. | {{begin-eqn}}
{{eqn | l = \map \tan {\pi - \theta}
| r = \frac {\map \sin {\pi - \theta} } {\map \cos {\pi - \theta} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\sin \theta} {-\cos \theta}
| c = [[Sine of Supplementary Angle]] and [[Cosine of Supplementary Angle]]
}}
{{eqn | r ... | Tangent of Supplementary Angle | https://proofwiki.org/wiki/Tangent_of_Supplementary_Angle | https://proofwiki.org/wiki/Tangent_of_Supplementary_Angle | [
"Tangent Function",
"Supplementary Angles"
] | [
"Definition:Tangent Function",
"Definition:Tangent Function",
"Definition:Angle",
"Definition:Supplementary Angles"
] | [
"Tangent is Sine divided by Cosine",
"Sine of Supplementary Angle",
"Cosine of Supplementary Angle"
] |
proofwiki-7976 | Cotangent of Supplementary Angle | :$\map \cot {\pi - \theta} = -\cot \theta$
where $\cot$ denotes tangent.
That is, the cotangent of an angle is the negative of its supplement. | {{begin-eqn}}
{{eqn | l = \map \cot {\pi - \theta}
| r = \frac {\map \cos {\pi - \theta} } {\map \sin {\pi - \theta} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {-\cos \theta} {\sin \theta}
| c = Cosine of Supplementary Angle and Sine of Supplementary Angle
}}
{{eqn | r = -\cot \... | :$\map \cot {\pi - \theta} = -\cot \theta$
where $\cot$ denotes [[Definition:Tangent Function|tangent]].
That is, the [[Definition:Cotangent|cotangent]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Supplement of Angle|supplement]]. | {{begin-eqn}}
{{eqn | l = \map \cot {\pi - \theta}
| r = \frac {\map \cos {\pi - \theta} } {\map \sin {\pi - \theta} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {-\cos \theta} {\sin \theta}
| c = [[Cosine of Supplementary Angle]] and [[Sine of Supplementary Angle]]
}}
{{eqn |... | Cotangent of Supplementary Angle | https://proofwiki.org/wiki/Cotangent_of_Supplementary_Angle | https://proofwiki.org/wiki/Cotangent_of_Supplementary_Angle | [
"Cotangent Function",
"Supplementary Angles"
] | [
"Definition:Tangent Function",
"Definition:Cotangent",
"Definition:Angle",
"Definition:Supplementary Angles"
] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Supplementary Angle",
"Sine of Supplementary Angle"
] |
proofwiki-7977 | Tangent of Conjugate Angle | :$\map \tan {2 \pi - \theta} = -\tan \theta$
where $\tan$ denotes tangent.
That is, the tangent of an angle is the negative of its conjugate. | {{begin-eqn}}
{{eqn | l = \map \tan {2 \pi - \theta}
| r = \frac {\map \sin {2 \pi - \theta} } {\map \cos {2 \pi - \theta} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\sin \theta} {\cos \theta}
| c = Sine of Conjugate Angle and Cosine of Conjugate Angle
}}
{{eqn | r = -\tan \theta... | :$\map \tan {2 \pi - \theta} = -\tan \theta$
where $\tan$ denotes [[Definition:Tangent Function|tangent]].
That is, the [[Definition:Tangent Function|tangent]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Conjugate Angle|conjugate]]. | {{begin-eqn}}
{{eqn | l = \map \tan {2 \pi - \theta}
| r = \frac {\map \sin {2 \pi - \theta} } {\map \cos {2 \pi - \theta} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {-\sin \theta} {\cos \theta}
| c = [[Sine of Conjugate Angle]] and [[Cosine of Conjugate Angle]]
}}
{{eqn | r = ... | Tangent of Conjugate Angle | https://proofwiki.org/wiki/Tangent_of_Conjugate_Angle | https://proofwiki.org/wiki/Tangent_of_Conjugate_Angle | [
"Tangent Function",
"Conjugate Angles"
] | [
"Definition:Tangent Function",
"Definition:Tangent Function",
"Definition:Angle",
"Definition:Conjugate Angles"
] | [
"Tangent is Sine divided by Cosine",
"Sine of Conjugate Angle",
"Cosine of Conjugate Angle",
"Tangent is Sine divided by Cosine"
] |
proofwiki-7978 | Cotangent of Conjugate Angle | :$\map \cot {2 \pi - \theta} = -\cot \theta$
where $\cot$ denotes cotangent.
That is, the cotangent of an angle is the negative of its conjugate. | {{begin-eqn}}
{{eqn | l = \map \cot {2 \pi - \theta}
| r = \frac {\map \cos {2 \pi - \theta} } {\map \sin {2 \pi - \theta} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\cos \theta} {-\sin \theta}
| c = Cosine of Conjugate Angle and Sine of Conjugate Angle
}}
{{eqn | r = -\cot \the... | :$\map \cot {2 \pi - \theta} = -\cot \theta$
where $\cot$ denotes [[Definition:Cotangent|cotangent]].
That is, the [[Definition:Cotangent|cotangent]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Conjugate Angle|conjugate]]. | {{begin-eqn}}
{{eqn | l = \map \cot {2 \pi - \theta}
| r = \frac {\map \cos {2 \pi - \theta} } {\map \sin {2 \pi - \theta} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\cos \theta} {-\sin \theta}
| c = [[Cosine of Conjugate Angle]] and [[Sine of Conjugate Angle]]
}}
{{eqn | r ... | Cotangent of Conjugate Angle | https://proofwiki.org/wiki/Cotangent_of_Conjugate_Angle | https://proofwiki.org/wiki/Cotangent_of_Conjugate_Angle | [
"Cotangent Function",
"Conjugate Angles"
] | [
"Definition:Cotangent",
"Definition:Cotangent",
"Definition:Angle",
"Definition:Conjugate Angles"
] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Conjugate Angle",
"Sine of Conjugate Angle",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-7979 | Tangent of Right Angle | :$\tan 90 \degrees = \tan \dfrac \pi 2$ is undefined | 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 $\theta = \dfrac \pi 2$, $\cos \t... | :$\tan 90 \degrees = \tan \dfrac \pi 2$ is undefined | 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 $\theta = \dfrac \pi 2$, $... | Tangent of Right Angle | https://proofwiki.org/wiki/Tangent_of_Right_Angle | https://proofwiki.org/wiki/Tangent_of_Right_Angle | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine"
] |
proofwiki-7980 | Tangent of 105 Degrees | :$\tan 105^\circ = \tan \dfrac {7 \pi} {12} = - \left({2 + \sqrt 3}\right)$ | {{begin-eqn}}
{{eqn | l = \tan 105^\circ
| r = \tan \left({90^\circ + 15^\circ}\right)
| c =
}}
{{eqn | r = - \cot 15^\circ
| c = Tangent of Angle plus Right Angle
}}
{{eqn | r = - \left({2 + \sqrt 3}\right)
| c = Cotangent of 15 Degrees
}}
{{end-eqn}}
{{qed}} | :$\tan 105^\circ = \tan \dfrac {7 \pi} {12} = - \left({2 + \sqrt 3}\right)$ | {{begin-eqn}}
{{eqn | l = \tan 105^\circ
| r = \tan \left({90^\circ + 15^\circ}\right)
| c =
}}
{{eqn | r = - \cot 15^\circ
| c = [[Tangent of Angle plus Right Angle]]
}}
{{eqn | r = - \left({2 + \sqrt 3}\right)
| c = [[Cotangent of 15 Degrees]]
}}
{{end-eqn}}
{{qed}} | Tangent of 105 Degrees | https://proofwiki.org/wiki/Tangent_of_105_Degrees | https://proofwiki.org/wiki/Tangent_of_105_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Angle plus Right Angle",
"Cotangent of 15 Degrees"
] |
proofwiki-7981 | Tangent of 120 Degrees | :$\tan 120 \degrees = \tan \dfrac {2 \pi} 3 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 120 \degrees
| r = \map \tan {90 \degrees + 30 \degrees}
| c =
}}
{{eqn | r = -\cot 30 \degrees
| c = Tangent of Angle plus Right Angle
}}
{{eqn | r = -\sqrt 3
| c = Cotangent of $30 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 120 \degrees = \tan \dfrac {2 \pi} 3 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 120 \degrees
| r = \map \tan {90 \degrees + 30 \degrees}
| c =
}}
{{eqn | r = -\cot 30 \degrees
| c = [[Tangent of Angle plus Right Angle]]
}}
{{eqn | r = -\sqrt 3
| c = [[Cotangent of 30 Degrees|Cotangent of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 120 Degrees | https://proofwiki.org/wiki/Tangent_of_120_Degrees | https://proofwiki.org/wiki/Tangent_of_120_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Angle plus Right Angle",
"Cotangent of 30 Degrees"
] |
proofwiki-7982 | Tangent of 135 Degrees | :$\tan 135^\circ = \tan \dfrac {3 \pi} 4 = - 1$ | {{begin-eqn}}
{{eqn | l = \tan 135^\circ
| r = \tan \left({90^\circ + 45^\circ}\right)
| c =
}}
{{eqn | r = - \cot 45^\circ
| c = Tangent of Angle plus Right Angle
}}
{{eqn | r = - 1
| c = Cotangent of 45 Degrees
}}
{{end-eqn}}
{{qed}} | :$\tan 135^\circ = \tan \dfrac {3 \pi} 4 = - 1$ | {{begin-eqn}}
{{eqn | l = \tan 135^\circ
| r = \tan \left({90^\circ + 45^\circ}\right)
| c =
}}
{{eqn | r = - \cot 45^\circ
| c = [[Tangent of Angle plus Right Angle]]
}}
{{eqn | r = - 1
| c = [[Cotangent of 45 Degrees]]
}}
{{end-eqn}}
{{qed}} | Tangent of 135 Degrees | https://proofwiki.org/wiki/Tangent_of_135_Degrees | https://proofwiki.org/wiki/Tangent_of_135_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Angle plus Right Angle",
"Cotangent of 45 Degrees"
] |
proofwiki-7983 | Tangent of 150 Degrees | :$\tan 150 \degrees = \tan \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 150 \degrees
| r = \map \tan {90 \degrees + 60 \degrees}
| c =
}}
{{eqn | r = -\cot 60 \degrees
| c = Tangent of Angle plus Right Angle
}}
{{eqn | r = -\dfrac {\sqrt 3} 3
| c = Cotangent of $60 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 150 \degrees = \tan \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 150 \degrees
| r = \map \tan {90 \degrees + 60 \degrees}
| c =
}}
{{eqn | r = -\cot 60 \degrees
| c = [[Tangent of Angle plus Right Angle]]
}}
{{eqn | r = -\dfrac {\sqrt 3} 3
| c = [[Cotangent of 60 Degrees|Cotangent of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 150 Degrees | https://proofwiki.org/wiki/Tangent_of_150_Degrees | https://proofwiki.org/wiki/Tangent_of_150_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Angle plus Right Angle",
"Cotangent of 60 Degrees"
] |
proofwiki-7984 | Tangent of 165 Degrees | :$\tan 165 \degrees = \tan \dfrac {11 \pi} {12} = -\paren {2 - \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \tan 165 \degrees
| r = \map \tan {90 \degrees + 75 \degrees}
| c =
}}
{{eqn | r = -\cot 75 \degrees
| c = Tangent of Angle plus Right Angle
}}
{{eqn | r = -\paren {2 - \sqrt 3}
| c = Cotangent of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 165 \degrees = \tan \dfrac {11 \pi} {12} = -\paren {2 - \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \tan 165 \degrees
| r = \map \tan {90 \degrees + 75 \degrees}
| c =
}}
{{eqn | r = -\cot 75 \degrees
| c = [[Tangent of Angle plus Right Angle]]
}}
{{eqn | r = -\paren {2 - \sqrt 3}
| c = [[Cotangent of 75 Degrees|Cotangent of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 165 Degrees | https://proofwiki.org/wiki/Tangent_of_165_Degrees | https://proofwiki.org/wiki/Tangent_of_165_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Angle plus Right Angle",
"Cotangent of 75 Degrees"
] |
proofwiki-7985 | Tangent of Straight Angle | :$\tan 180 \degrees = \tan \pi = 0$ | {{begin-eqn}}
{{eqn | l = \tan 180 \degrees
| r = \frac {\sin 180 \degrees} {\cos 180 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac 0 {-1}
| c = Sine of Straight Angle and Cosine of Straight Angle
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | :$\tan 180 \degrees = \tan \pi = 0$ | {{begin-eqn}}
{{eqn | l = \tan 180 \degrees
| r = \frac {\sin 180 \degrees} {\cos 180 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac 0 {-1}
| c = [[Sine of Straight Angle]] and [[Cosine of Straight Angle]]
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | Tangent of Straight Angle | https://proofwiki.org/wiki/Tangent_of_Straight_Angle | https://proofwiki.org/wiki/Tangent_of_Straight_Angle | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Straight Angle",
"Cosine of Straight Angle"
] |
proofwiki-7986 | Tangent of 195 Degrees | :$\tan 195 \degrees = \tan \dfrac {13 \pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 195 \degrees
| r = \map \tan {360 \degrees - 165 \degrees}
| c =
}}
{{eqn | r = -\tan 165 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = 2 - \sqrt 3
| c = Tangent of $165 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 195 \degrees = \tan \dfrac {13 \pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 195 \degrees
| r = \map \tan {360 \degrees - 165 \degrees}
| c =
}}
{{eqn | r = -\tan 165 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = 2 - \sqrt 3
| c = [[Tangent of 165 Degrees|Tangent of $165 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 195 Degrees | https://proofwiki.org/wiki/Tangent_of_195_Degrees | https://proofwiki.org/wiki/Tangent_of_195_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 165 Degrees"
] |
proofwiki-7987 | Tangent of 210 Degrees | :$\tan 210 \degrees = \tan \dfrac {7 \pi} 6 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 210 \degrees
| r = \map \tan {360 \degrees - 150 \degrees}
| c =
}}
{{eqn | r = -\tan 150 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = \frac {\sqrt 3} 3
| c = Tangent of $150 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 210 \degrees = \tan \dfrac {7 \pi} 6 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 210 \degrees
| r = \map \tan {360 \degrees - 150 \degrees}
| c =
}}
{{eqn | r = -\tan 150 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = \frac {\sqrt 3} 3
| c = [[Tangent of 150 Degrees|Tangent of $150 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 210 Degrees | https://proofwiki.org/wiki/Tangent_of_210_Degrees | https://proofwiki.org/wiki/Tangent_of_210_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 150 Degrees"
] |
proofwiki-7988 | Tangent of 225 Degrees | :$\tan 225 \degrees = \tan \dfrac {5 \pi} 4 = 1$ | {{begin-eqn}}
{{eqn | l = \tan 225 \degrees
| r = \map \tan {360 \degrees - 135 \degrees}
| c =
}}
{{eqn | r = -\tan 135 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = 1
| c = {{tan|135}}
}}
{{end-eqn}}
{{qed}} | :$\tan 225 \degrees = \tan \dfrac {5 \pi} 4 = 1$ | {{begin-eqn}}
{{eqn | l = \tan 225 \degrees
| r = \map \tan {360 \degrees - 135 \degrees}
| c =
}}
{{eqn | r = -\tan 135 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = 1
| c = {{tan|135}}
}}
{{end-eqn}}
{{qed}} | Tangent of 225 Degrees | https://proofwiki.org/wiki/Tangent_of_225_Degrees | https://proofwiki.org/wiki/Tangent_of_225_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle"
] |
proofwiki-7989 | Tangent of 240 Degrees | :$\tan 240 \degrees = \tan \dfrac {4 \pi} 3 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 240 \degrees
| r = \map \tan {360 \degrees - 120 \degrees}
| c =
}}
{{eqn | r = -\tan 120 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = \sqrt 3
| c = Tangent of $120 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 240 \degrees = \tan \dfrac {4 \pi} 3 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 240 \degrees
| r = \map \tan {360 \degrees - 120 \degrees}
| c =
}}
{{eqn | r = -\tan 120 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = \sqrt 3
| c = [[Tangent of 120 Degrees|Tangent of $120 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 240 Degrees | https://proofwiki.org/wiki/Tangent_of_240_Degrees | https://proofwiki.org/wiki/Tangent_of_240_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 120 Degrees"
] |
proofwiki-7990 | Tangent of 255 Degrees | :$\tan 255 \degrees = \tan \dfrac {17 \pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 255 \degrees
| r = \map \tan {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = -\tan 105 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = 2 + \sqrt 3
| c = Tangent of $105 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 255 \degrees = \tan \dfrac {17 \pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 255 \degrees
| r = \map \tan {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = -\tan 105 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = 2 + \sqrt 3
| c = [[Tangent of 105 Degrees|Tangent of $105 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 255 Degrees | https://proofwiki.org/wiki/Tangent_of_255_Degrees | https://proofwiki.org/wiki/Tangent_of_255_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 105 Degrees"
] |
proofwiki-7991 | Tangent of Three Right Angles | :$\tan 270 \degrees = \tan \dfrac {3 \pi} 2$ is undefined | We have:
{{begin-eqn}}
{{eqn | l = \tan 270 \degrees
| r = \map \tan {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = -\tan 90 \degrees
| c = Tangent of Conjugate Angle
}}
{{end-eqn}}
But from Tangent of Right Angle, $\tan 90 \degrees$ is undefined.
Hence so is $\tan 270 \degrees$.
{{qed}} | :$\tan 270 \degrees = \tan \dfrac {3 \pi} 2$ is undefined | We have:
{{begin-eqn}}
{{eqn | l = \tan 270 \degrees
| r = \map \tan {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = -\tan 90 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{end-eqn}}
But from [[Tangent of Right Angle]], $\tan 90 \degrees$ is undefined.
Hence so is $\tan 270 \degrees$.
{{qed}} | Tangent of Three Right Angles | https://proofwiki.org/wiki/Tangent_of_Three_Right_Angles | https://proofwiki.org/wiki/Tangent_of_Three_Right_Angles | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of Right Angle"
] |
proofwiki-7992 | Tangent of 285 Degrees | :$\tan 285 \degrees = \tan \dfrac {19 \pi} {12} = -\paren {2 + \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \tan 285 \degrees
| r = \map \tan {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = -\tan 75 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = -\paren {2 + \sqrt 3}
| c = Tangent of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 285 \degrees = \tan \dfrac {19 \pi} {12} = -\paren {2 + \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \tan 285 \degrees
| r = \map \tan {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = -\tan 75 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = -\paren {2 + \sqrt 3}
| c = [[Tangent of 75 Degrees|Tangent of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 285 Degrees | https://proofwiki.org/wiki/Tangent_of_285_Degrees | https://proofwiki.org/wiki/Tangent_of_285_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 75 Degrees"
] |
proofwiki-7993 | Tangent of 300 Degrees | :$\tan 300 \degrees = \tan \dfrac {5 \pi} 3 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 300 \degrees
| r = \map \tan {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = -\tan 60 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = -\sqrt 3
| c = Tangent of $60 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 300 \degrees = \tan \dfrac {5 \pi} 3 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \tan 300 \degrees
| r = \map \tan {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = -\tan 60 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = -\sqrt 3
| c = [[Tangent of 60 Degrees|Tangent of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 300 Degrees | https://proofwiki.org/wiki/Tangent_of_300_Degrees | https://proofwiki.org/wiki/Tangent_of_300_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 60 Degrees"
] |
proofwiki-7994 | Tangent of 315 Degrees | :$\tan 315 \degrees = \tan \dfrac {7 \pi} 4 = -1$ | {{begin-eqn}}
{{eqn | l = \tan 315 \degrees
| r = \map \tan {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = -\tan 45 \degrees
| c = Tangent of Conjugate Angle
}}
{{eqn | r = -1
| c = Tangent of $45 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\tan 315 \degrees = \tan \dfrac {7 \pi} 4 = -1$ | {{begin-eqn}}
{{eqn | l = \tan 315 \degrees
| r = \map \tan {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = -\tan 45 \degrees
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = -1
| c = [[Tangent of 45 Degrees|Tangent of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Tangent of 315 Degrees | https://proofwiki.org/wiki/Tangent_of_315_Degrees | https://proofwiki.org/wiki/Tangent_of_315_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 45 Degrees"
] |
proofwiki-7995 | Tangent of 330 Degrees | :$\tan 330^\circ = \tan \dfrac {11 \pi} 6 = -\dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 330^\circ
| r = \tan \left({360^\circ - 30^\circ}\right)
| c =
}}
{{eqn | r = -\tan 30^\circ
| c = Tangent of Conjugate Angle
}}
{{eqn | r = -\frac {\sqrt 3} 3
| c = Tangent of 30 Degrees
}}
{{end-eqn}}
{{qed}} | :$\tan 330^\circ = \tan \dfrac {11 \pi} 6 = -\dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \tan 330^\circ
| r = \tan \left({360^\circ - 30^\circ}\right)
| c =
}}
{{eqn | r = -\tan 30^\circ
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = -\frac {\sqrt 3} 3
| c = [[Tangent of 30 Degrees]]
}}
{{end-eqn}}
{{qed}} | Tangent of 330 Degrees | https://proofwiki.org/wiki/Tangent_of_330_Degrees | https://proofwiki.org/wiki/Tangent_of_330_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 30 Degrees"
] |
proofwiki-7996 | Tangent of 345 Degrees | :$\tan 345^\circ = \tan \dfrac {23 \pi} {12} = -\left({2 - \sqrt 3}\right)$ | {{begin-eqn}}
{{eqn | l = \tan 345^\circ
| r = \tan \left({360^\circ - 15^\circ}\right)
| c =
}}
{{eqn | r = -\tan 15^\circ
| c = Tangent of Conjugate Angle
}}
{{eqn | r = -\left({2 - \sqrt 3}\right)
| c = Tangent of 15 Degrees
}}
{{end-eqn}}
{{qed}} | :$\tan 345^\circ = \tan \dfrac {23 \pi} {12} = -\left({2 - \sqrt 3}\right)$ | {{begin-eqn}}
{{eqn | l = \tan 345^\circ
| r = \tan \left({360^\circ - 15^\circ}\right)
| c =
}}
{{eqn | r = -\tan 15^\circ
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = -\left({2 - \sqrt 3}\right)
| c = [[Tangent of 15 Degrees]]
}}
{{end-eqn}}
{{qed}} | Tangent of 345 Degrees | https://proofwiki.org/wiki/Tangent_of_345_Degrees | https://proofwiki.org/wiki/Tangent_of_345_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of 15 Degrees"
] |
proofwiki-7997 | Tangent of Full Angle | :$\tan 360^\circ = \tan 2 \pi = 0$ | {{begin-eqn}}
{{eqn | l = \tan 360^\circ
| r = \tan \left({360^\circ - 0^\circ}\right)
| c =
}}
{{eqn | r = -\tan 0
| c = Tangent of Conjugate Angle
}}
{{eqn | r = 0
| c = Tangent of Zero
}}
{{end-eqn}}
{{qed}} | :$\tan 360^\circ = \tan 2 \pi = 0$ | {{begin-eqn}}
{{eqn | l = \tan 360^\circ
| r = \tan \left({360^\circ - 0^\circ}\right)
| c =
}}
{{eqn | r = -\tan 0
| c = [[Tangent of Conjugate Angle]]
}}
{{eqn | r = 0
| c = [[Tangent of Zero]]
}}
{{end-eqn}}
{{qed}} | Tangent of Full Angle | https://proofwiki.org/wiki/Tangent_of_Full_Angle | https://proofwiki.org/wiki/Tangent_of_Full_Angle | [
"Tangent Function"
] | [] | [
"Tangent of Conjugate Angle",
"Tangent of Zero"
] |
proofwiki-7998 | Cotangent of Zero | :$\cot 0$ is undefined | From Cotangent is Cosine divided by Sine:
:$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = 0$, $\sin \theta = 0$... | :$\cot 0$ is undefined | From [[Cotangent is Cosine divided by Sine]]:
:$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = 0$, $\sin \the... | Cotangent of Zero | https://proofwiki.org/wiki/Cotangent_of_Zero | https://proofwiki.org/wiki/Cotangent_of_Zero | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine"
] |
proofwiki-7999 | Cotangent of 30 Degrees | :$\cot 30 \degrees = \cot \dfrac \pi 6 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 30 \degrees
| r = \frac {\cos 30 \degrees} {\sin 30 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\frac {\sqrt 3} 2} {\frac 1 2}
| c = {{cos|30}} and {{sin|30}}
}}
{{eqn | r = \sqrt 3
| c = multiplying top and bottom by $2$
}}
{{end-eqn... | :$\cot 30 \degrees = \cot \dfrac \pi 6 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 30 \degrees
| r = \frac {\cos 30 \degrees} {\sin 30 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\frac {\sqrt 3} 2} {\frac 1 2}
| c = {{cos|30}} and {{sin|30}}
}}
{{eqn | r = \sqrt 3
| c = multiplying [[Definition:Numerator|top]] a... | Cotangent of 30 Degrees | https://proofwiki.org/wiki/Cotangent_of_30_Degrees | https://proofwiki.org/wiki/Cotangent_of_30_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
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