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-8000 | Cotangent of 15 Degrees | :$\cot 15 \degrees = \cot \dfrac {\pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 15^\circ
| r = \frac {\cos 15 \degrees} {\sin 15 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{cos|15}} and {{sin|15}}
}}
{{eqn | r = \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - \sqr... | :$\cot 15 \degrees = \cot \dfrac {\pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 15^\circ
| r = \frac {\cos 15 \degrees} {\sin 15 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{cos|15}} and {{sin|15}}
}}
{{eqn | r = \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - ... | Cotangent of 15 Degrees | https://proofwiki.org/wiki/Cotangent_of_15_Degrees | https://proofwiki.org/wiki/Cotangent_of_15_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-8001 | Cotangent of 45 Degrees | :$\cot 45 \degrees = \cot \dfrac \pi 4 = 1$ | {{begin-eqn}}
{{eqn | l = \cot 45 \degrees
| r = \frac {\cos 45 \degrees} {\sin 45 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
| c = {{cos|45}} and {{sin|45}}
}}
{{eqn | r = 1
| c = dividing top and bottom by $\dfrac {\sqrt 2}... | :$\cot 45 \degrees = \cot \dfrac \pi 4 = 1$ | {{begin-eqn}}
{{eqn | l = \cot 45 \degrees
| r = \frac {\cos 45 \degrees} {\sin 45 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
| c = {{cos|45}} and {{sin|45}}
}}
{{eqn | r = 1
| c = [[Definition:Real Division|dividing]] [[... | Cotangent of 45 Degrees | https://proofwiki.org/wiki/Cotangent_of_45_Degrees | https://proofwiki.org/wiki/Cotangent_of_45_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Definition:Division/Field/Real Numbers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8002 | Cotangent of 60 Degrees | :$\cot 60 \degrees = \cot \dfrac \pi 3 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 60 \degrees
| r = \frac {\cos 60 \degrees} {\sin 60 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\frac 1 2} {\frac {\sqrt 3} 2}
| c = {{cos|60}} and {{sin|60}}
}}
{{eqn | r = \frac {\sqrt 3} 3
| c = multiplying top and bottom by $2 \sq... | :$\cot 60 \degrees = \cot \dfrac \pi 3 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 60 \degrees
| r = \frac {\cos 60 \degrees} {\sin 60 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\frac 1 2} {\frac {\sqrt 3} 2}
| c = {{cos|60}} and {{sin|60}}
}}
{{eqn | r = \frac {\sqrt 3} 3
| c = multiplying [[Definition:Numerat... | Cotangent of 60 Degrees | https://proofwiki.org/wiki/Cotangent_of_60_Degrees | https://proofwiki.org/wiki/Cotangent_of_60_Degrees | [
"Cotangent of 60 Degrees",
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8003 | Cotangent of 75 Degrees | :$\cot 75 \degrees = \cot \dfrac {5 \pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 75 \degrees
| r = \frac {\cos 75 \degrees} {\sin 75 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}
| c = {{cos|75}} and {{sin|75}}
}}
{{eqn | r = \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + \s... | :$\cot 75 \degrees = \cot \dfrac {5 \pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 75 \degrees
| r = \frac {\cos 75 \degrees} {\sin 75 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}
| c = {{cos|75}} and {{sin|75}}
}}
{{eqn | r = \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 ... | Cotangent of 75 Degrees | https://proofwiki.org/wiki/Cotangent_of_75_Degrees | https://proofwiki.org/wiki/Cotangent_of_75_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8004 | Cotangent of Right Angle | :$\cot 90 \degrees = \cot \dfrac \pi 2 = 0$ | {{begin-eqn}}
{{eqn | l = \cot 90 \degrees
| r = \frac {\cos 90 \degrees} {\sin 90 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac 0 1
| c = Cosine of Right Angle and Sine of Right Angle
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | :$\cot 90 \degrees = \cot \dfrac \pi 2 = 0$ | {{begin-eqn}}
{{eqn | l = \cot 90 \degrees
| r = \frac {\cos 90 \degrees} {\sin 90 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac 0 1
| c = [[Cosine of Right Angle]] and [[Sine of Right Angle]]
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}} | Cotangent of Right Angle | https://proofwiki.org/wiki/Cotangent_of_Right_Angle | https://proofwiki.org/wiki/Cotangent_of_Right_Angle | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Right Angle",
"Sine of Right Angle"
] |
proofwiki-8005 | Cotangent of 105 Degrees | :$\cot 105^\circ = \cot \dfrac {7 \pi} {12} = -\left({2 - \sqrt 3}\right)$ | {{begin-eqn}}
{{eqn | l = \cot 105^\circ
| r = \cot \left({90^\circ + 15^\circ}\right)
| c =
}}
{{eqn | r = - \tan 15^\circ
| c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = - \left({2 - \sqrt 3}\right)
| c = Tangent of 15 Degrees
}}
{{end-eqn}}
{{qed}} | :$\cot 105^\circ = \cot \dfrac {7 \pi} {12} = -\left({2 - \sqrt 3}\right)$ | {{begin-eqn}}
{{eqn | l = \cot 105^\circ
| r = \cot \left({90^\circ + 15^\circ}\right)
| c =
}}
{{eqn | r = - \tan 15^\circ
| c = [[Cotangent of Angle plus Right Angle]]
}}
{{eqn | r = - \left({2 - \sqrt 3}\right)
| c = [[Tangent of 15 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 105 Degrees | https://proofwiki.org/wiki/Cotangent_of_105_Degrees | https://proofwiki.org/wiki/Cotangent_of_105_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Angle plus Right Angle",
"Tangent of 15 Degrees"
] |
proofwiki-8006 | Cotangent of 120 Degrees | :$\cot 120 \degrees = \cot \dfrac {2 \pi} 3 = -\dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 120 \degrees
| r = \map \cot {90 \degrees + 30 \degrees}
| c =
}}
{{eqn | r = -\tan 30 \degrees
| c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = -\frac {\sqrt 3} 3
| c = Tangent of 30 Degrees
}}
{{end-eqn}}
{{qed}} | :$\cot 120 \degrees = \cot \dfrac {2 \pi} 3 = -\dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 120 \degrees
| r = \map \cot {90 \degrees + 30 \degrees}
| c =
}}
{{eqn | r = -\tan 30 \degrees
| c = [[Cotangent of Angle plus Right Angle]]
}}
{{eqn | r = -\frac {\sqrt 3} 3
| c = [[Tangent of 30 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 120 Degrees | https://proofwiki.org/wiki/Cotangent_of_120_Degrees | https://proofwiki.org/wiki/Cotangent_of_120_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Angle plus Right Angle",
"Tangent of 30 Degrees"
] |
proofwiki-8007 | Cotangent of 135 Degrees | :$\cot 135 \degrees = \cot \dfrac {3 \pi} 4 = -1$ | {{begin-eqn}}
{{eqn | l = \cot 135 \degrees
| r = \cot \paren {90 \degrees + 45 \degrees}
| c =
}}
{{eqn | r = -\tan 45 \degrees
| c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = - 1
| c = Tangent of $45 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cot 135 \degrees = \cot \dfrac {3 \pi} 4 = -1$ | {{begin-eqn}}
{{eqn | l = \cot 135 \degrees
| r = \cot \paren {90 \degrees + 45 \degrees}
| c =
}}
{{eqn | r = -\tan 45 \degrees
| c = [[Cotangent of Angle plus Right Angle]]
}}
{{eqn | r = - 1
| c = [[Tangent of 45 Degrees|Tangent of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 135 Degrees | https://proofwiki.org/wiki/Cotangent_of_135_Degrees | https://proofwiki.org/wiki/Cotangent_of_135_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Angle plus Right Angle",
"Tangent of 45 Degrees"
] |
proofwiki-8008 | Cotangent of 150 Degrees | :$\cot 150 \degrees = \cot \dfrac {5 \pi} 6 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 150 \degrees
| r = \map \cot {90 \degrees + 60 \degrees}
| c =
}}
{{eqn | r = -\tan 60 \degrees
| c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = -\sqrt 3
| c = Tangent of $60 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cot 150 \degrees = \cot \dfrac {5 \pi} 6 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 150 \degrees
| r = \map \cot {90 \degrees + 60 \degrees}
| c =
}}
{{eqn | r = -\tan 60 \degrees
| c = [[Cotangent of Angle plus Right Angle]]
}}
{{eqn | r = -\sqrt 3
| c = [[Tangent of 60 Degrees|Tangent of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 150 Degrees | https://proofwiki.org/wiki/Cotangent_of_150_Degrees | https://proofwiki.org/wiki/Cotangent_of_150_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Angle plus Right Angle",
"Tangent of 60 Degrees"
] |
proofwiki-8009 | Cotangent of 165 Degrees | :$\cot 165 \degrees = \cot \dfrac {11 \pi} {12} = -\paren {2 + \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \cot 165 \degrees
| r = \map \cot {90 \degrees + 75 \degrees}
| c =
}}
{{eqn | r = -\tan 75 \degrees
| c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = -\paren {2 + \sqrt 3}
| c = {{tan|75}}
}}
{{end-eqn}}
{{qed}} | :$\cot 165 \degrees = \cot \dfrac {11 \pi} {12} = -\paren {2 + \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \cot 165 \degrees
| r = \map \cot {90 \degrees + 75 \degrees}
| c =
}}
{{eqn | r = -\tan 75 \degrees
| c = [[Cotangent of Angle plus Right Angle]]
}}
{{eqn | r = -\paren {2 + \sqrt 3}
| c = {{tan|75}}
}}
{{end-eqn}}
{{qed}} | Cotangent of 165 Degrees | https://proofwiki.org/wiki/Cotangent_of_165_Degrees | https://proofwiki.org/wiki/Cotangent_of_165_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Angle plus Right Angle"
] |
proofwiki-8010 | Cotangent of Straight Angle | :$\cot 180^\circ = \cot \pi$ 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 $\theta = \pi$, $\sin \theta =... | :$\cot 180^\circ = \cot \pi$ 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 $\theta = \pi$, $\sin \... | Cotangent of Straight Angle | https://proofwiki.org/wiki/Cotangent_of_Straight_Angle | https://proofwiki.org/wiki/Cotangent_of_Straight_Angle | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine"
] |
proofwiki-8011 | Cotangent of 195 Degrees | :$\cot 195 \degrees = \cot \dfrac {13 \pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 195 \degrees
| r = \map \cot {360 \degrees - 165 \degrees}
| c =
}}
{{eqn | r = -\cot 165 \degrees
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = 2 + \sqrt 3
| c = {{cot|165}}
}}
{{end-eqn}}
{{qed}} | :$\cot 195 \degrees = \cot \dfrac {13 \pi} {12} = 2 + \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 195 \degrees
| r = \map \cot {360 \degrees - 165 \degrees}
| c =
}}
{{eqn | r = -\cot 165 \degrees
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = 2 + \sqrt 3
| c = {{cot|165}}
}}
{{end-eqn}}
{{qed}} | Cotangent of 195 Degrees | https://proofwiki.org/wiki/Cotangent_of_195_Degrees | https://proofwiki.org/wiki/Cotangent_of_195_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle"
] |
proofwiki-8012 | Cotangent of 210 Degrees | :$\cot 210^\circ = \cot \dfrac {7 \pi} 6 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 210^\circ
| r = \cot \left({360^\circ - 150^\circ}\right)
| c =
}}
{{eqn | r = -\cot 150^\circ
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = \sqrt 3
| c = Cotangent of 150 Degrees
}}
{{end-eqn}}
{{qed}} | :$\cot 210^\circ = \cot \dfrac {7 \pi} 6 = \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 210^\circ
| r = \cot \left({360^\circ - 150^\circ}\right)
| c =
}}
{{eqn | r = -\cot 150^\circ
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = \sqrt 3
| c = [[Cotangent of 150 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 210 Degrees | https://proofwiki.org/wiki/Cotangent_of_210_Degrees | https://proofwiki.org/wiki/Cotangent_of_210_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 150 Degrees"
] |
proofwiki-8013 | Cotangent of 225 Degrees | :$\cot 225^\circ = \cot \dfrac {5 \pi} 4 = 1$ | {{begin-eqn}}
{{eqn | l = \cot 225^\circ
| r = \cot \left({360^\circ - 135^\circ}\right)
| c =
}}
{{eqn | r = -\cot 135^\circ
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = 1
| c = Cotangent of 135 Degrees
}}
{{end-eqn}}
{{qed}} | :$\cot 225^\circ = \cot \dfrac {5 \pi} 4 = 1$ | {{begin-eqn}}
{{eqn | l = \cot 225^\circ
| r = \cot \left({360^\circ - 135^\circ}\right)
| c =
}}
{{eqn | r = -\cot 135^\circ
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = 1
| c = [[Cotangent of 135 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 225 Degrees | https://proofwiki.org/wiki/Cotangent_of_225_Degrees | https://proofwiki.org/wiki/Cotangent_of_225_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 135 Degrees"
] |
proofwiki-8014 | Cotangent of 240 Degrees | :$\cot 240^\circ = \cot \dfrac {4 \pi} 3 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 240^\circ
| r = \cot \left({360^\circ - 120^\circ}\right)
| c =
}}
{{eqn | r = -\cot 120^\circ
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = \frac {\sqrt 3} 3
| c = Cotangent of 120 Degrees
}}
{{end-eqn}}
{{qed}} | :$\cot 240^\circ = \cot \dfrac {4 \pi} 3 = \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 240^\circ
| r = \cot \left({360^\circ - 120^\circ}\right)
| c =
}}
{{eqn | r = -\cot 120^\circ
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = \frac {\sqrt 3} 3
| c = [[Cotangent of 120 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 240 Degrees | https://proofwiki.org/wiki/Cotangent_of_240_Degrees | https://proofwiki.org/wiki/Cotangent_of_240_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 120 Degrees"
] |
proofwiki-8015 | Cotangent of 255 Degrees | :$\cot 255 \degrees = \cot \dfrac {17 \pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 255 \degrees
| r = \map \cot {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = -\cot 105 \degrees
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = 2 - \sqrt 3
| c = Cotangent of $105 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cot 255 \degrees = \cot \dfrac {17 \pi} {12} = 2 - \sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 255 \degrees
| r = \map \cot {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = -\cot 105 \degrees
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = 2 - \sqrt 3
| c = [[Cotangent of 105 Degrees|Cotangent of $105 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 255 Degrees | https://proofwiki.org/wiki/Cotangent_of_255_Degrees | https://proofwiki.org/wiki/Cotangent_of_255_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 105 Degrees"
] |
proofwiki-8016 | Cotangent of Three Right Angles | :$\cot 270 \degrees = \cot \dfrac {3 \pi} 2 = 0$ | {{begin-eqn}}
{{eqn | l = \cot 270 \degrees
| r = \map \cot {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = -\cot 90 \degrees
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = 0
| c = Cotangent of Right Angle
}}
{{end-eqn}}
{{qed}} | :$\cot 270 \degrees = \cot \dfrac {3 \pi} 2 = 0$ | {{begin-eqn}}
{{eqn | l = \cot 270 \degrees
| r = \map \cot {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = -\cot 90 \degrees
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = 0
| c = [[Cotangent of Right Angle]]
}}
{{end-eqn}}
{{qed}} | Cotangent of Three Right Angles | https://proofwiki.org/wiki/Cotangent_of_Three_Right_Angles | https://proofwiki.org/wiki/Cotangent_of_Three_Right_Angles | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of Right Angle"
] |
proofwiki-8017 | Cotangent of 285 Degrees | :$\cot 285 \degrees = \cot \dfrac {19 \pi} {12} = -\paren {2 - \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \cot 285 \degrees
| r = \cot \paren {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = -\cot 75^\circ
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\paren {2 - \sqrt 3}
| c = Cotangent of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cot 285 \degrees = \cot \dfrac {19 \pi} {12} = -\paren {2 - \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \cot 285 \degrees
| r = \cot \paren {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = -\cot 75^\circ
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = -\paren {2 - \sqrt 3}
| c = [[Cotangent of 75 Degrees|Cotangent of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 285 Degrees | https://proofwiki.org/wiki/Cotangent_of_285_Degrees | https://proofwiki.org/wiki/Cotangent_of_285_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 75 Degrees"
] |
proofwiki-8018 | Cotangent of 300 Degrees | :$\cot 300 \degrees = \cot \dfrac {5 \pi} 3 = - \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 300 \degrees
| r = \map \cot {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = -\cot 60 \degrees
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\frac {\sqrt 3} 3
| c = {{cot|60}}
}}
{{end-eqn}}
{{qed}} | :$\cot 300 \degrees = \cot \dfrac {5 \pi} 3 = - \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \cot 300 \degrees
| r = \map \cot {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = -\cot 60 \degrees
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = -\frac {\sqrt 3} 3
| c = {{cot|60}}
}}
{{end-eqn}}
{{qed}} | Cotangent of 300 Degrees | https://proofwiki.org/wiki/Cotangent_of_300_Degrees | https://proofwiki.org/wiki/Cotangent_of_300_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle"
] |
proofwiki-8019 | Cotangent of 315 Degrees | :$\cot 315 \degrees = \cot \dfrac {7 \pi} 4 = -1$ | {{begin-eqn}}
{{eqn | l = \cot 315 \degrees
| r = \map \cot {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = -\cot 45 \degrees
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = -1
| c = Cotangent of $45 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cot 315 \degrees = \cot \dfrac {7 \pi} 4 = -1$ | {{begin-eqn}}
{{eqn | l = \cot 315 \degrees
| r = \map \cot {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = -\cot 45 \degrees
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = -1
| c = [[Cotangent of 45 Degrees|Cotangent of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 315 Degrees | https://proofwiki.org/wiki/Cotangent_of_315_Degrees | https://proofwiki.org/wiki/Cotangent_of_315_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 45 Degrees"
] |
proofwiki-8020 | Cotangent of 330 Degrees | :$\cot 330^\circ = \cot \dfrac {11 \pi} 6 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 330^\circ
| r = \cot \left({360^\circ - 30^\circ}\right)
| c =
}}
{{eqn | r = -\cot 30^\circ
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\sqrt 3
| c = Cotangent of 30 Degrees
}}
{{end-eqn}}
{{qed}} | :$\cot 330^\circ = \cot \dfrac {11 \pi} 6 = -\sqrt 3$ | {{begin-eqn}}
{{eqn | l = \cot 330^\circ
| r = \cot \left({360^\circ - 30^\circ}\right)
| c =
}}
{{eqn | r = -\cot 30^\circ
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = -\sqrt 3
| c = [[Cotangent of 30 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 330 Degrees | https://proofwiki.org/wiki/Cotangent_of_330_Degrees | https://proofwiki.org/wiki/Cotangent_of_330_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 30 Degrees"
] |
proofwiki-8021 | Cotangent of 345 Degrees | :$\cot 345 \degrees = \cot \dfrac {23 \pi} {12} = -\paren {2 + \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \cot 345 \degrees
| r = \map \cot {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = -\cot 15 \degrees
| c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\paren {2 + \sqrt 3}
| c = Cotangent of $15 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\cot 345 \degrees = \cot \dfrac {23 \pi} {12} = -\paren {2 + \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \cot 345 \degrees
| r = \map \cot {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = -\cot 15 \degrees
| c = [[Cotangent of Conjugate Angle]]
}}
{{eqn | r = -\paren {2 + \sqrt 3}
| c = [[Cotangent of 15 Degrees|Cotangent of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cotangent of 345 Degrees | https://proofwiki.org/wiki/Cotangent_of_345_Degrees | https://proofwiki.org/wiki/Cotangent_of_345_Degrees | [
"Cotangent Function"
] | [] | [
"Cotangent of Conjugate Angle",
"Cotangent of 15 Degrees"
] |
proofwiki-8022 | Cotangent of Full Angle | :$\cot 360^\circ = \cot 2 \pi$ is undefined | From Cotangent is Cosine divided by Sine:
: $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
From Cosine of Full Angle:
: $\cos 2 \pi = 1$
From Sine of Full Angle:
: $\sin 2 \pi = 0$
Thus $\cot \theta$ is undefined at this value.
{{qed}} | :$\cot 360^\circ = \cot 2 \pi$ is undefined | From [[Cotangent is Cosine divided by Sine]]:
: $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
From [[Cosine of Full Angle]]:
: $\cos 2 \pi = 1$
From [[Sine of Full Angle]]:
: $\sin 2 \pi = 0$
Thus $\cot \theta$ is undefined at this value.
{{qed}} | Cotangent of Full Angle | https://proofwiki.org/wiki/Cotangent_of_Full_Angle | https://proofwiki.org/wiki/Cotangent_of_Full_Angle | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Full Angle",
"Sine of Full Angle"
] |
proofwiki-8023 | Secant of Zero | :$\sec 0 = 0$ | {{begin-eqn}}
{{eqn | l = \sec 0 \degrees
| r = \frac 1 {\cos 0 \degrees}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 1
| c = Cosine of Zero is One
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
{{qed}} | :$\sec 0 = 0$ | {{begin-eqn}}
{{eqn | l = \sec 0 \degrees
| r = \frac 1 {\cos 0 \degrees}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 1
| c = [[Cosine of Zero is One]]
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
{{qed}} | Secant of Zero | https://proofwiki.org/wiki/Secant_of_Zero | https://proofwiki.org/wiki/Secant_of_Zero | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of Zero is One"
] |
proofwiki-8024 | Secant of 15 Degrees | :$\sec 15 \degrees = \sec \dfrac \pi {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 15 \degrees
| r = \frac 1 {\cos 15 \degrees}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 4 {\sqrt 6 + \sqrt 2}
| c = {{cos|15}}
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 - \sqrt 2} } {\paren {\sqrt 6 + \sqrt 2} \paren {\sqrt 6 - \sqrt 2} }
| c = multipl... | :$\sec 15 \degrees = \sec \dfrac \pi {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 15 \degrees
| r = \frac 1 {\cos 15 \degrees}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 4 {\sqrt 6 + \sqrt 2}
| c = {{cos|15}}
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 - \sqrt 2} } {\paren {\sqrt 6 + \sqrt 2} \paren {\sqrt 6 - \sqrt 2} }
| c = mul... | Secant of 15 Degrees | https://proofwiki.org/wiki/Secant_of_15_Degrees | https://proofwiki.org/wiki/Secant_of_15_Degrees | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Difference of Two Squares"
] |
proofwiki-8025 | Secant of 30 Degrees | :$\sec 30 \degrees = \sec \dfrac \pi 6 = \dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 30 \degrees
| r = \frac 1 {\cos 30 \degrees}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 3} 2}
| c = {{cos|30}}
}}
{{eqn | r = \frac {2 \sqrt 3} 3
| c = multiplying top and bottom by $2 \sqrt 3$
}}
{{end-eqn}}
{{qed}} | :$\sec 30 \degrees = \sec \dfrac \pi 6 = \dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 30 \degrees
| r = \frac 1 {\cos 30 \degrees}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 3} 2}
| c = {{cos|30}}
}}
{{eqn | r = \frac {2 \sqrt 3} 3
| c = multiplying top and bottom by $2 \sqrt 3$
}}
{{end-eqn}}
{{qed}} | Secant of 30 Degrees | https://proofwiki.org/wiki/Secant_of_30_Degrees | https://proofwiki.org/wiki/Secant_of_30_Degrees | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine"
] |
proofwiki-8026 | Secant of 45 Degrees | :$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 45 \degrees
| r = \frac 1 {\cos 45 \degrees}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 2} 2}
| c = Cosine of $45 \degrees$
}}
{{eqn | r = \sqrt 2
| c = multiplying top and bottom by $2 \sqrt 2$
}}
{{end-eqn}}
{{qed}} | :$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 45 \degrees
| r = \frac 1 {\cos 45 \degrees}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 2} 2}
| c = [[Cosine of 45 Degrees|Cosine of $45 \degrees$]]
}}
{{eqn | r = \sqrt 2
| c = multiplying top and bottom by $2 \sqrt 2$
}}
{{end-e... | Secant of 45 Degrees | https://proofwiki.org/wiki/Secant_of_45_Degrees | https://proofwiki.org/wiki/Secant_of_45_Degrees | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of 45 Degrees"
] |
proofwiki-8027 | Secant of 60 Degrees | :$\sec 60 \degrees = \sec \dfrac \pi 3 = 2$ | {{begin-eqn}}
{{eqn | l = \sec 60 \degrees
| r = \frac 1 {\cos 60 \degrees}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac 1 2}
| c = Cosine of $60 \degrees$
}}
{{eqn | r = 2
| c = multiplying top and bottom by $2$
}}
{{end-eqn}}
{{qed}} | :$\sec 60 \degrees = \sec \dfrac \pi 3 = 2$ | {{begin-eqn}}
{{eqn | l = \sec 60 \degrees
| r = \frac 1 {\cos 60 \degrees}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\frac 1 2}
| c = [[Cosine of 60 Degrees|Cosine of $60 \degrees$]]
}}
{{eqn | r = 2
| c = multiplying top and bottom by $2$
}}
{{end-eqn}}
{{qed}} | Secant of 60 Degrees | https://proofwiki.org/wiki/Secant_of_60_Degrees | https://proofwiki.org/wiki/Secant_of_60_Degrees | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of 60 Degrees"
] |
proofwiki-8028 | Secant of 75 Degrees | :$\sec 75 \degrees = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 75 \degrees
| r = \frac 1 {\cos 75 \degrees}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{cos|75}}
}}
{{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}
| c = multiplying top and bottom by $4$
}}
{{eqn | r = \frac {4 \paren... | :$\sec 75 \degrees = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 75 \degrees
| r = \frac 1 {\cos 75 \degrees}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{cos|75}}
}}
{{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:De... | Secant of 75 Degrees | https://proofwiki.org/wiki/Secant_of_75_Degrees | https://proofwiki.org/wiki/Secant_of_75_Degrees | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-8029 | Secant of Right Angle | :$\sec 90 \degrees = \sec \dfrac \pi 2$ is undefined | From Secant is Reciprocal of Cosine:
:$\sec \theta = \dfrac 1 {\cos \theta}$
From Cosine of Right Angle:
:$\cos \dfrac \pi 2 = 0$
Thus $\sec \theta$ is undefined at this value.
{{qed}} | :$\sec 90 \degrees = \sec \dfrac \pi 2$ is undefined | From [[Secant is Reciprocal of Cosine]]:
:$\sec \theta = \dfrac 1 {\cos \theta}$
From [[Cosine of Right Angle]]:
:$\cos \dfrac \pi 2 = 0$
Thus $\sec \theta$ is undefined at this value.
{{qed}} | Secant of Right Angle | https://proofwiki.org/wiki/Secant_of_Right_Angle | https://proofwiki.org/wiki/Secant_of_Right_Angle | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of Right Angle"
] |
proofwiki-8030 | Secant of Angle plus Right Angle | : $\sec \left({x + \dfrac \pi 2}\right) = -\csc x$ | {{begin-eqn}}
{{eqn | l = \sec \left({x + \frac \pi 2}\right)
| r = \frac 1 {\cos \left({x + \frac \pi 2}\right)}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {- \sin x}
| c = Cosine of Angle plus Right Angle
}}
{{eqn | r = -\csc x
| c = Cosecant is Reciprocal of Sine
}}
{{end-eq... | : $\sec \left({x + \dfrac \pi 2}\right) = -\csc x$ | {{begin-eqn}}
{{eqn | l = \sec \left({x + \frac \pi 2}\right)
| r = \frac 1 {\cos \left({x + \frac \pi 2}\right)}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {- \sin x}
| c = [[Cosine of Angle plus Right Angle]]
}}
{{eqn | r = -\csc x
| c = [[Cosecant is Reciprocal of Sine]]... | Secant of Angle plus Right Angle | https://proofwiki.org/wiki/Secant_of_Angle_plus_Right_Angle | https://proofwiki.org/wiki/Secant_of_Angle_plus_Right_Angle | [
"Secant Function",
"Reduction Formulae (Trigonometry)"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of Angle plus Right Angle",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8031 | Cosecant of Angle plus Right Angle | : $\csc \left({x + \dfrac \pi 2}\right) = \sec x$ | {{begin-eqn}}
{{eqn | l = \csc \left({x + \frac \pi 2}\right)
| r = \frac 1 {\sin \left({x + \frac \pi 2}\right)}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\cos x}
| c = Sine of Angle plus Right Angle
}}
{{eqn | r = \sec x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{... | : $\csc \left({x + \dfrac \pi 2}\right) = \sec x$ | {{begin-eqn}}
{{eqn | l = \csc \left({x + \frac \pi 2}\right)
| r = \frac 1 {\sin \left({x + \frac \pi 2}\right)}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\cos x}
| c = [[Sine of Angle plus Right Angle]]
}}
{{eqn | r = \sec x
| c = [[Secant is Reciprocal of Cosine]]
}}
{... | Cosecant of Angle plus Right Angle | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Right_Angle | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Right_Angle | [
"Cosecant Function",
"Reduction Formulae (Trigonometry)"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Angle plus Right Angle",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8032 | Secant of Complement equals Cosecant | :$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$
where $\sec$ and $\csc$ are secant and cosecant respectively.
That is, the cosecant of an angle is the secant of its complement.
This relation is defined wherever $\sin \theta \ne 0$. | {{begin-eqn}}
{{eqn | l = \map \sec {\frac \pi 2 - \theta}
| r = \frac 1 {\map \cos {\frac \pi 2 - \theta} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\sin \theta}
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \csc \theta
| c = Cosecant is Reciprocal of Sine
}}
{{end-e... | :$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$
where $\sec$ and $\csc$ are [[Definition:Secant Function|secant]] and [[Definition:Cosecant|cosecant]] respectively.
That is, the [[Definition:Cosecant|cosecant]] of an [[Definition:Angle|angle]] is the [[Definition:Secant Function|secant]] of ... | {{begin-eqn}}
{{eqn | l = \map \sec {\frac \pi 2 - \theta}
| r = \frac 1 {\map \cos {\frac \pi 2 - \theta} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\sin \theta}
| c = [[Cosine of Complement equals Sine]]
}}
{{eqn | r = \csc \theta
| c = [[Cosecant is Reciprocal of Sine]... | Secant of Complement equals Cosecant | https://proofwiki.org/wiki/Secant_of_Complement_equals_Cosecant | https://proofwiki.org/wiki/Secant_of_Complement_equals_Cosecant | [
"Secant of Complement equals Cosecant",
"Secant Function",
"Cosecant Function",
"Complementary Angles"
] | [
"Definition:Secant Function",
"Definition:Cosecant",
"Definition:Cosecant",
"Definition:Angle",
"Definition:Secant Function",
"Definition:Complementary Angles"
] | [
"Secant is Reciprocal of Cosine",
"Cosine of Complement equals Sine",
"Cosecant is Reciprocal of Sine",
"Sine of Integer Multiple of Pi"
] |
proofwiki-8033 | Cosecant of Complement equals Secant | :$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
where $\csc$ and $\sec$ are cosecant and secant respectively.
That is, the secant of an angle is the cosecant of its complement.
This relation is defined wherever $\cos \theta \ne 0$. | {{begin-eqn}}
{{eqn | l = \map \csc {\frac \pi 2 - \theta}
| r = \frac 1 {\map \sin {\frac \pi 2 - \theta} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\cos \theta}
| c = Sine of Complement equals Cosine
}}
{{eqn | r = \sec \theta
| c = Secant is Reciprocal of Cosine
}}
{{end-e... | :$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
where $\csc$ and $\sec$ are [[Definition:Cosecant|cosecant]] and [[Definition:Secant Function|secant]] respectively.
That is, the [[Definition:Secant Function|secant]] of an [[Definition:Angle|angle]] is the [[Definition... | {{begin-eqn}}
{{eqn | l = \map \csc {\frac \pi 2 - \theta}
| r = \frac 1 {\map \sin {\frac \pi 2 - \theta} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\cos \theta}
| c = [[Sine of Complement equals Cosine]]
}}
{{eqn | r = \sec \theta
| c = [[Secant is Reciprocal of Cosine]... | Cosecant of Complement equals Secant | https://proofwiki.org/wiki/Cosecant_of_Complement_equals_Secant | https://proofwiki.org/wiki/Cosecant_of_Complement_equals_Secant | [
"Secant Function",
"Cosecant Function",
"Complementary Angles"
] | [
"Definition:Cosecant",
"Definition:Secant Function",
"Definition:Secant Function",
"Definition:Angle",
"Definition:Cosecant",
"Definition:Complementary Angles"
] | [
"Cosecant is Reciprocal of Sine",
"Sine of Complement equals Cosine",
"Secant is Reciprocal of Cosine",
"Cosine of Half-Integer Multiple of Pi"
] |
proofwiki-8034 | Secant of Supplementary Angle | :$\map \sec {\pi - \theta} = -\sec \theta$
where $\sec$ denotes secant.
That is, the secant of an angle is the negative of its supplement. | {{begin-eqn}}
{{eqn | l = \map \sec {\pi - \theta}
| r = \frac 1 {\map \cos {\pi - \theta} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {-\cos \theta}
| c = Cosine of Supplementary Angle
}}
{{eqn | r = -\sec \theta
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map \sec {\pi - \theta} = -\sec \theta$
where $\sec$ denotes [[Definition:Secant Function|secant]].
That is, the [[Definition:Secant Function|secant]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Supplement of Angle|supplement]]. | {{begin-eqn}}
{{eqn | l = \map \sec {\pi - \theta}
| r = \frac 1 {\map \cos {\pi - \theta} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {-\cos \theta}
| c = [[Cosine of Supplementary Angle]]
}}
{{eqn | r = -\sec \theta
| c = [[Secant is Reciprocal of Cosine]]
}}
{{end-eqn}}... | Secant of Supplementary Angle | https://proofwiki.org/wiki/Secant_of_Supplementary_Angle | https://proofwiki.org/wiki/Secant_of_Supplementary_Angle | [
"Secant Function",
"Supplementary Angles"
] | [
"Definition:Secant Function",
"Definition:Secant Function",
"Definition:Angle",
"Definition:Supplementary Angles"
] | [
"Secant is Reciprocal of Cosine",
"Cosine of Supplementary Angle",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8035 | Cosecant of Supplementary Angle | :$\map \csc {\pi - \theta} = \csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle equals its supplement. | {{begin-eqn}}
{{eqn | l = \map \csc {\pi - \theta}
| r = \frac 1 {\map \sin {\pi - \theta} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\sin \theta}
| c = Sine of Supplementary Angle
}}
{{eqn | r = \csc \theta
| c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}} | :$\map \csc {\pi - \theta} = \csc \theta$
where $\csc$ denotes [[Definition:Cosecant|cosecant]].
That is, the [[Definition:Cosecant|cosecant]] of an [[Definition:Angle|angle]] equals its [[Definition:Supplement of Angle|supplement]]. | {{begin-eqn}}
{{eqn | l = \map \csc {\pi - \theta}
| r = \frac 1 {\map \sin {\pi - \theta} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\sin \theta}
| c = [[Sine of Supplementary Angle]]
}}
{{eqn | r = \csc \theta
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{end-eqn}}
{{q... | Cosecant of Supplementary Angle | https://proofwiki.org/wiki/Cosecant_of_Supplementary_Angle | https://proofwiki.org/wiki/Cosecant_of_Supplementary_Angle | [
"Cosecant of Supplementary Angle",
"Cosecant Function",
"Supplementary Angles"
] | [
"Definition:Cosecant",
"Definition:Cosecant",
"Definition:Angle",
"Definition:Supplementary Angles"
] | [
"Cosecant is Reciprocal of Sine",
"Sine of Supplementary Angle",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8036 | Secant of Conjugate Angle | :$\map \sec {2 \pi - \theta} = \sec \theta$
where $\sec$ denotes secant.
That is, the secant of an angle equals its conjugate. | {{begin-eqn}}
{{eqn | l = \map \sec {2 \pi - \theta}
| r = \frac 1 {\map \cos {2 \pi - \theta} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\cos \theta}
| c = Cosine of Conjugate Angle
}}
{{eqn | r = \sec \theta
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map \sec {2 \pi - \theta} = \sec \theta$
where $\sec$ denotes [[Definition:Secant Function|secant]].
That is, the [[Definition:Secant Function|secant]] of an [[Definition:Angle|angle]] equals its [[Definition:Conjugate Angle|conjugate]]. | {{begin-eqn}}
{{eqn | l = \map \sec {2 \pi - \theta}
| r = \frac 1 {\map \cos {2 \pi - \theta} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\cos \theta}
| c = [[Cosine of Conjugate Angle]]
}}
{{eqn | r = \sec \theta
| c = [[Secant is Reciprocal of Cosine]]
}}
{{end-eqn}}
{{... | Secant of Conjugate Angle | https://proofwiki.org/wiki/Secant_of_Conjugate_Angle | https://proofwiki.org/wiki/Secant_of_Conjugate_Angle | [
"Secant Function",
"Conjugate Angles"
] | [
"Definition:Secant Function",
"Definition:Secant Function",
"Definition:Angle",
"Definition:Conjugate Angles"
] | [
"Secant is Reciprocal of Cosine",
"Cosine of Conjugate Angle",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8037 | Cosecant of Conjugate Angle | :$\map \csc {2 \pi - \theta} = -\csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle is the negative of its conjugate. | {{begin-eqn}}
{{eqn | l = \map \csc {2 \pi - \theta}
| r = \frac 1 {\map \sin {2 \pi - \theta} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\sin \theta}
| c = Cosine of Conjugate Angle and Sine of Conjugate Angle
}}
{{eqn | r = -\csc \theta
| c = Cosecant is Reciprocal of Sine... | :$\map \csc {2 \pi - \theta} = -\csc \theta$
where $\csc$ denotes [[Definition:Cosecant|cosecant]].
That is, the [[Definition:Cosecant|cosecant]] of an [[Definition:Angle|angle]] is the negative of its [[Definition:Conjugate Angle|conjugate]]. | {{begin-eqn}}
{{eqn | l = \map \csc {2 \pi - \theta}
| r = \frac 1 {\map \sin {2 \pi - \theta} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {-\sin \theta}
| c = [[Cosine of Conjugate Angle]] and [[Sine of Conjugate Angle]]
}}
{{eqn | r = -\csc \theta
| c = [[Cosecant is Reci... | Cosecant of Conjugate Angle | https://proofwiki.org/wiki/Cosecant_of_Conjugate_Angle | https://proofwiki.org/wiki/Cosecant_of_Conjugate_Angle | [
"Cosecant Function",
"Conjugate Angles"
] | [
"Definition:Cosecant",
"Definition:Cosecant",
"Definition:Angle",
"Definition:Conjugate Angles"
] | [
"Cosecant is Reciprocal of Sine",
"Cosine of Conjugate Angle",
"Sine of Conjugate Angle",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8038 | Secant of 105 Degrees | :$\sec 105 \degrees = \sec \dfrac {7 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 105 \degrees
| r = \map \sec {90 \degrees + 15 \degrees}
| c =
}}
{{eqn | r = -\csc 15 \degrees
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
| c = Cosecant of $15 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 105 \degrees = \sec \dfrac {7 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 105 \degrees
| r = \map \sec {90 \degrees + 15 \degrees}
| c =
}}
{{eqn | r = -\csc 15 \degrees
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
| c = [[Cosecant of 15 Degrees|Cosecant of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 105 Degrees | https://proofwiki.org/wiki/Secant_of_105_Degrees | https://proofwiki.org/wiki/Secant_of_105_Degrees | [
"Secant Function"
] | [] | [
"Secant of Angle plus Right Angle",
"Cosecant of 15 Degrees"
] |
proofwiki-8039 | Secant of 120 Degrees | :$\sec 120 \degrees = \sec \dfrac {2 \pi} 3 = -2$ | {{begin-eqn}}
{{eqn | l = \sec 120 \degrees
| r = \map \sec {90 \degrees + 30 \degrees}
| c =
}}
{{eqn | r = -\csc 30 \degrees
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = -2
| c = Cosecant of $30 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 120 \degrees = \sec \dfrac {2 \pi} 3 = -2$ | {{begin-eqn}}
{{eqn | l = \sec 120 \degrees
| r = \map \sec {90 \degrees + 30 \degrees}
| c =
}}
{{eqn | r = -\csc 30 \degrees
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = -2
| c = [[Cosecant of 30 Degrees|Cosecant of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 120 Degrees | https://proofwiki.org/wiki/Secant_of_120_Degrees | https://proofwiki.org/wiki/Secant_of_120_Degrees | [
"Secant Function"
] | [] | [
"Secant of Angle plus Right Angle",
"Cosecant of 30 Degrees"
] |
proofwiki-8040 | Secant of 135 Degrees | :$\sec 135 \degrees = \sec \dfrac {3 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 135 \degrees
| r = \map \sec {90 \degrees + 45 \degrees}
| c =
}}
{{eqn | r = -\csc 45 \degrees
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\sqrt 2
| c = Cosecant of $45 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 135 \degrees = \sec \dfrac {3 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 135 \degrees
| r = \map \sec {90 \degrees + 45 \degrees}
| c =
}}
{{eqn | r = -\csc 45 \degrees
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = -\sqrt 2
| c = [[Cosecant of 45 Degrees|Cosecant of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 135 Degrees | https://proofwiki.org/wiki/Secant_of_135_Degrees | https://proofwiki.org/wiki/Secant_of_135_Degrees | [
"Secant Function"
] | [] | [
"Secant of Angle plus Right Angle",
"Cosecant of 45 Degrees"
] |
proofwiki-8041 | Secant of 150 Degrees | :$\sec 150 \degrees = \sec \dfrac {5 \pi} 6 = -\dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 150 \degrees
| r = \map \sec {90 \degrees + 60 \degrees}
| c =
}}
{{eqn | r = -\csc 60 \degrees
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\dfrac {2 \sqrt 3} 3
| c = Cosecant of $60 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 150 \degrees = \sec \dfrac {5 \pi} 6 = -\dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 150 \degrees
| r = \map \sec {90 \degrees + 60 \degrees}
| c =
}}
{{eqn | r = -\csc 60 \degrees
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = -\dfrac {2 \sqrt 3} 3
| c = [[Cosecant of 60 Degrees|Cosecant of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 150 Degrees | https://proofwiki.org/wiki/Secant_of_150_Degrees | https://proofwiki.org/wiki/Secant_of_150_Degrees | [
"Secant Function"
] | [] | [
"Secant of Angle plus Right Angle",
"Cosecant of 60 Degrees"
] |
proofwiki-8042 | Secant of 165 Degrees | :$\sec 165 \degrees = \sec \dfrac {11 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 165 \degrees
| r = \map \sec {90 \degrees + 75 \degrees}
| c =
}}
{{eqn | r = -\csc 75 \degrees
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = Cosecant of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 165 \degrees = \sec \dfrac {11 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 165 \degrees
| r = \map \sec {90 \degrees + 75 \degrees}
| c =
}}
{{eqn | r = -\csc 75 \degrees
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = [[Cosecant of 75 Degrees|Cosecant of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 165 Degrees | https://proofwiki.org/wiki/Secant_of_165_Degrees | https://proofwiki.org/wiki/Secant_of_165_Degrees | [
"Secant Function"
] | [] | [
"Secant of Angle plus Right Angle",
"Cosecant of 75 Degrees"
] |
proofwiki-8043 | Secant of Straight Angle | :$\sec 180 \degrees = \sec \pi = -1$ | {{begin-eqn}}
{{eqn | l = \sec 180 \degrees
| r = \map \sec {90 \degrees + 90 \degrees}
| c =
}}
{{eqn | r = -\csc 90 \degrees
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = -1
| c = Cosecant of Right Angle
}}
{{end-eqn}}
{{qed}} | :$\sec 180 \degrees = \sec \pi = -1$ | {{begin-eqn}}
{{eqn | l = \sec 180 \degrees
| r = \map \sec {90 \degrees + 90 \degrees}
| c =
}}
{{eqn | r = -\csc 90 \degrees
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = -1
| c = [[Cosecant of Right Angle]]
}}
{{end-eqn}}
{{qed}} | Secant of Straight Angle | https://proofwiki.org/wiki/Secant_of_Straight_Angle | https://proofwiki.org/wiki/Secant_of_Straight_Angle | [
"Secant Function"
] | [] | [
"Secant of Angle plus Right Angle",
"Cosecant of Right Angle"
] |
proofwiki-8044 | Secant of 195 Degrees | :$\sec 195 \degrees = \sec \dfrac {13 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 195 \degrees
| r = \map \sec {360 \degrees - 165 \degrees}
| c =
}}
{{eqn | r = \sec 165 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = Secant of $165 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 195 \degrees = \sec \dfrac {13 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 195 \degrees
| r = \map \sec {360 \degrees - 165 \degrees}
| c =
}}
{{eqn | r = \sec 165 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = [[Secant of 165 Degrees|Secant of $165 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 195 Degrees | https://proofwiki.org/wiki/Secant_of_195_Degrees | https://proofwiki.org/wiki/Secant_of_195_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 165 Degrees"
] |
proofwiki-8045 | Secant of 210 Degrees | :$\sec 210 \degrees = \sec \dfrac {7 \pi} 6 = -2 \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 210 \degrees
| r = \map \sec {360 \degrees - 150 \degrees}
| c =
}}
{{eqn | r = \sec 150 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = -2 \frac {\sqrt 3} 3
| c = Secant of $150 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 210 \degrees = \sec \dfrac {7 \pi} 6 = -2 \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 210 \degrees
| r = \map \sec {360 \degrees - 150 \degrees}
| c =
}}
{{eqn | r = \sec 150 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = -2 \frac {\sqrt 3} 3
| c = [[Secant of 150 Degrees|Secant of $150 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 210 Degrees | https://proofwiki.org/wiki/Secant_of_210_Degrees | https://proofwiki.org/wiki/Secant_of_210_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 150 Degrees"
] |
proofwiki-8046 | Secant of 225 Degrees | :$\sec 225 \degrees = \sec \dfrac {5 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 225 \degrees
| r = \map \sec {360 \degrees - 135 \degrees}
| c =
}}
{{eqn | r = \sec 135 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = -\sqrt 2
| c = Secant of $135 degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 225 \degrees = \sec \dfrac {5 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 225 \degrees
| r = \map \sec {360 \degrees - 135 \degrees}
| c =
}}
{{eqn | r = \sec 135 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = -\sqrt 2
| c = [[Secant of 135 Degrees|Secant of $135 degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 225 Degrees | https://proofwiki.org/wiki/Secant_of_225_Degrees | https://proofwiki.org/wiki/Secant_of_225_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 135 Degrees"
] |
proofwiki-8047 | Secant of 240 Degrees | :$\sec 240 \degrees = \sec \dfrac {4 \pi} 3 = - 2$ | {{begin-eqn}}
{{eqn | l = \sec 240 \degrees
| r = \map \sec {360 \degrees - 120 \degrees}
| c =
}}
{{eqn | r = \sec 120 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = -2
| c = Secant of $120 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 240 \degrees = \sec \dfrac {4 \pi} 3 = - 2$ | {{begin-eqn}}
{{eqn | l = \sec 240 \degrees
| r = \map \sec {360 \degrees - 120 \degrees}
| c =
}}
{{eqn | r = \sec 120 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = -2
| c = [[Secant of 120 Degrees|Secant of $120 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 240 Degrees | https://proofwiki.org/wiki/Secant_of_240_Degrees | https://proofwiki.org/wiki/Secant_of_240_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 120 Degrees"
] |
proofwiki-8048 | Secant of 255 Degrees | :$\sec 255 \degrees = \sec \dfrac {17 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 255 \degrees
| r = \map \sec {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = \sec 105 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
| c = Secant of $105 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 255 \degrees = \sec \dfrac {17 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sec 255 \degrees
| r = \map \sec {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = \sec 105 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
| c = [[Secant of 105 Degrees|Secant of $105 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 255 Degrees | https://proofwiki.org/wiki/Secant_of_255_Degrees | https://proofwiki.org/wiki/Secant_of_255_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 105 Degrees"
] |
proofwiki-8049 | Secant of Three Right Angles | :$\sec 270 \degrees = \sec \dfrac {3 \pi} 2$ is undefined | {{begin-eqn}}
{{eqn | l = \sec 270 \degrees
| r = \map \sec {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = \sec 90 \degrees
| c = Secant of Conjugate Angle
}}
{{end-eqn}}
But from Secant of Right Angle, $\sec 90 \degrees$ is undefined.
{{qed}} | :$\sec 270 \degrees = \sec \dfrac {3 \pi} 2$ is undefined | {{begin-eqn}}
{{eqn | l = \sec 270 \degrees
| r = \map \sec {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = \sec 90 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{end-eqn}}
But from [[Secant of Right Angle]], $\sec 90 \degrees$ is undefined.
{{qed}} | Secant of Three Right Angles | https://proofwiki.org/wiki/Secant_of_Three_Right_Angles | https://proofwiki.org/wiki/Secant_of_Three_Right_Angles | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of Right Angle"
] |
proofwiki-8050 | Secant of 285 Degrees | :$\sec 285 \degrees = \sec \dfrac {19 \pi} {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 285 \degrees
| r = \map \sec {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = \sec 75 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = \sqrt 6 + \sqrt 2
| c = Secant of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 285 \degrees = \sec \dfrac {19 \pi} {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 285 \degrees
| r = \map \sec {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = \sec 75 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = \sqrt 6 + \sqrt 2
| c = [[Secant of 75 Degrees|Secant of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 285 Degrees | https://proofwiki.org/wiki/Secant_of_285_Degrees | https://proofwiki.org/wiki/Secant_of_285_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 75 Degrees"
] |
proofwiki-8051 | Secant of 300 Degrees | :$\sec 300 \degrees = \sec \dfrac {5 \pi} 3 = 2$ | {{begin-eqn}}
{{eqn | l = \sec 300 \degrees
| r = \map \sec {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = \sec 60 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = 2
| c = Secant of $60 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 300 \degrees = \sec \dfrac {5 \pi} 3 = 2$ | {{begin-eqn}}
{{eqn | l = \sec 300 \degrees
| r = \map \sec {360 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = \sec 60 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = 2
| c = [[Secant of 60 Degrees|Secant of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 300 Degrees | https://proofwiki.org/wiki/Secant_of_300_Degrees | https://proofwiki.org/wiki/Secant_of_300_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 60 Degrees"
] |
proofwiki-8052 | Secant of 315 Degrees | :$\sec 315 \degrees = \sec \dfrac {7 \pi} 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 315 \degrees
| r = \map \sec {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = \sec 45 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = \sqrt 2
| c = {{sec|45}}
}}
{{end-eqn}}
{{qed}} | :$\sec 315 \degrees = \sec \dfrac {7 \pi} 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 315 \degrees
| r = \map \sec {360 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = \sec 45 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = \sqrt 2
| c = {{sec|45}}
}}
{{end-eqn}}
{{qed}} | Secant of 315 Degrees | https://proofwiki.org/wiki/Secant_of_315_Degrees | https://proofwiki.org/wiki/Secant_of_315_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle"
] |
proofwiki-8053 | Secant of 330 Degrees | :$\sec 330 \degrees = \sec \dfrac {11 \pi} 6 = 2 \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 330 \degrees
| r = \map \sec {360 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \sec 30 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = 2 \frac {\sqrt 3} 3
| c = Secant of $30 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 330 \degrees = \sec \dfrac {11 \pi} 6 = 2 \dfrac {\sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \sec 330 \degrees
| r = \map \sec {360 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \sec 30 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = 2 \frac {\sqrt 3} 3
| c = [[Secant of 30 Degrees|Secant of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 330 Degrees | https://proofwiki.org/wiki/Secant_of_330_Degrees | https://proofwiki.org/wiki/Secant_of_330_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 30 Degrees"
] |
proofwiki-8054 | Secant of 345 Degrees | :$\sec 345 \degrees = \sec \dfrac {23 \pi} {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 345 \degrees
| r = \map \sec {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = \sec 15 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = \sqrt 6 - \sqrt 2
| c = Secant of $15 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\sec 345 \degrees = \sec \dfrac {23 \pi} {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \sec 345 \degrees
| r = \map \sec {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = \sec 15 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = \sqrt 6 - \sqrt 2
| c = [[Secant of 15 Degrees|Secant of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Secant of 345 Degrees | https://proofwiki.org/wiki/Secant_of_345_Degrees | https://proofwiki.org/wiki/Secant_of_345_Degrees | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of 15 Degrees"
] |
proofwiki-8055 | Secant of Full Angle | :$\sec 360 \degrees = \sec 2 \pi = 1$ | {{begin-eqn}}
{{eqn | l = \sec 360 \degrees
| r = \map \sec {360 \degrees - 0 \degrees}
| c =
}}
{{eqn | r = \sec 0 \degrees
| c = Secant of Conjugate Angle
}}
{{eqn | r = 1
| c = Secant of Zero
}}
{{end-eqn}}
{{qed}} | :$\sec 360 \degrees = \sec 2 \pi = 1$ | {{begin-eqn}}
{{eqn | l = \sec 360 \degrees
| r = \map \sec {360 \degrees - 0 \degrees}
| c =
}}
{{eqn | r = \sec 0 \degrees
| c = [[Secant of Conjugate Angle]]
}}
{{eqn | r = 1
| c = [[Secant of Zero]]
}}
{{end-eqn}}
{{qed}} | Secant of Full Angle | https://proofwiki.org/wiki/Secant_of_Full_Angle | https://proofwiki.org/wiki/Secant_of_Full_Angle | [
"Secant Function"
] | [] | [
"Secant of Conjugate Angle",
"Secant of Zero"
] |
proofwiki-8056 | Cosecant of Zero | :$\csc 0$ is undefined | From Cosecant is Reciprocal of Sine:
: $\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Zero is Zero:
: $\sin 0 = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}} | :$\csc 0$ is undefined | From [[Cosecant is Reciprocal of Sine]]:
: $\csc \theta = \dfrac 1 {\sin \theta}$
From [[Sine of Zero is Zero]]:
: $\sin 0 = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}} | Cosecant of Zero | https://proofwiki.org/wiki/Cosecant_of_Zero | https://proofwiki.org/wiki/Cosecant_of_Zero | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Zero is Zero"
] |
proofwiki-8057 | Cosecant of 15 Degrees | :$\csc 15 \degrees = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 15 \degrees
| r = \frac 1 {\sin 15 \degrees}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{sin|15}}
}}
{{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}
| c = multiplying top and bottom by $4$
}}
{{eqn | r = \frac {4 \paren... | :$\csc 15 \degrees = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 15 \degrees
| r = \frac 1 {\sin 15 \degrees}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}
| c = {{sin|15}}
}}
{{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:De... | Cosecant of 15 Degrees | https://proofwiki.org/wiki/Cosecant_of_15_Degrees | https://proofwiki.org/wiki/Cosecant_of_15_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-8058 | Cosecant of 30 Degrees | :$\csc 30^\circ = \csc \dfrac \pi 6 = 2$ | {{begin-eqn}}
{{eqn | l = \csc 30^\circ
| r = \frac 1 {\sin 30^\circ}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac 1 2}
| c = Sine of $30^\circ$
}}
{{eqn | r = 2
| c = multiplying top and bottom by $2$
}}
{{end-eqn}}
{{qed}} | :$\csc 30^\circ = \csc \dfrac \pi 6 = 2$ | {{begin-eqn}}
{{eqn | l = \csc 30^\circ
| r = \frac 1 {\sin 30^\circ}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\frac 1 2}
| c = [[Sine of 30 Degrees|Sine of $30^\circ$]]
}}
{{eqn | r = 2
| c = multiplying top and bottom by $2$
}}
{{end-eqn}}
{{qed}} | Cosecant of 30 Degrees | https://proofwiki.org/wiki/Cosecant_of_30_Degrees | https://proofwiki.org/wiki/Cosecant_of_30_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of 30 Degrees"
] |
proofwiki-8059 | Cosecant of 45 Degrees | :$\csc 45 \degrees = \csc \dfrac \pi 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 45 \degrees
| r = \frac 1 {\sin 45 \degrees}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 2} 2}
| c = {{sin|45}}
}}
{{eqn | r = \sqrt 2
| c = multiplying top and bottom by $2 \sqrt 2$
}}
{{end-eqn}}
{{qed}} | :$\csc 45 \degrees = \csc \dfrac \pi 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 45 \degrees
| r = \frac 1 {\sin 45 \degrees}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 2} 2}
| c = {{sin|45}}
}}
{{eqn | r = \sqrt 2
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $2 \sqrt... | Cosecant of 45 Degrees | https://proofwiki.org/wiki/Cosecant_of_45_Degrees | https://proofwiki.org/wiki/Cosecant_of_45_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8060 | Cosecant of 60 Degrees | :$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 60^\circ
| r = \frac 1 {\sin 60^\circ}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 3} 2}
| c = Sine of 60 Degrees
}}
{{eqn | r = \frac {2 \sqrt 3} 3
| c = multiplying top and bottom by $2 \sqrt 3$
}}
{{end-eqn}}
{{qed}} | :$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 60^\circ
| r = \frac 1 {\sin 60^\circ}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 3} 2}
| c = [[Sine of 60 Degrees]]
}}
{{eqn | r = \frac {2 \sqrt 3} 3
| c = multiplying top and bottom by $2 \sqrt 3$
}}
{{end-eqn}}
{{qed}} | Cosecant of 60 Degrees | https://proofwiki.org/wiki/Cosecant_of_60_Degrees | https://proofwiki.org/wiki/Cosecant_of_60_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of 60 Degrees"
] |
proofwiki-8061 | Cosecant of 75 Degrees | :$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 75^\circ
| r = \frac 1 {\sin 75 \degrees}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 + \sqrt 2} 4}
| c = {{sin|75}}
}}
{{eqn | r = \frac 4 {\sqrt 6 + \sqrt 2}
| c = multiplying top and bottom by $4$
}}
{{eqn | r = \frac {4 \paren {\... | :$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 75^\circ
| r = \frac 1 {\sin 75 \degrees}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 + \sqrt 2} 4}
| c = {{sin|75}}
}}
{{eqn | r = \frac 4 {\sqrt 6 + \sqrt 2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denom... | Cosecant of 75 Degrees | https://proofwiki.org/wiki/Cosecant_of_75_Degrees | https://proofwiki.org/wiki/Cosecant_of_75_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-8062 | Cosecant of Right Angle | :$\csc 90 \degrees = \csc \dfrac \pi 2 = 1$ | {{begin-eqn}}
{{eqn | l = \csc 90 \degrees
| r = \frac 1 {\sin 90 \degrees}
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 1
| c = Sine of Right Angle
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
{{qed}} | :$\csc 90 \degrees = \csc \dfrac \pi 2 = 1$ | {{begin-eqn}}
{{eqn | l = \csc 90 \degrees
| r = \frac 1 {\sin 90 \degrees}
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 1
| c = [[Sine of Right Angle]]
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
{{qed}} | Cosecant of Right Angle | https://proofwiki.org/wiki/Cosecant_of_Right_Angle | https://proofwiki.org/wiki/Cosecant_of_Right_Angle | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Right Angle"
] |
proofwiki-8063 | Cosecant of 105 Degrees | :$\csc 105 \degrees = \csc \dfrac {7 \pi} {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 105 \degrees
| r = \map \csc {180 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = \csc 75 \degrees
| c = Cosecant of Supplementary Angle
}}
{{eqn | r = \sqrt 6 - \sqrt 2
| c = Cosecant of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 105 \degrees = \csc \dfrac {7 \pi} {12} = \sqrt 6 - \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 105 \degrees
| r = \map \csc {180 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = \csc 75 \degrees
| c = [[Cosecant of Supplementary Angle]]
}}
{{eqn | r = \sqrt 6 - \sqrt 2
| c = [[Cosecant of 75 Degrees|Cosecant of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 105 Degrees | https://proofwiki.org/wiki/Cosecant_of_105_Degrees | https://proofwiki.org/wiki/Cosecant_of_105_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Supplementary Angle",
"Cosecant of 75 Degrees"
] |
proofwiki-8064 | Cosecant of 120 Degrees | :$\csc 120 \degrees = \csc \dfrac {2 \pi} 3 = \dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 120 \degrees
| r = \map \csc {180 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = \csc 60 \degrees
| c = Cosecant of Supplementary Angle
}}
{{eqn | r = \frac {2 \sqrt 3} 3
| c = Cosecant of $60 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 120 \degrees = \csc \dfrac {2 \pi} 3 = \dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 120 \degrees
| r = \map \csc {180 \degrees - 60 \degrees}
| c =
}}
{{eqn | r = \csc 60 \degrees
| c = [[Cosecant of Supplementary Angle]]
}}
{{eqn | r = \frac {2 \sqrt 3} 3
| c = [[Cosecant of 60 Degrees|Cosecant of $60 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 120 Degrees | https://proofwiki.org/wiki/Cosecant_of_120_Degrees | https://proofwiki.org/wiki/Cosecant_of_120_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Supplementary Angle",
"Cosecant of 60 Degrees"
] |
proofwiki-8065 | Cosecant of 135 Degrees | :$\csc 135 \degrees = \csc \dfrac {3 \pi} 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 135 \degrees
| r = \map \csc {180 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = \csc 45 \degrees
| c = Cosecant of Supplementary Angle
}}
{{eqn | r = \sqrt 2
| c = Cosecant of $45 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 135 \degrees = \csc \dfrac {3 \pi} 4 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 135 \degrees
| r = \map \csc {180 \degrees - 45 \degrees}
| c =
}}
{{eqn | r = \csc 45 \degrees
| c = [[Cosecant of Supplementary Angle]]
}}
{{eqn | r = \sqrt 2
| c = [[Cosecant of 45 Degrees|Cosecant of $45 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 135 Degrees | https://proofwiki.org/wiki/Cosecant_of_135_Degrees | https://proofwiki.org/wiki/Cosecant_of_135_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Supplementary Angle",
"Cosecant of 45 Degrees"
] |
proofwiki-8066 | Cosecant of 150 Degrees | :$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 150 \degrees
| r = \map \csc {180 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \csc 30 \degrees
| c = Cosecant of Supplementary Angle
}}
{{eqn | r = 2
| c = Cosecant of $30 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 150 \degrees
| r = \map \csc {180 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = \csc 30 \degrees
| c = [[Cosecant of Supplementary Angle]]
}}
{{eqn | r = 2
| c = [[Cosecant of 30 Degrees|Cosecant of $30 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 150 Degrees | https://proofwiki.org/wiki/Cosecant_of_150_Degrees | https://proofwiki.org/wiki/Cosecant_of_150_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Supplementary Angle",
"Cosecant of 30 Degrees"
] |
proofwiki-8067 | Cosecant of 165 Degrees | :$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 165 \degrees
| r = \map \csc {180 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = \csc 15 \degrees
| c = Cosecant of Supplementary Angle
}}
{{eqn | r = \sqrt 6 + \sqrt 2
| c = Cosecant of 15 Degrees
}}
{{end-eqn}}
{{qed}} | :$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 165 \degrees
| r = \map \csc {180 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = \csc 15 \degrees
| c = [[Cosecant of Supplementary Angle]]
}}
{{eqn | r = \sqrt 6 + \sqrt 2
| c = [[Cosecant of 15 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 165 Degrees | https://proofwiki.org/wiki/Cosecant_of_165_Degrees | https://proofwiki.org/wiki/Cosecant_of_165_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Supplementary Angle",
"Cosecant of 15 Degrees"
] |
proofwiki-8068 | Cosecant of Straight Angle | :$\csc 180 \degrees = \csc \pi$ is undefined | From Cosecant is Reciprocal of Sine:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Straight Angle:
:$\sin \pi = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}} | :$\csc 180 \degrees = \csc \pi$ is undefined | From [[Cosecant is Reciprocal of Sine]]:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From [[Sine of Straight Angle]]:
:$\sin \pi = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}} | Cosecant of Straight Angle | https://proofwiki.org/wiki/Cosecant_of_Straight_Angle | https://proofwiki.org/wiki/Cosecant_of_Straight_Angle | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Straight Angle"
] |
proofwiki-8069 | Cosecant of 195 Degrees | :$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$ | {{begin-eqn}}
{{eqn | l = \csc 195^\circ
| r = \csc \left({360^\circ - 165^\circ}\right)
| c =
}}
{{eqn | r = -\csc 165^\circ
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = - \left({\sqrt 6 + \sqrt 2}\right)
| c = Cosecant of 165 Degrees
}}
{{end-eqn}}
{{qed}} | :$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$ | {{begin-eqn}}
{{eqn | l = \csc 195^\circ
| r = \csc \left({360^\circ - 165^\circ}\right)
| c =
}}
{{eqn | r = -\csc 165^\circ
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = - \left({\sqrt 6 + \sqrt 2}\right)
| c = [[Cosecant of 165 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 195 Degrees | https://proofwiki.org/wiki/Cosecant_of_195_Degrees | https://proofwiki.org/wiki/Cosecant_of_195_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 165 Degrees"
] |
proofwiki-8070 | Cosecant of 210 Degrees | :$\csc 210 \degrees = \csc \dfrac {7 \pi} 6 = -2$ | {{begin-eqn}}
{{eqn | l = \csc 210 \degrees
| r = \map \csc {360 \degrees - 150 \degrees}
| c =
}}
{{eqn | r = -\csc 150 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -2
| c = Cosecant of $150 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 210 \degrees = \csc \dfrac {7 \pi} 6 = -2$ | {{begin-eqn}}
{{eqn | l = \csc 210 \degrees
| r = \map \csc {360 \degrees - 150 \degrees}
| c =
}}
{{eqn | r = -\csc 150 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -2
| c = [[Cosecant of 150 Degrees|Cosecant of $150 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 210 Degrees | https://proofwiki.org/wiki/Cosecant_of_210_Degrees | https://proofwiki.org/wiki/Cosecant_of_210_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 150 Degrees"
] |
proofwiki-8071 | Cosecant of 225 Degrees | :$\csc 225 \degrees = \csc \dfrac {5 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 225 \degrees
| r = \map \csc {360 \degrees - 135 \degrees}
| c =
}}
{{eqn | r = -\csc 135 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\sqrt 2
| c = Cosecant of $135 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 225 \degrees = \csc \dfrac {5 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 225 \degrees
| r = \map \csc {360 \degrees - 135 \degrees}
| c =
}}
{{eqn | r = -\csc 135 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\sqrt 2
| c = [[Cosecant of 135 Degrees|Cosecant of $135 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 225 Degrees | https://proofwiki.org/wiki/Cosecant_of_225_Degrees | https://proofwiki.org/wiki/Cosecant_of_225_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 135 Degrees"
] |
proofwiki-8072 | Cosecant of 240 Degrees | :$\csc 240 \degrees = \csc \dfrac {4 \pi} 3 = -\dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 240 \degrees
| r = \map \csc {360 \degrees - 120 \degrees}
| c =
}}
{{eqn | r = -\csc 120 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\frac {2 \sqrt 3} 3
| c = Cosecant of $120 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 240 \degrees = \csc \dfrac {4 \pi} 3 = -\dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 240 \degrees
| r = \map \csc {360 \degrees - 120 \degrees}
| c =
}}
{{eqn | r = -\csc 120 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\frac {2 \sqrt 3} 3
| c = [[Cosecant of 120 Degrees|Cosecant of $120 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 240 Degrees | https://proofwiki.org/wiki/Cosecant_of_240_Degrees | https://proofwiki.org/wiki/Cosecant_of_240_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 120 Degrees"
] |
proofwiki-8073 | Cosecant of 255 Degrees | :$\csc 255 \degrees = \csc \dfrac {17 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \csc 255 \degrees
| r = \map \csc {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = -\csc 105 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = Cosecant of $105 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 255 \degrees = \csc \dfrac {17 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \csc 255 \degrees
| r = \map \csc {360 \degrees - 105 \degrees}
| c =
}}
{{eqn | r = -\csc 105 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = [[Cosecant of 105 Degrees|Cosecant of $105 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 255 Degrees | https://proofwiki.org/wiki/Cosecant_of_255_Degrees | https://proofwiki.org/wiki/Cosecant_of_255_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 105 Degrees"
] |
proofwiki-8074 | Cosecant of Three Right Angles | :$\csc 270 \degrees = \csc \dfrac {3 \pi} 2 = -1$ | {{begin-eqn}}
{{eqn | l = \csc 270 \degrees
| r = \map \csc {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = -\csc 90 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -1
| c = Cosecant of Right Angle
}}
{{end-eqn}}
{{qed}} | :$\csc 270 \degrees = \csc \dfrac {3 \pi} 2 = -1$ | {{begin-eqn}}
{{eqn | l = \csc 270 \degrees
| r = \map \csc {360 \degrees - 90 \degrees}
| c =
}}
{{eqn | r = -\csc 90 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -1
| c = [[Cosecant of Right Angle]]
}}
{{end-eqn}}
{{qed}} | Cosecant of Three Right Angles | https://proofwiki.org/wiki/Cosecant_of_Three_Right_Angles | https://proofwiki.org/wiki/Cosecant_of_Three_Right_Angles | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of Right Angle"
] |
proofwiki-8075 | Cosecant of 285 Degrees | :$\csc 285 \degrees = \csc \dfrac {19 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \csc 285 \degrees
| r = \map \csc {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = -\csc 75 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = Cosecant of $75 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 285 \degrees = \csc \dfrac {19 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \csc 285 \degrees
| r = \map \csc {360 \degrees - 75 \degrees}
| c =
}}
{{eqn | r = -\csc 75 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
| c = [[Cosecant of 75 Degrees|Cosecant of $75 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 285 Degrees | https://proofwiki.org/wiki/Cosecant_of_285_Degrees | https://proofwiki.org/wiki/Cosecant_of_285_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 75 Degrees"
] |
proofwiki-8076 | Cosecant of 300 Degrees | :$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 300^\circ
| r = \csc \left({360^\circ - 60^\circ}\right)
| c =
}}
{{eqn | r = -\csc 60^\circ
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\frac {2 \sqrt 3} 3
| c = Cosecant of 60 Degrees
}}
{{end-eqn}}
{{qed}} | :$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$ | {{begin-eqn}}
{{eqn | l = \csc 300^\circ
| r = \csc \left({360^\circ - 60^\circ}\right)
| c =
}}
{{eqn | r = -\csc 60^\circ
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\frac {2 \sqrt 3} 3
| c = [[Cosecant of 60 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 300 Degrees | https://proofwiki.org/wiki/Cosecant_of_300_Degrees | https://proofwiki.org/wiki/Cosecant_of_300_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 60 Degrees"
] |
proofwiki-8077 | Cosecant of 315 Degrees | :$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 315^\circ
| r = \csc \left({360^\circ - 45^\circ}\right)
| c =
}}
{{eqn | r = -\csc 45^\circ
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\sqrt 2
| c = Cosecant of 45 Degrees
}}
{{end-eqn}}
{{qed}} | :$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$ | {{begin-eqn}}
{{eqn | l = \csc 315^\circ
| r = \csc \left({360^\circ - 45^\circ}\right)
| c =
}}
{{eqn | r = -\csc 45^\circ
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\sqrt 2
| c = [[Cosecant of 45 Degrees]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 315 Degrees | https://proofwiki.org/wiki/Cosecant_of_315_Degrees | https://proofwiki.org/wiki/Cosecant_of_315_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 45 Degrees"
] |
proofwiki-8078 | Cosecant of 330 Degrees | :$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$ | {{begin-eqn}}
{{eqn | l = \csc 330 \degrees
| r = \map \csc {360 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = -\csc 30 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -2
| c = {{csc|30}}
}}
{{end-eqn}}
{{qed}} | :$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$ | {{begin-eqn}}
{{eqn | l = \csc 330 \degrees
| r = \map \csc {360 \degrees - 30 \degrees}
| c =
}}
{{eqn | r = -\csc 30 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -2
| c = {{csc|30}}
}}
{{end-eqn}}
{{qed}} | Cosecant of 330 Degrees | https://proofwiki.org/wiki/Cosecant_of_330_Degrees | https://proofwiki.org/wiki/Cosecant_of_330_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle"
] |
proofwiki-8079 | Cosecant of 345 Degrees | :$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \csc 345 \degrees
| r = \map \csc {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = -\csc 15 \degrees
| c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
| c = Cosecant of $15 \degrees$
}}
{{end-eqn}}
{{qed}} | :$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \csc 345 \degrees
| r = \map \csc {360 \degrees - 15 \degrees}
| c =
}}
{{eqn | r = -\csc 15 \degrees
| c = [[Cosecant of Conjugate Angle]]
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
| c = [[Cosecant of 15 Degrees|Cosecant of $15 \degrees$]]
}}
{{end-eqn}}
{{qed}} | Cosecant of 345 Degrees | https://proofwiki.org/wiki/Cosecant_of_345_Degrees | https://proofwiki.org/wiki/Cosecant_of_345_Degrees | [
"Cosecant Function"
] | [] | [
"Cosecant of Conjugate Angle",
"Cosecant of 15 Degrees"
] |
proofwiki-8080 | Cosecant of Full Angle | :$\csc 360 \degrees = \csc 2 \pi$ is undefined | From Cosecant is Reciprocal of Sine:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Full Angle:
:$\sin 360 \degrees = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}} | :$\csc 360 \degrees = \csc 2 \pi$ is undefined | From [[Cosecant is Reciprocal of Sine]]:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From [[Sine of Full Angle]]:
:$\sin 360 \degrees = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}} | Cosecant of Full Angle | https://proofwiki.org/wiki/Cosecant_of_Full_Angle | https://proofwiki.org/wiki/Cosecant_of_Full_Angle | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Full Angle"
] |
proofwiki-8081 | Cosecant Function is Odd | :$\map \csc {-x} = -\csc x$
That is, the cosecant function is odd. | {{begin-eqn}}
{{eqn | l = \map \csc {-x}
| r = \frac 1 {\map \sin {-x} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\sin x}
| c = Sine Function is Odd
}}
{{eqn | r = -\csc x
| c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}} | :$\map \csc {-x} = -\csc x$
That is, the [[Definition:Cosecant|cosecant function]] is [[Definition:Odd Function|odd]]. | {{begin-eqn}}
{{eqn | l = \map \csc {-x}
| r = \frac 1 {\map \sin {-x} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {-\sin x}
| c = [[Sine Function is Odd]]
}}
{{eqn | r = -\csc x
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{end-eqn}}
{{qed}} | Cosecant Function is Odd | https://proofwiki.org/wiki/Cosecant_Function_is_Odd | https://proofwiki.org/wiki/Cosecant_Function_is_Odd | [
"Cosecant Function",
"Examples of Odd Functions"
] | [
"Definition:Cosecant",
"Definition:Odd Function"
] | [
"Cosecant is Reciprocal of Sine",
"Sine Function is Odd",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8082 | Secant Function is Even | :$\map \sec {-x} = \sec x$
That is, the secant function is even. | {{begin-eqn}}
{{eqn | l = \map \sech {-x}
| r = \frac 1 {\map \cosh {-x} }
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {\cosh x}
| c = Hyperbolic Cosine Function is Even
}}
{{eqn | r = \sech x
}}
{{end-eqn}}
{{qed}} | :$\map \sec {-x} = \sec x$
That is, the [[Definition:Secant Function|secant function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \map \sech {-x}
| r = \frac 1 {\map \cosh {-x} }
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {\cosh x}
| c = [[Hyperbolic Cosine Function is Even]]
}}
{{eqn | r = \sech x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Secant Function is Even/Proof 1 | https://proofwiki.org/wiki/Secant_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Secant_Function_is_Even/Proof_1 | [
"Secant Function",
"Examples of Even Functions"
] | [
"Definition:Secant Function",
"Definition:Even Function"
] | [
"Hyperbolic Cosine Function is Even"
] |
proofwiki-8083 | Secant Function is Even | :$\map \sec {-x} = \sec x$
That is, the secant function is even. | {{begin-eqn}}
{{eqn | l = \sech \paren {-x}
| r = \frac 1 {\cosh \paren {-x} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cos \paren {-i x} }
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac 1 {\cos \paren {i x} }
| c = Cosine Function is Even
}}
{{eqn | r = \frac 1 {\cosh x}
| c = H... | :$\map \sec {-x} = \sec x$
That is, the [[Definition:Secant Function|secant function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \sech \paren {-x}
| r = \frac 1 {\cosh \paren {-x} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cos \paren {-i x} }
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \frac 1 {\cos \paren {i x} }
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \frac 1 {\cosh x}
... | Hyperbolic Secant Function is Even/Proof 2 | https://proofwiki.org/wiki/Secant_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Secant_Function_is_Even/Proof_2 | [
"Secant Function",
"Examples of Even Functions"
] | [
"Definition:Secant Function",
"Definition:Even Function"
] | [
"Hyperbolic Cosine in terms of Cosine",
"Cosine Function is Even",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8084 | Secant Function is Even | :$\map \sec {-x} = \sec x$
That is, the secant function is even. | {{begin-eqn}}
{{eqn | l = \map \sec {-x}
| r = \frac 1 {\map \cos {-x} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\cos x}
| c = Cosine Function is Even
}}
{{eqn | r = \sec x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map \sec {-x} = \sec x$
That is, the [[Definition:Secant Function|secant function]] is [[Definition:Even Function|even]]. | {{begin-eqn}}
{{eqn | l = \map \sec {-x}
| r = \frac 1 {\map \cos {-x} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\cos x}
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \sec x
| c = [[Secant is Reciprocal of Cosine]]
}}
{{end-eqn}}
{{qed}} | Secant Function is Even | https://proofwiki.org/wiki/Secant_Function_is_Even | https://proofwiki.org/wiki/Secant_Function_is_Even | [
"Secant Function",
"Examples of Even Functions"
] | [
"Definition:Secant Function",
"Definition:Even Function"
] | [
"Secant is Reciprocal of Cosine",
"Cosine Function is Even",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8085 | Cotangent Function is Odd | :$\map \cot {-x} = -\cot x$
That is, the cotangent function is odd. | {{begin-eqn}}
{{eqn | l = \map \cot {-x}
| r = \frac {\map \cos {-x} } {\map \sin {-x} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {-\sin x} {\cos x}
| c = Cosine Function is Even and Sine Function is Odd
}}
{{eqn | r = -\cot x
| c = Cotangent is Cosine divided by Sine
}}
{{... | :$\map \cot {-x} = -\cot x$
That is, the [[Definition:Cotangent|cotangent function]] is [[Definition:Odd Function|odd]]. | {{begin-eqn}}
{{eqn | l = \map \cot {-x}
| r = \frac {\map \cos {-x} } {\map \sin {-x} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {-\sin x} {\cos x}
| c = [[Cosine Function is Even]] and [[Sine Function is Odd]]
}}
{{eqn | r = -\cot x
| c = [[Cotangent is Cosine divided... | Cotangent Function is Odd | https://proofwiki.org/wiki/Cotangent_Function_is_Odd | https://proofwiki.org/wiki/Cotangent_Function_is_Odd | [
"Cotangent Function is Odd",
"Cotangent Function",
"Examples of Odd Functions"
] | [
"Definition:Cotangent",
"Definition:Odd Function"
] | [
"Cotangent is Cosine divided by Sine",
"Cosine Function is Even",
"Sine Function is Odd",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-8086 | Cotangent of Sum | :$\map \cot {a + b} = \dfrac {\cot a \cot b - 1} {\cot b + \cot a}$
where $\cot $ is cotangent.
=== Corollary ===
{{:Cotangent of Sum/Corollary}} | {{begin-eqn}}
{{eqn | l = \map \cot {a + b}
| r = \frac {\map \cos {a + b} } {\map \sin {a + b} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\cos a \cos b - \sin a \sin b} {\sin a \cos b + \cos a \sin b}
| c = Cosine of Sum and Sine of Sum
}}
{{eqn | r = \frac {\frac {\cos a \cos... | :$\map \cot {a + b} = \dfrac {\cot a \cot b - 1} {\cot b + \cot a}$
where $\cot $ is [[Definition:Cotangent|cotangent]].
=== [[Cotangent of Sum/Corollary|Corollary]] ===
{{:Cotangent of Sum/Corollary}} | {{begin-eqn}}
{{eqn | l = \map \cot {a + b}
| r = \frac {\map \cos {a + b} } {\map \sin {a + b} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\cos a \cos b - \sin a \sin b} {\sin a \cos b + \cos a \sin b}
| c = [[Cosine of Sum]] and [[Sine of Sum]]
}}
{{eqn | r = \frac {\frac ... | Cotangent of Sum | https://proofwiki.org/wiki/Cotangent_of_Sum | https://proofwiki.org/wiki/Cotangent_of_Sum | [
"Cotangent of Sum",
"Cotangent Function",
"Trigonometric Addition Formulas"
] | [
"Definition:Cotangent",
"Cotangent of Difference"
] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Sum",
"Sine of Sum",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-8087 | Half-Integer is Half Odd Integer | Let $r$ be a number.
Then $r$ is a half-integer {{iff}} $r = \dfrac n 2$ where $n$ is an odd integer. | === Necessary Condition ===
Let $r$ be a half-integer.
Then by definition $r = n + \dfrac 1 2$ for some $n \in \Z$.
Thus:
{{begin-eqn}}
{{eqn | l = 2 r
| r = 2 \paren {n + \dfrac 1 2}
| c =
}}
{{eqn | r = 2 n + 2 \paren {\frac 1 2}
| c =
}}
{{eqn | r = 2 n + 1
| c =
}}
{{end-eqn}}
thus showin... | Let $r$ be a [[Definition:Number|number]].
Then $r$ is a [[Definition:Half-Integer|half-integer]] {{iff}} $r = \dfrac n 2$ where $n$ is an [[Definition:Odd Integer|odd integer]]. | === Necessary Condition ===
Let $r$ be a [[Definition:Half-Integer|half-integer]].
Then by definition $r = n + \dfrac 1 2$ for some $n \in \Z$.
Thus:
{{begin-eqn}}
{{eqn | l = 2 r
| r = 2 \paren {n + \dfrac 1 2}
| c =
}}
{{eqn | r = 2 n + 2 \paren {\frac 1 2}
| c =
}}
{{eqn | r = 2 n + 1
|... | Half-Integer is Half Odd Integer | https://proofwiki.org/wiki/Half-Integer_is_Half_Odd_Integer | https://proofwiki.org/wiki/Half-Integer_is_Half_Odd_Integer | [
"Integers",
"Odd Integers",
"Numbers"
] | [
"Definition:Number",
"Definition:Half-Integer",
"Definition:Odd Integer"
] | [
"Definition:Half-Integer",
"Odd Integer 2n + 1",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Odd Integer 2n + 1",
"Definition:Half-Integer"
] |
proofwiki-8088 | Sine of Angle plus Straight Angle | :$\map \sin {x + \pi} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \map \sin {x + \pi}
| r = \sin x \cos \pi + \cos x \sin \pi
| c = Sine of Sum
}}
{{eqn | r = \sin x \cdot \paren {-1} + \cos x \cdot 0
| c = Cosine of Straight Angle and Sine of Straight Angle
}}
{{eqn | r = -\sin x
| c =
}}
{{end-eqn}}
{{qed}} | :$\map \sin {x + \pi} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \map \sin {x + \pi}
| r = \sin x \cos \pi + \cos x \sin \pi
| c = [[Sine of Sum]]
}}
{{eqn | r = \sin x \cdot \paren {-1} + \cos x \cdot 0
| c = [[Cosine of Straight Angle]] and [[Sine of Straight Angle]]
}}
{{eqn | r = -\sin x
| c =
}}
{{end-eqn}}
{{qed}} | Sine of Angle plus Straight Angle | https://proofwiki.org/wiki/Sine_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Sine_of_Angle_plus_Straight_Angle | [
"Sine Function"
] | [] | [
"Sine of Sum",
"Cosine of Straight Angle",
"Sine of Straight Angle"
] |
proofwiki-8089 | Cosine of Angle plus Straight Angle | :$\map \cos {x + \pi} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
| r = \cos x \cos \pi - \sin x \sin \pi
| c = Cosine of Sum
}}
{{eqn | r = \cos x \cdot \paren {-1} - \sin x \cdot 0
| c = Cosine of Straight Angle and Sine of Straight Angle
}}
{{eqn | r = -\cos x
| c =
}}
{{end-eqn}}
{{qed}} | :$\map \cos {x + \pi} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
| r = \cos x \cos \pi - \sin x \sin \pi
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos x \cdot \paren {-1} - \sin x \cdot 0
| c = [[Cosine of Straight Angle]] and [[Sine of Straight Angle]]
}}
{{eqn | r = -\cos x
| c =
}}
{{end-eqn}}
{{qed}} | Cosine of Angle plus Straight Angle/Proof 1 | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle/Proof_1 | [
"Cosine of Angle plus Straight Angle",
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Cosine of Straight Angle",
"Sine of Straight Angle"
] |
proofwiki-8090 | Cosine of Angle plus Straight Angle | :$\map \cos {x + \pi} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
| r = \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }
| c =
}}
{{eqn | r = \map \Re {e^{i \paren {x + \pi} } }
| c = Euler's Formula
}}
{{eqn | r = \map \Re {e^{i x + i \pi} }
| c =
}}
{{eqn | r = \map \Re {e^{i x} e^{i \pi} }
| c ... | :$\map \cos {x + \pi} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
| r = \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }
| c =
}}
{{eqn | r = \map \Re {e^{i \paren {x + \pi} } }
| c = [[Euler's Formula]]
}}
{{eqn | r = \map \Re {e^{i x + i \pi} }
| c =
}}
{{eqn | r = \map \Re {e^{i x} e^{i \pi} }
... | Cosine of Angle plus Straight Angle/Proof 2 | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle/Proof_2 | [
"Cosine of Angle plus Straight Angle",
"Cosine Function"
] | [] | [
"Euler's Formula",
"Exponential of Sum/Complex Numbers",
"Euler's Identity",
"Euler's Formula"
] |
proofwiki-8091 | Cosine of Angle plus Straight Angle | :$\map \cos {x + \pi} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
| r = \frac 1 2 \paren {e^{i \paren {x + \pi} } + e^{-i \paren {x + \pi} } }
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac 1 2 \paren {e^{i x} e^{i \pi} + e^{-i x} e^{-i \pi} }
| c = Exponential of Sum: Complex Numbers
}}
{{eqn | r = \frac 1 2 \paren ... | :$\map \cos {x + \pi} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
| r = \frac 1 2 \paren {e^{i \paren {x + \pi} } + e^{-i \paren {x + \pi} } }
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 1 2 \paren {e^{i x} e^{i \pi} + e^{-i x} e^{-i \pi} }
| c = [[Exponential of Sum/Complex Numbers|Exponential of Sum: Complex... | Cosine of Angle plus Straight Angle/Proof 3 | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle/Proof_3 | [
"Cosine of Angle plus Straight Angle",
"Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Exponential of Sum/Complex Numbers",
"Euler's Identity",
"Euler's Cosine Identity"
] |
proofwiki-8092 | Cosine of Angle plus Straight Angle | :$\map \cos {x + \pi} = -\cos x$ | From the discussion in the proof of Real Cosine Function is Periodic:
:$\map \sin {x + \eta} = \cos x$
:$\map \cos {x + \eta} = -\sin x$
for $\eta \in \R_{>0}$.
From Sine and Cosine are Periodic on Reals: Pi, we define $\pi \in \R$ as $\pi := 2 \eta$.
It follows that $\eta = \dfrac \pi 2$, thus:
:$\map \cos {x + \pi} =... | :$\map \cos {x + \pi} = -\cos x$ | From the discussion in the proof of [[Real Cosine Function is Periodic]]:
:$\map \sin {x + \eta} = \cos x$
:$\map \cos {x + \eta} = -\sin x$
for $\eta \in \R_{>0}$.
From [[Sine and Cosine are Periodic on Reals/Pi|Sine and Cosine are Periodic on Reals: Pi]], we define $\pi \in \R$ as $\pi := 2 \eta$.
It follows that... | Cosine of Angle plus Straight Angle/Proof 4 | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Straight_Angle/Proof_4 | [
"Cosine of Angle plus Straight Angle",
"Cosine Function"
] | [] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Sine and Cosine are Periodic on Reals/Pi"
] |
proofwiki-8093 | Sine of Angle plus Full Angle | :$\map \sin {x + 2 \pi} = \sin x$ | {{begin-eqn}}
{{eqn | l = \map \sin {x + 2 \pi}
| r = \sin x \cos 2 \pi + \cos x \sin 2 \pi
| c = Sine of Sum
}}
{{eqn | r = \sin x \cdot 1 + \cos x \cdot 0
| c = Cosine of Full Angle and Sine of Full Angle
}}
{{eqn | r = \sin x
| c =
}}
{{end-eqn}}
{{qed}} | :$\map \sin {x + 2 \pi} = \sin x$ | {{begin-eqn}}
{{eqn | l = \map \sin {x + 2 \pi}
| r = \sin x \cos 2 \pi + \cos x \sin 2 \pi
| c = [[Sine of Sum]]
}}
{{eqn | r = \sin x \cdot 1 + \cos x \cdot 0
| c = [[Cosine of Full Angle]] and [[Sine of Full Angle]]
}}
{{eqn | r = \sin x
| c =
}}
{{end-eqn}}
{{qed}} | Sine of Angle plus Full Angle | https://proofwiki.org/wiki/Sine_of_Angle_plus_Full_Angle | https://proofwiki.org/wiki/Sine_of_Angle_plus_Full_Angle | [
"Sine Function"
] | [] | [
"Sine of Sum",
"Cosine of Full Angle",
"Sine of Full Angle"
] |
proofwiki-8094 | Cosine of Angle plus Full Angle | :$\map \cos {x + 2 \pi} = \cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + 2 \pi}
| r = \cos x \cos 2 \pi - \sin x \sin 2 \pi
| c = Cosine of Sum
}}
{{eqn | r = \cos x \cdot 1 - \sin x \cdot 0
| c = Cosine of Full Angle and Sine of Full Angle
}}
{{eqn | r = \cos x
| c =
}}
{{end-eqn}}
{{qed}} | :$\map \cos {x + 2 \pi} = \cos x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + 2 \pi}
| r = \cos x \cos 2 \pi - \sin x \sin 2 \pi
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos x \cdot 1 - \sin x \cdot 0
| c = [[Cosine of Full Angle]] and [[Sine of Full Angle]]
}}
{{eqn | r = \cos x
| c =
}}
{{end-eqn}}
{{qed}} | Cosine of Angle plus Full Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Full_Angle | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Full_Angle | [
"Cosine of Angle plus Full Angle",
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Cosine of Full Angle",
"Sine of Full Angle"
] |
proofwiki-8095 | Tangent of Angle plus Full Angle | :$\map \tan {x + 2 \pi} = \tan x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x + 2 \pi}
| r = \frac {\map \sin {x + 2 \pi} } {\map \cos {x + 2 \pi} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\sin x} {\cos x}
| c = Sine of Angle plus Full Angle and Cosine of Angle plus Full Angle
}}
{{eqn | r = \tan x
| c = Tangen... | :$\map \tan {x + 2 \pi} = \tan x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x + 2 \pi}
| r = \frac {\map \sin {x + 2 \pi} } {\map \cos {x + 2 \pi} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\sin x} {\cos x}
| c = [[Sine of Angle plus Full Angle]] and [[Cosine of Angle plus Full Angle]]
}}
{{eqn | r = \tan x
... | Tangent of Angle plus Full Angle | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Full_Angle | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Full_Angle | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Angle plus Full Angle",
"Cosine of Angle plus Full Angle",
"Tangent is Sine divided by Cosine"
] |
proofwiki-8096 | Tangent of Angle plus Straight Angle | : $\tan \left({x + \pi}\right) = \tan x$ | {{begin-eqn}}
{{eqn | l = \tan \left({x + \pi}\right)
| r = \frac {\sin \left({x + \pi}\right)} {\cos \left({x + \pi}\right)}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\sin x} {-\cos x}
| c = Sine of Angle plus Straight Angle and Cosine of Angle plus Straight Angle
}}
{{eqn | r =... | : $\tan \left({x + \pi}\right) = \tan x$ | {{begin-eqn}}
{{eqn | l = \tan \left({x + \pi}\right)
| r = \frac {\sin \left({x + \pi}\right)} {\cos \left({x + \pi}\right)}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {-\sin x} {-\cos x}
| c = [[Sine of Angle plus Straight Angle]] and [[Cosine of Angle plus Straight Angle]]
}}... | Tangent of Angle plus Straight Angle | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Straight_Angle | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Angle plus Straight Angle",
"Cosine of Angle plus Straight Angle",
"Tangent is Sine divided by Cosine"
] |
proofwiki-8097 | Cotangent of Angle plus Straight Angle | :$\map \cot {x + \pi} = \cot x$ | {{begin-eqn}}
{{eqn | l = \map \cot {x + \pi}
| r = \frac {\map \cos {x + \pi} } {\map \sin {x + \pi} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
| c = Cosine of Angle plus Straight Angle and Sine of Angle plus Straight Angle
}}
{{eqn | r = \cot x
| c = C... | :$\map \cot {x + \pi} = \cot x$ | {{begin-eqn}}
{{eqn | l = \map \cot {x + \pi}
| r = \frac {\map \cos {x + \pi} } {\map \sin {x + \pi} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
| c = [[Cosine of Angle plus Straight Angle]] and [[Sine of Angle plus Straight Angle]]
}}
{{eqn | r = \cot x
... | Cotangent of Angle plus Straight Angle | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Straight_Angle | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Angle plus Straight Angle",
"Sine of Angle plus Straight Angle",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-8098 | Secant of Angle plus Straight Angle | :$\map \sec {x + \pi} = -\sec x$ | {{begin-eqn}}
{{eqn | l = \map \sec {x + \pi}
| r = \frac 1 {\map \cos {x + \pi} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {-\cos x}
| c = Cosine of Angle plus Straight Angle
}}
{{eqn | r = -\sec x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map \sec {x + \pi} = -\sec x$ | {{begin-eqn}}
{{eqn | l = \map \sec {x + \pi}
| r = \frac 1 {\map \cos {x + \pi} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {-\cos x}
| c = [[Cosine of Angle plus Straight Angle]]
}}
{{eqn | r = -\sec x
| c = [[Secant is Reciprocal of Cosine]]
}}
{{end-eqn}}
{{qed}} | Secant of Angle plus Straight Angle | https://proofwiki.org/wiki/Secant_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Secant_of_Angle_plus_Straight_Angle | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of Angle plus Straight Angle",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8099 | Cosecant of Angle plus Straight Angle | :$\map \csc {x + \pi} = -\csc x$ | {{begin-eqn}}
{{eqn | l = \map \csc {x + \pi}
| r = \frac 1 {\map \sin {x + \pi} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\sin x}
| c = Sine of Angle plus Straight Angle
}}
{{eqn | r = -\csc x
| c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}} | :$\map \csc {x + \pi} = -\csc x$ | {{begin-eqn}}
{{eqn | l = \map \csc {x + \pi}
| r = \frac 1 {\map \sin {x + \pi} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {-\sin x}
| c = [[Sine of Angle plus Straight Angle]]
}}
{{eqn | r = -\csc x
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{end-eqn}}
{{qed}} | Cosecant of Angle plus Straight Angle | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Straight_Angle | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Straight_Angle | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Angle plus Straight Angle",
"Cosecant is Reciprocal of Sine"
] |
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