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-8100 | Cotangent of Angle plus Full Angle | : $\cot \left({x + 2 \pi}\right) = \cot x$ | {{begin-eqn}}
{{eqn | l = \cot \left({x + 2 \pi}\right)
| r = \frac {\cos \left({x + 2 \pi}\right)} {\sin \left({x + 2 \pi}\right)}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\cos x} {\sin x}
| c = Cosine of Angle plus Full Angle and Sine of Angle plus Full Angle
}}
{{eqn | r = \... | : $\cot \left({x + 2 \pi}\right) = \cot x$ | {{begin-eqn}}
{{eqn | l = \cot \left({x + 2 \pi}\right)
| r = \frac {\cos \left({x + 2 \pi}\right)} {\sin \left({x + 2 \pi}\right)}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\cos x} {\sin x}
| c = [[Cosine of Angle plus Full Angle]] and [[Sine of Angle plus Full Angle]]
}}
{... | Cotangent of Angle plus Full Angle | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Full_Angle | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Full_Angle | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Angle plus Full Angle",
"Sine of Angle plus Full Angle",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-8101 | Secant of Angle plus Full Angle | : $\sec \left({x + 2 \pi}\right) = \sec x$ | {{begin-eqn}}
{{eqn | l = \sec \left({x + 2 \pi}\right)
| r = \frac 1 {\cos \left({x + 2 \pi}\right)}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\cos x}
| c = Cosine of Angle plus Full Angle
}}
{{eqn | r = \sec x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}} | : $\sec \left({x + 2 \pi}\right) = \sec x$ | {{begin-eqn}}
{{eqn | l = \sec \left({x + 2 \pi}\right)
| r = \frac 1 {\cos \left({x + 2 \pi}\right)}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\cos x}
| c = [[Cosine of Angle plus Full Angle]]
}}
{{eqn | r = \sec x
| c = [[Secant is Reciprocal of Cosine]]
}}
{{end-eqn}}
... | Secant of Angle plus Full Angle | https://proofwiki.org/wiki/Secant_of_Angle_plus_Full_Angle | https://proofwiki.org/wiki/Secant_of_Angle_plus_Full_Angle | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of Angle plus Full Angle",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8102 | Cosecant of Angle plus Full Angle | : $\csc \left({x + 2 \pi}\right) = \csc x$ | {{begin-eqn}}
{{eqn | l = \csc \left({x + 2 \pi}\right)
| r = \frac 1 {\sin \left({x + 2 \pi}\right)}
| 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}} | : $\csc \left({x + 2 \pi}\right) = \csc x$ | {{begin-eqn}}
{{eqn | l = \csc \left({x + 2 \pi}\right)
| r = \frac 1 {\sin \left({x + 2 \pi}\right)}
| 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}... | Cosecant of Angle plus Full Angle | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Full_Angle | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Full_Angle | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Angle plus Straight Angle",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8103 | Sine of Angle plus Three Right Angles | :$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \sin {x + \frac {3 \pi} 2}
| r = \sin x \cos \frac {3 \pi} 2 + \cos x \sin \frac {3 \pi} 2
| c = Sine of Sum
}}
{{eqn | r = \sin x \cdot 0 + \cos x \cdot \paren {-1}
| c = Cosine of Three Right Angles and Sine of Three Right Angles
}}
{{eqn | r = -\cos x
| c =
}}
... | :$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$ | {{begin-eqn}}
{{eqn | l = \map \sin {x + \frac {3 \pi} 2}
| r = \sin x \cos \frac {3 \pi} 2 + \cos x \sin \frac {3 \pi} 2
| c = [[Sine of Sum]]
}}
{{eqn | r = \sin x \cdot 0 + \cos x \cdot \paren {-1}
| c = [[Cosine of Three Right Angles]] and [[Sine of Three Right Angles]]
}}
{{eqn | r = -\cos x
... | Sine of Angle plus Three Right Angles | https://proofwiki.org/wiki/Sine_of_Angle_plus_Three_Right_Angles | https://proofwiki.org/wiki/Sine_of_Angle_plus_Three_Right_Angles | [
"Sine Function"
] | [] | [
"Sine of Sum",
"Cosine of Three Right Angles",
"Sine of Three Right Angles"
] |
proofwiki-8104 | Cosine of Angle plus Three Right Angles | :$\map \cos {x + \dfrac {3 \pi} 2} = \sin x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \frac {3 \pi} 2}
| r = \cos x \cos \frac {3 \pi} 2 - \sin x \sin \frac {3 \pi} 2
| c = Cosine of Sum
}}
{{eqn | r = \cos x \cdot 0 - \sin x \cdot \paren {-1}
| c = Cosine of Three Right Angles and Sine of Three Right Angles
}}
{{eqn | r = \sin x
| c =
}}... | :$\map \cos {x + \dfrac {3 \pi} 2} = \sin x$ | {{begin-eqn}}
{{eqn | l = \map \cos {x + \frac {3 \pi} 2}
| r = \cos x \cos \frac {3 \pi} 2 - \sin x \sin \frac {3 \pi} 2
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos x \cdot 0 - \sin x \cdot \paren {-1}
| c = [[Cosine of Three Right Angles]] and [[Sine of Three Right Angles]]
}}
{{eqn | r = \sin x
... | Cosine of Angle plus Three Right Angles | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Three_Right_Angles | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Three_Right_Angles | [
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Cosine of Three Right Angles",
"Sine of Three Right Angles"
] |
proofwiki-8105 | Tangent of Angle plus Three Right Angles | :$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x + \frac {3 \pi} 2}
| r = \frac {\map \sin {x + \frac {3 \pi} 2} } {\map \cos {x + \frac {3 \pi} 2} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\cos x} {\sin x}
| c = Sine of Angle plus Three Right Angles and Cosine of Angle plus Three Right ... | :$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x + \frac {3 \pi} 2}
| r = \frac {\map \sin {x + \frac {3 \pi} 2} } {\map \cos {x + \frac {3 \pi} 2} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {-\cos x} {\sin x}
| c = [[Sine of Angle plus Three Right Angles]] and [[Cosine of Angle plus Th... | Tangent of Angle plus Three Right Angles | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Three_Right_Angles | https://proofwiki.org/wiki/Tangent_of_Angle_plus_Three_Right_Angles | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Angle plus Three Right Angles",
"Cosine of Angle plus Three Right Angles",
"Cotangent is Cosine divided by Sine"
] |
proofwiki-8106 | Cotangent of Angle plus Three Right Angles | :$\map \cot {x + \dfrac {3 \pi} 2} = -\tan x$ | {{begin-eqn}}
{{eqn | l = \map \cot {x + \frac {3 \pi} 2}
| r = \frac {\map \cos {x + \frac {3 \pi} 2} } {\map \sin {x + \frac {3 \pi} 2} }
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\sin x} {-\cos x}
| c = Cosine of Angle plus Right Angle and Sine of Angle plus Right Angle
}}
{{e... | :$\map \cot {x + \dfrac {3 \pi} 2} = -\tan x$ | {{begin-eqn}}
{{eqn | l = \map \cot {x + \frac {3 \pi} 2}
| r = \frac {\map \cos {x + \frac {3 \pi} 2} } {\map \sin {x + \frac {3 \pi} 2} }
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\sin x} {-\cos x}
| c = [[Cosine of Angle plus Right Angle]] and [[Sine of Angle plus Right An... | Cotangent of Angle plus Three Right Angles | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Three_Right_Angles | https://proofwiki.org/wiki/Cotangent_of_Angle_plus_Three_Right_Angles | [
"Cotangent Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Cosine of Angle plus Right Angle",
"Sine of Angle plus Right Angle",
"Tangent is Sine divided by Cosine"
] |
proofwiki-8107 | Secant of Angle plus Three Right Angles | :$\map \sec {x + \dfrac {3 \pi} 2} = \csc x$ | {{begin-eqn}}
{{eqn | l = \map \sec {x + \frac {3 \pi} 2}
| r = \frac 1 {\map \cos {x + \frac {3 \pi} 2} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\sin x}
| c = Cosine of Angle plus Three Right Angles
}}
{{eqn | r = \csc x
| c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
... | :$\map \sec {x + \dfrac {3 \pi} 2} = \csc x$ | {{begin-eqn}}
{{eqn | l = \map \sec {x + \frac {3 \pi} 2}
| r = \frac 1 {\map \cos {x + \frac {3 \pi} 2} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {\sin x}
| c = [[Cosine of Angle plus Three Right Angles]]
}}
{{eqn | r = \csc x
| c = [[Cosecant is Reciprocal of Sine]]
}}
... | Secant of Angle plus Three Right Angles | https://proofwiki.org/wiki/Secant_of_Angle_plus_Three_Right_Angles | https://proofwiki.org/wiki/Secant_of_Angle_plus_Three_Right_Angles | [
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Cosine of Angle plus Three Right Angles",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8108 | Cosecant of Angle plus Three Right Angles | :$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$ | {{begin-eqn}}
{{eqn | l = \map \csc {x + \frac {3 \pi} 2}
| r = \frac 1 {\map \sin {x + \frac {3 \pi} 2} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\cos x}
| c = Sine of Angle plus Three Right Angles
}}
{{eqn | r = -\sec x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
... | :$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$ | {{begin-eqn}}
{{eqn | l = \map \csc {x + \frac {3 \pi} 2}
| r = \frac 1 {\map \sin {x + \frac {3 \pi} 2} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {-\cos x}
| c = [[Sine of Angle plus Three Right Angles]]
}}
{{eqn | r = -\sec x
| c = [[Secant is Reciprocal of Cosine]]
}}
... | Cosecant of Angle plus Three Right Angles | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Three_Right_Angles | https://proofwiki.org/wiki/Cosecant_of_Angle_plus_Three_Right_Angles | [
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Sine of Angle plus Three Right Angles",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8109 | Sine of Three Right Angles less Angle | :$\map \sin {\dfrac {3 \pi} 2 - \theta} = -\cos \theta$
where $\sin$ and $\cos$ are sine and cosine respectively. | {{begin-eqn}}
{{eqn | l = \map \sin {\frac {3 \pi} 2 - \theta}
| r = \sin \frac {3 \pi} 2 \cos \theta - \cos \frac {3 \pi} 2 \sin \theta
| c = Sine of Difference
}}
{{eqn | r = \paren {-1} \times \cos \theta - 0 \times \sin \theta
| c = Sine of Three Right Angles and Cosine of Three Right Angles
}}
{{... | :$\map \sin {\dfrac {3 \pi} 2 - \theta} = -\cos \theta$
where $\sin$ and $\cos$ are [[Definition:Sine|sine]] and [[Definition:Cosine|cosine]] respectively. | {{begin-eqn}}
{{eqn | l = \map \sin {\frac {3 \pi} 2 - \theta}
| r = \sin \frac {3 \pi} 2 \cos \theta - \cos \frac {3 \pi} 2 \sin \theta
| c = [[Sine of Difference]]
}}
{{eqn | r = \paren {-1} \times \cos \theta - 0 \times \sin \theta
| c = [[Sine of Three Right Angles]] and [[Cosine of Three Right An... | Sine of Three Right Angles less Angle | https://proofwiki.org/wiki/Sine_of_Three_Right_Angles_less_Angle | https://proofwiki.org/wiki/Sine_of_Three_Right_Angles_less_Angle | [
"Sine Function"
] | [
"Definition:Sine",
"Definition:Cosine"
] | [
"Sine of Difference",
"Sine of Three Right Angles",
"Cosine of Three Right Angles"
] |
proofwiki-8110 | Cosine of Three Right Angles less Angle | :$\map \cos {\dfrac {3 \pi} 2 - \theta} = -\sin \theta$
where $\cos$ and $\sin$ are cosine and sine respectively. | {{begin-eqn}}
{{eqn | l = \map \cos {\frac {3 \pi} 2 - \theta}
| r = \cos \frac {3 \pi} 2 \cos \theta + \sin \frac {3 \pi} 2 \sin \theta
| c = Cosine of Difference
}}
{{eqn | r = 0 \times \cos \theta + \paren {-1} \times \sin \theta
| c = Cosine of Three Right Angles and Sine of Three Right Angles
}}
... | :$\map \cos {\dfrac {3 \pi} 2 - \theta} = -\sin \theta$
where $\cos$ and $\sin$ are [[Definition:Cosine|cosine]] and [[Definition:Sine|sine]] respectively. | {{begin-eqn}}
{{eqn | l = \map \cos {\frac {3 \pi} 2 - \theta}
| r = \cos \frac {3 \pi} 2 \cos \theta + \sin \frac {3 \pi} 2 \sin \theta
| c = [[Cosine of Difference]]
}}
{{eqn | r = 0 \times \cos \theta + \paren {-1} \times \sin \theta
| c = [[Cosine of Three Right Angles]] and [[Sine of Three Right ... | Cosine of Three Right Angles less Angle | https://proofwiki.org/wiki/Cosine_of_Three_Right_Angles_less_Angle | https://proofwiki.org/wiki/Cosine_of_Three_Right_Angles_less_Angle | [
"Cosine Function"
] | [
"Definition:Cosine",
"Definition:Sine"
] | [
"Cosine of Difference",
"Cosine of Three Right Angles",
"Sine of Three Right Angles"
] |
proofwiki-8111 | Secant of Three Right Angles less Angle | :$\map \sec {\dfrac {3 \pi} 2 - \theta} = -\csc \theta$
where $\sec$ and $\csc$ are secant and cosecant respectively. | {{begin-eqn}}
{{eqn | l = \map \sec {\frac {3 \pi} 2 - \theta}
| r = \frac 1 {\map \cos {\frac {3 \pi} 2 - \theta} }
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {-\sin \theta}
| c = Cosine of Three Right Angles less Angle
}}
{{eqn | r = -\csc \theta
| c = Cosecant is Reciprocal o... | :$\map \sec {\dfrac {3 \pi} 2 - \theta} = -\csc \theta$
where $\sec$ and $\csc$ are [[Definition:Secant Function|secant]] and [[Definition:Cosecant|cosecant]] respectively. | {{begin-eqn}}
{{eqn | l = \map \sec {\frac {3 \pi} 2 - \theta}
| r = \frac 1 {\map \cos {\frac {3 \pi} 2 - \theta} }
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \frac 1 {-\sin \theta}
| c = [[Cosine of Three Right Angles less Angle]]
}}
{{eqn | r = -\csc \theta
| c = [[Cosecant is Re... | Secant of Three Right Angles less Angle | https://proofwiki.org/wiki/Secant_of_Three_Right_Angles_less_Angle | https://proofwiki.org/wiki/Secant_of_Three_Right_Angles_less_Angle | [
"Secant Function"
] | [
"Definition:Secant Function",
"Definition:Cosecant"
] | [
"Secant is Reciprocal of Cosine",
"Cosine of Three Right Angles less Angle",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8112 | Cosecant of Three Right Angles less Angle | :$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$
where $\csc$ and $\sec$ are cosecant and secant respectively. | {{begin-eqn}}
{{eqn | l = \map \csc {\frac {3 \pi} 2 - \theta}
| r = \frac 1 {\map \sin {\frac {3 \pi} 2 - \theta} }
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\cos \theta}
| c = Sine of Three Right Angles less Angle
}}
{{eqn | r = -\sec \theta
| c = Secant is Reciprocal of Co... | :$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$
where $\csc$ and $\sec$ are [[Definition:Cosecant|cosecant]] and [[Definition:Secant Function|secant]] respectively. | {{begin-eqn}}
{{eqn | l = \map \csc {\frac {3 \pi} 2 - \theta}
| r = \frac 1 {\map \sin {\frac {3 \pi} 2 - \theta} }
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | r = \frac 1 {-\cos \theta}
| c = [[Sine of Three Right Angles less Angle]]
}}
{{eqn | r = -\sec \theta
| c = [[Secant is Recipr... | Cosecant of Three Right Angles less Angle | https://proofwiki.org/wiki/Cosecant_of_Three_Right_Angles_less_Angle | https://proofwiki.org/wiki/Cosecant_of_Three_Right_Angles_less_Angle | [
"Cosecant Function"
] | [
"Definition:Cosecant",
"Definition:Secant Function"
] | [
"Cosecant is Reciprocal of Sine",
"Sine of Three Right Angles less Angle",
"Secant is Reciprocal of Cosine"
] |
proofwiki-8113 | Triple Angle Formulas/Sine | :$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$ | {{begin-eqn}}
{{eqn | l = \sin 3 \theta
| r = \map \sin {2 \theta + \theta}
}}
{{eqn | r = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta
| c = Sine of Sum
}}
{{eqn | r = \paren {2 \sin \theta \cos \theta} \cos \theta + \paren {\cos^2 \theta - \sin^2 \theta} \sin \theta
| c = Double Angle Formu... | :$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$ | {{begin-eqn}}
{{eqn | l = \sin 3 \theta
| r = \map \sin {2 \theta + \theta}
}}
{{eqn | r = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta
| c = [[Sine of Sum]]
}}
{{eqn | r = \paren {2 \sin \theta \cos \theta} \cos \theta + \paren {\cos^2 \theta - \sin^2 \theta} \sin \theta
| c = [[Double Angle... | Triple Angle Formulas/Sine/Proof 1 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Sine/Proof_1 | [
"Triple Angle Formula for Sine",
"Triple Angle Formulas",
"Sine Function"
] | [] | [
"Sine of Sum",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Cosine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8114 | Triple Angle Formulas/Sine | :$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 3 \theta + i \sin 3 \theta
| r = \paren {\cos \theta + i \sin \theta}^3
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}
}}
{{eqn | o =
| ro=+
| r = \binom 3 2 \paren {\cos \theta} ... | :$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 3 \theta + i \sin 3 \theta
| r = \paren {\cos \theta + i \sin \theta}^3
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}
}}
{{eqn | o =
| ro=+
| r = \binom 3 2 \paren {\cos \th... | Triple Angle Formulas/Sine/Proof 2 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Sine/Proof_2 | [
"Triple Angle Formula for Sine",
"Triple Angle Formulas",
"Sine Function"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Imaginary Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8115 | Triple Angle Formulas/Cosine | :$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 3 \theta
| r = \cos \paren {2 \theta + \theta}
}}
{{eqn | r = \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta
| c = Cosine of Sum
}}
{{eqn | r = \paren {\cos^2 \theta - \sin^2 \theta} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta
| c = Double Angle F... | :$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 3 \theta
| r = \cos \paren {2 \theta + \theta}
}}
{{eqn | r = \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta
| c = [[Cosine of Sum]]
}}
{{eqn | r = \paren {\cos^2 \theta - \sin^2 \theta} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta
| c = [[Double A... | Triple Angle Formulas/Cosine/Proof 1 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine/Proof_1 | [
"Triple Angle Formula for Cosine",
"Triple Angle Formulas",
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Double Angle Formulas/Cosine",
"Double Angle Formulas/Sine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8116 | Triple Angle Formulas/Cosine | :$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 3 \theta + i \sin 3 \theta
| r = \paren {\cos \theta + i \sin \theta}^3
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}
| c = Binomial Theorem
}}
{{eqn | o =
| ro=+
| r = \bi... | :$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 3 \theta + i \sin 3 \theta
| r = \paren {\cos \theta + i \sin \theta}^3
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}
| c = [[Binomial Theorem]]
}}
{{eqn | o =
| ro=+
... | Triple Angle Formulas/Cosine/Proof 2 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine/Proof_2 | [
"Triple Angle Formula for Cosine",
"Triple Angle Formulas",
"Cosine Function"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Real Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8117 | Triple Angle Formulas/Cosine | :$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 3 \theta
| r = \cos \paren {2 \theta + \theta}
}}
{{eqn | r = \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta
| c = Cosine of Sum
}}
{{eqn | r = \paren {2 \cos^2 \theta - 1} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta
| c = Double Angle Formula for... | :$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 3 \theta
| r = \cos \paren {2 \theta + \theta}
}}
{{eqn | r = \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta
| c = [[Cosine of Sum]]
}}
{{eqn | r = \paren {2 \cos^2 \theta - 1} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta
| c = [[Double Angle Formu... | Triple Angle Formulas/Cosine/Proof 3 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine/Proof_3 | [
"Triple Angle Formula for Cosine",
"Triple Angle Formulas",
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Double Angle Formulas/Cosine",
"Double Angle Formulas/Sine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8118 | Triple Angle Formulas/Tangent | :$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$ | {{begin-eqn}}
{{eqn | l = \tan 3 \theta
| r = \frac {\sin 3 \theta} {\cos 3 \theta}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {3 \sin \theta - 4 \sin^3 \theta} {4 \cos^3 \theta - 3 \cos \theta}
| c = Triple Angle Formula for Sine and Triple Angle Formula for Cosine
}}
{{eqn | r = \f... | :$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$ | {{begin-eqn}}
{{eqn | l = \tan 3 \theta
| r = \frac {\sin 3 \theta} {\cos 3 \theta}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {3 \sin \theta - 4 \sin^3 \theta} {4 \cos^3 \theta - 3 \cos \theta}
| c = [[Triple Angle Formula for Sine]] and [[Triple Angle Formula for Cosine]]
}}
{{... | Triple Angle Formulas/Tangent/Proof 1 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Tangent | https://proofwiki.org/wiki/Triple_Angle_Formulas/Tangent/Proof_1 | [
"Tangent Function",
"Triple Angle Formula for Tangent",
"Triple Angle Formulas"
] | [] | [
"Tangent is Sine divided by Cosine",
"Triple Angle Formulas/Sine",
"Triple Angle Formulas/Cosine",
"Tangent is Sine divided by Cosine",
"Secant is Reciprocal of Cosine",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-8119 | Triple Angle Formulas/Tangent | :$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$ | Let $\theta$ be such that $\tan 2 \theta$ is defined.
Then:
{{begin-eqn}}
{{eqn | l = \tan 3 \theta
| r = \dfrac {\tan \theta + \tan 2 \theta} {1 - \tan \theta \tan 2 \theta}
| c = Tangent of Sum
}}
{{eqn | r = \dfrac {\tan \theta + \dfrac {2 \tan \theta} {1 - \tan^2 \theta} } {1 - \tan \theta \dfrac {2 \ta... | :$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$ | Let $\theta$ be such that $\tan 2 \theta$ is defined.
Then:
{{begin-eqn}}
{{eqn | l = \tan 3 \theta
| r = \dfrac {\tan \theta + \tan 2 \theta} {1 - \tan \theta \tan 2 \theta}
| c = [[Tangent of Sum]]
}}
{{eqn | r = \dfrac {\tan \theta + \dfrac {2 \tan \theta} {1 - \tan^2 \theta} } {1 - \tan \theta \dfrac ... | Triple Angle Formulas/Tangent/Proof 2 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Tangent | https://proofwiki.org/wiki/Triple_Angle_Formulas/Tangent/Proof_2 | [
"Tangent Function",
"Triple Angle Formula for Tangent",
"Triple Angle Formulas"
] | [] | [
"Tangent of Sum",
"Double Angle Formulas/Tangent",
"Definition:Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Triple Angle Formulas/Tangent"
] |
proofwiki-8120 | Triple Angle Formulas/Tangent | :$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$ | From Tangent of Sum of Three Angles:
{{:Tangent of Sum of Three Angles}}
The result follows by setting $\theta = A = B = C$.
{{qed}} | :$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$ | From [[Tangent of Sum of Three Angles]]:
{{:Tangent of Sum of Three Angles}}
The result follows by setting $\theta = A = B = C$.
{{qed}} | Triple Angle Formulas/Tangent/Proof 3 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Tangent | https://proofwiki.org/wiki/Triple_Angle_Formulas/Tangent/Proof_3 | [
"Tangent Function",
"Triple Angle Formula for Tangent",
"Triple Angle Formulas"
] | [] | [
"Tangent of Sum of Three Angles"
] |
proofwiki-8121 | Quadruple Angle Formulas/Sine | :$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$ | {{begin-eqn}}
{{eqn | l = \map \sin {4 \theta}
| r = \map \sin {3 \theta + \theta}
}}
{{eqn | r = \sin 3 \theta \cos \theta + \cos 3 \theta \sin \theta
| c = Sine of Sum
}}
{{eqn | r = \paren {3 \sin \theta - 4 \sin^3 \theta} \cos \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin \theta
| c = Tri... | :$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$ | {{begin-eqn}}
{{eqn | l = \map \sin {4 \theta}
| r = \map \sin {3 \theta + \theta}
}}
{{eqn | r = \sin 3 \theta \cos \theta + \cos 3 \theta \sin \theta
| c = [[Sine of Sum]]
}}
{{eqn | r = \paren {3 \sin \theta - 4 \sin^3 \theta} \cos \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin \theta
| c =... | Quadruple Angle Formulas/Sine/Proof 1 | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Sine/Proof_1 | [
"Quadruple Angle Formula for Sine",
"Sine Function",
"Quadruple Angle Formulas"
] | [] | [
"Sine of Sum",
"Triple Angle Formulas/Sine",
"Triple Angle Formulas/Cosine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8122 | Quadruple Angle Formulas/Sine | :$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 4 \theta + i \sin 4 \theta
| r = \paren {\cos \theta + i \sin \theta}^4
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^4 + \binom 4 1 \paren {\cos \theta}^3 \paren {i \sin \theta} + \binom 4 2 \paren {\cos \theta}^2 \paren {i \sin \theta}^2
}}
{{eqn | o... | :$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 4 \theta + i \sin 4 \theta
| r = \paren {\cos \theta + i \sin \theta}^4
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^4 + \binom 4 1 \paren {\cos \theta}^3 \paren {i \sin \theta} + \binom 4 2 \paren {\cos \theta}^2 \paren {i \sin \theta}^2
}}
{{eq... | Quadruple Angle Formulas/Sine/Proof 2 | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Sine/Proof_2 | [
"Quadruple Angle Formula for Sine",
"Sine Function",
"Quadruple Angle Formulas"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Imaginary Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8123 | Quadruple Angle Formulas/Sine | :$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$ | {{begin-eqn}}
{{eqn | l = \sin {4 \theta}
| r = \map \sin {2 \times 2 \theta}
}}
{{eqn | r = 2 \sin 2 \theta \cos 2 \theta
| c = Double Angle Formula for Sine
}}
{{eqn | r = 2 \paren {2 \sin \theta \cos \theta} \paren {\cos^2 \theta - \sin^2 \theta}
| c = Double Angle Formula for Sine, Double Angle Fo... | :$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$ | {{begin-eqn}}
{{eqn | l = \sin {4 \theta}
| r = \map \sin {2 \times 2 \theta}
}}
{{eqn | r = 2 \sin 2 \theta \cos 2 \theta
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = 2 \paren {2 \sin \theta \cos \theta} \paren {\cos^2 \theta - \sin^2 \theta}
| c = [[Double Angle Formula for Sine]], [[Doubl... | Quadruple Angle Formulas/Sine/Proof 3 | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Sine/Proof_3 | [
"Quadruple Angle Formula for Sine",
"Sine Function",
"Quadruple Angle Formulas"
] | [] | [
"Double Angle Formulas/Sine",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Cosine",
"Distributive Laws/Arithmetic",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8124 | Quadruple Angle Formulas/Cosine | :$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$ | {{begin-eqn}}
{{eqn | l = \cos 4 \theta
| r = \cos \paren {2 \theta + 2 \theta}
}}
{{eqn | r = \cos 2 \theta \cos 2 \theta - \sin 2 \theta \sin 2 \theta
| c = Cosine of Sum
}}
{{eqn | r = \paren {\cos^2 \theta - \sin^2 \theta} \paren {\cos^2 \theta - \sin^2 \theta} - \paren {2 \sin \theta \cos \theta} \pare... | :$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$ | {{begin-eqn}}
{{eqn | l = \cos 4 \theta
| r = \cos \paren {2 \theta + 2 \theta}
}}
{{eqn | r = \cos 2 \theta \cos 2 \theta - \sin 2 \theta \sin 2 \theta
| c = [[Cosine of Sum]]
}}
{{eqn | r = \paren {\cos^2 \theta - \sin^2 \theta} \paren {\cos^2 \theta - \sin^2 \theta} - \paren {2 \sin \theta \cos \theta} \... | Quadruple Angle Formulas/Cosine/Proof 1 | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Cosine/Proof_1 | [
"Quadruple Angle Formula for Cosine",
"Cosine Function",
"Quadruple Angle Formulas"
] | [] | [
"Cosine of Sum",
"Double Angle Formulas",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8125 | Quadruple Angle Formulas/Cosine | :$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 4 \theta + i \sin 4 \theta
| r = \paren {\cos \theta + i \sin \theta}^4
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^4 + \binom 4 1 \paren {\cos \theta}^3 \paren {i \sin \theta} + \binom 4 2 \paren {\cos \theta}^2 \paren {i \sin \theta}^2
}}
{{eqn | o... | :$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 4 \theta + i \sin 4 \theta
| r = \paren {\cos \theta + i \sin \theta}^4
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^4 + \binom 4 1 \paren {\cos \theta}^3 \paren {i \sin \theta} + \binom 4 2 \paren {\cos \theta}^2 \paren {i \sin \theta}^2
}}
{{eq... | Quadruple Angle Formulas/Cosine/Proof 2 | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Cosine/Proof_2 | [
"Quadruple Angle Formula for Cosine",
"Cosine Function",
"Quadruple Angle Formulas"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Real Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8126 | Quadruple Angle Formulas/Tangent | :$\tan 4 \theta = \dfrac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}$ | {{begin-eqn}}
{{eqn | l = \tan 4 \theta
| r = \frac {\sin 4 \theta} {\cos 4 \theta}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta} {8 \cos^4 \theta - 8 \cos^2 \theta + 1}
| c = Quadruple Angle Formula for Sine and Quadruple Angle ... | :$\tan 4 \theta = \dfrac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}$ | {{begin-eqn}}
{{eqn | l = \tan 4 \theta
| r = \frac {\sin 4 \theta} {\cos 4 \theta}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta} {8 \cos^4 \theta - 8 \cos^2 \theta + 1}
| c = [[Quadruple Angle Formula for Sine]] and [[Quadru... | Quadruple Angle Formulas/Tangent | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Tangent | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Tangent | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Quadruple Angle Formulas/Sine",
"Quadruple Angle Formulas/Cosine",
"Secant is Reciprocal of Cosine",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-8127 | Quintuple Angle Formulas/Sine | :$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$ | {{begin-eqn}}
{{eqn | l = \sin 5 \theta
| r = \map \sin {3 \theta + 2 \theta}
}}
{{eqn | r = \sin 3 \theta \cos 2 \theta + \cos 3 \theta \sin 2 \theta
| c = Sine of Sum
}}
{{eqn | r = \paren {3 \sin \theta - 4 \sin^3 \theta} \cos 2 \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin 2 \theta
| c = ... | :$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$ | {{begin-eqn}}
{{eqn | l = \sin 5 \theta
| r = \map \sin {3 \theta + 2 \theta}
}}
{{eqn | r = \sin 3 \theta \cos 2 \theta + \cos 3 \theta \sin 2 \theta
| c = [[Sine of Sum]]
}}
{{eqn | r = \paren {3 \sin \theta - 4 \sin^3 \theta} \cos 2 \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin 2 \theta
| ... | Quintuple Angle Formulas/Sine/Proof 1 | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Sine/Proof_1 | [
"Sine Function",
"Quintuple Angle Formula for Sine",
"Quintuple Angle Formulas"
] | [] | [
"Sine of Sum",
"Triple Angle Formulas",
"Double Angle Formulas",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8128 | Quintuple Angle Formulas/Sine | :$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 5 \theta + i \sin 5 \theta
| r = \paren {\cos \theta + i \sin \theta}^5
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^5 + \binom 5 1 \paren {\cos \theta}^4 \paren {i \sin \theta} + \binom 5 2 \paren {\cos \theta}^3 \paren {i \sin \theta}^2
}}
{{eqn | o... | :$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 5 \theta + i \sin 5 \theta
| r = \paren {\cos \theta + i \sin \theta}^5
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^5 + \binom 5 1 \paren {\cos \theta}^4 \paren {i \sin \theta} + \binom 5 2 \paren {\cos \theta}^3 \paren {i \sin \theta}^2
}}
{{eq... | Quintuple Angle Formulas/Sine/Proof 2 | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Sine/Proof_2 | [
"Sine Function",
"Quintuple Angle Formula for Sine",
"Quintuple Angle Formulas"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Imaginary Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8129 | Quintuple Angle Formulas/Cosine | :$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 5 \theta
| r = \map \cos {4 \theta + \theta}
}}
{{eqn | r = \cos 4 \theta \cos \theta - \sin 4 \theta \sin \theta
| c = Cosine of Sum
}}
{{eqn | r = \paren {8 \cos^4 \theta - 8 \cos^2 \theta + 1} \cos \theta - \paren {4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta} \s... | :$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 5 \theta
| r = \map \cos {4 \theta + \theta}
}}
{{eqn | r = \cos 4 \theta \cos \theta - \sin 4 \theta \sin \theta
| c = [[Cosine of Sum]]
}}
{{eqn | r = \paren {8 \cos^4 \theta - 8 \cos^2 \theta + 1} \cos \theta - \paren {4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta... | Quintuple Angle Formulas/Cosine/Proof 1 | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Cosine/Proof_1 | [
"Cosine Function",
"Quintuple Angle Formula for Cosine",
"Quintuple Angle Formulas"
] | [] | [
"Cosine of Sum",
"Quadruple Angle Formulas",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8130 | Quintuple Angle Formulas/Cosine | :$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 5 \theta + i \sin 5 \theta
| r = \paren {\cos \theta + i \sin \theta}^5
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^5 + \binom 5 1 \paren {\cos \theta}^4 \paren {i \sin \theta} + \binom 5 2 \paren {\cos \theta}^3 \paren {i \sin \theta}^2
| c = ... | :$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$ | We have:
{{begin-eqn}}
{{eqn | l = \cos 5 \theta + i \sin 5 \theta
| r = \paren {\cos \theta + i \sin \theta}^5
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^5 + \binom 5 1 \paren {\cos \theta}^4 \paren {i \sin \theta} + \binom 5 2 \paren {\cos \theta}^3 \paren {i \sin \theta}^2
|... | Quintuple Angle Formulas/Cosine/Proof 2 | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Cosine/Proof_2 | [
"Cosine Function",
"Quintuple Angle Formula for Cosine",
"Quintuple Angle Formulas"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Real Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8131 | Quintuple Angle Formulas/Tangent | :$\tan 5 \theta = \dfrac {\tan^5 \theta - 10 \tan^3 \theta + 5 \tan \theta} {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$ | {{begin-eqn}}
{{eqn | l = \tan 5 \theta
| r = \frac {\sin 5 \theta} {\cos 5 \theta}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta} {16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta}
| c = Quintuple Angle Formulas
}}
{{eqn | r = \fr... | :$\tan 5 \theta = \dfrac {\tan^5 \theta - 10 \tan^3 \theta + 5 \tan \theta} {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$ | {{begin-eqn}}
{{eqn | l = \tan 5 \theta
| r = \frac {\sin 5 \theta} {\cos 5 \theta}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta} {16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta}
| c = [[Quintuple Angle Formulas]]
}}
{{eqn |... | Quintuple Angle Formulas/Tangent | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Tangent | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Tangent | [
"Tangent Function",
"Quintuple Angle Formulas"
] | [] | [
"Tangent is Sine divided by Cosine",
"Quintuple Angle Formulas",
"Secant is Reciprocal of Cosine",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-8132 | Power Reduction Formulas/Sine Squared | :$\sin^2 x = \dfrac {1 - \cos 2 x} 2$ | {{begin-eqn}}
{{eqn | l = 1 - 2 \sin^2 x
| r = \cos 2 x
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = \sin^2 x
| r = \frac {\cos 2 x - 1} {-2}
| c = solving for $\sin^2x$
}}
{{eqn | r = \frac {1 - \cos 2 x} 2
| c = multiplying top and bottom by $... | :$\sin^2 x = \dfrac {1 - \cos 2 x} 2$ | {{begin-eqn}}
{{eqn | l = 1 - 2 \sin^2 x
| r = \cos 2 x
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = \sin^2 x
| r = \frac {\cos 2 x - 1} {-2}
| c = solving for $\sin^2x$
}}
{{eqn | r = \frac {1 - \cos 2 x} 2
| c = multiplying top and bottom by $... | Power Reduction Formulas/Sine Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_Squared | [
"Sine Function"
] | [] | [] |
proofwiki-8133 | Power Reduction Formulas/Cosine Squared | :$\cos^2 x = \dfrac {1 + \cos 2 x} 2$ | {{begin-eqn}}
{{eqn | l = 2 \cos^2 x - 1
| r = \cos 2 x
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | l = \cos^2 x
| r = \frac {1 + \cos 2 x} 2
| c = solving for $\cos^2 x$
}}
{{end-eqn}}
{{qed}} | :$\cos^2 x = \dfrac {1 + \cos 2 x} 2$ | {{begin-eqn}}
{{eqn | l = 2 \cos^2 x - 1
| r = \cos 2 x
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | l = \cos^2 x
| r = \frac {1 + \cos 2 x} 2
| c = solving for $\cos^2 x$
}}
{{end-eqn}}
{{qed}} | Power Reduction Formulas/Cosine Squared/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_Squared/Proof_1 | [
"Cosine Function"
] | [] | [] |
proofwiki-8134 | Power Reduction Formulas/Cosine Squared | :$\cos^2 x = \dfrac {1 + \cos 2 x} 2$ | {{begin-eqn}}
{{eqn | l = \dfrac {1 + \cos 2 x} 2
| r = \dfrac 1 2 \paren {1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}
| c = Euler's Cosine Identity
}}
{{eqn | r = \dfrac 1 4 \paren {e^{2 i x} + 2 + e^{-2 i x} }
| c = simplifying
}}
{{eqn | r = \dfrac 1 4 \paren {e^{2 i x} + 2 \paren {e^{i x} } \paren {e^... | :$\cos^2 x = \dfrac {1 + \cos 2 x} 2$ | {{begin-eqn}}
{{eqn | l = \dfrac {1 + \cos 2 x} 2
| r = \dfrac 1 2 \paren {1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \dfrac 1 4 \paren {e^{2 i x} + 2 + e^{-2 i x} }
| c = simplifying
}}
{{eqn | r = \dfrac 1 4 \paren {e^{2 i x} + 2 \paren {e^{i x} } \paren... | Power Reduction Formulas/Cosine Squared/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_Squared/Proof_2 | [
"Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Square of Sum",
"Euler's Cosine Identity"
] |
proofwiki-8135 | Power Reduction Formulas/Tangent Squared | :$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$ | {{begin-eqn}}
{{eqn | l = \tan^2 x
| r = \frac {\sin^2 x} {\cos^2 x}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\frac {1 - \cos 2 x} 2} {\frac {\cos 2 x + 1} 2}
| c = Square of Sine and Square of Cosine
}}
{{eqn | r = \frac {1 - \cos 2 x} {1 + \cos 2 x}
| c = multiplying top a... | :$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$ | {{begin-eqn}}
{{eqn | l = \tan^2 x
| r = \frac {\sin^2 x} {\cos^2 x}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\frac {1 - \cos 2 x} 2} {\frac {\cos 2 x + 1} 2}
| c = [[Square of Sine]] and [[Square of Cosine]]
}}
{{eqn | r = \frac {1 - \cos 2 x} {1 + \cos 2 x}
| c = multi... | Power Reduction Formulas/Tangent Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Tangent_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Tangent_Squared | [
"Tangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Power Reduction Formulas/Sine Squared",
"Power Reduction Formulas/Cosine Squared",
"Category:Tangent Function"
] |
proofwiki-8136 | Power Reduction Formulas/Sine Cubed | :$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$ | {{begin-eqn}}
{{eqn | l = \sin 3 x
| r = 3 \sin x - 4 \sin^3 x
| c = Triple Angle Formula for Sine
}}
{{eqn | ll= \leadsto
| l = 4 \sin^3 x
| r = 3 \sin x - \sin 3 x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \sin^3 x
| r = \frac {3 \sin x - \sin 3 x} 4
| c = dividin... | :$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$ | {{begin-eqn}}
{{eqn | l = \sin 3 x
| r = 3 \sin x - 4 \sin^3 x
| c = [[Triple Angle Formula for Sine]]
}}
{{eqn | ll= \leadsto
| l = 4 \sin^3 x
| r = 3 \sin x - \sin 3 x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \sin^3 x
| r = \frac {3 \sin x - \sin 3 x} 4
| c = div... | Power Reduction Formulas/Sine Cubed/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_Cubed/Proof_1 | [
"Sine Function",
"Cube of Sine"
] | [] | [
"Triple Angle Formulas/Sine"
] |
proofwiki-8137 | Power Reduction Formulas/Sine Cubed | :$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$ | {{begin-eqn}}
{{eqn | l = \sin^3 x
| r = \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^3
| c = Euler's Sine Identity
}}
{{eqn | r = \frac {\paren {e^{i x} - e^{-i x} }^3} {8 i^3}
| c = rearranging
}}
{{eqn | r = -\frac 1 {8 i} \paren {\paren {e^{i x} }^3 - 3 \paren {e^{i x} }^2 \paren {e^{-i x} } + 3 \p... | :$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$ | {{begin-eqn}}
{{eqn | l = \sin^3 x
| r = \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^3
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac {\paren {e^{i x} - e^{-i x} }^3} {8 i^3}
| c = rearranging
}}
{{eqn | r = -\frac 1 {8 i} \paren {\paren {e^{i x} }^3 - 3 \paren {e^{i x} }^2 \paren {e^{-i x} } + ... | Power Reduction Formulas/Sine Cubed/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_Cubed/Proof_2 | [
"Sine Function",
"Cube of Sine"
] | [] | [
"Euler's Sine Identity",
"Euler's Sine Identity"
] |
proofwiki-8138 | Power Reduction Formulas/Cosine Cubed | :$\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$ | {{begin-eqn}}
{{eqn | l = \cos 3 x
| r = 4 \cos^3 x - 3 \cos x
| c = Triple Angle Formula for Cosine
}}
{{eqn | ll= \leadsto
| l = 4 \cos^3 x
| r = 3 \cos x + \cos 3 x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \cos^3 x
| r = \dfrac {3 \cos x + \cos 3 x} 4
| c = divi... | :$\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$ | {{begin-eqn}}
{{eqn | l = \cos 3 x
| r = 4 \cos^3 x - 3 \cos x
| c = [[Triple Angle Formula for Cosine]]
}}
{{eqn | ll= \leadsto
| l = 4 \cos^3 x
| r = 3 \cos x + \cos 3 x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \cos^3 x
| r = \dfrac {3 \cos x + \cos 3 x} 4
| c = ... | Power Reduction Formulas/Cosine Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_Cubed | [
"Cosine Function"
] | [] | [
"Triple Angle Formulas/Cosine"
] |
proofwiki-8139 | Power Reduction Formulas/Sine to 4th | :$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \sin^4 x
| r = \paren {\sin^2 x}^2
}}
{{eqn | r = \paren {\frac {1 - \cos 2 x} 2}^2
| c = Square of Sine
}}
{{eqn | r = \frac {1 - 2 \cos 2 x + \cos^2 2 x} 4
| c = multiplying out
}}
{{eqn | r = \frac {1 - 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4
| c = Square of Cosine
}}... | :$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \sin^4 x
| r = \paren {\sin^2 x}^2
}}
{{eqn | r = \paren {\frac {1 - \cos 2 x} 2}^2
| c = [[Square of Sine]]
}}
{{eqn | r = \frac {1 - 2 \cos 2 x + \cos^2 2 x} 4
| c = multiplying out
}}
{{eqn | r = \frac {1 - 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4
| c = [[Square of Cos... | Power Reduction Formulas/Sine to 4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_to_4th | [
"Sine Function"
] | [] | [
"Power Reduction Formulas/Sine Squared",
"Power Reduction Formulas/Cosine Squared"
] |
proofwiki-8140 | Power Reduction Formulas/Cosine to 4th | :$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cos^4 x
| r = \paren {\cos^2 x}^2
}}
{{eqn | r = \paren {\frac {1 + \cos 2 x} 2}^2
| c = Square of Cosine
}}
{{eqn | r = \frac {1 + 2 \cos 2 x + \cos^2 2 x} 4
| c = multiplying out
}}
{{eqn | r = \frac {1 + 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4
| c = Square of Cosine
... | :$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cos^4 x
| r = \paren {\cos^2 x}^2
}}
{{eqn | r = \paren {\frac {1 + \cos 2 x} 2}^2
| c = [[Square of Cosine]]
}}
{{eqn | r = \frac {1 + 2 \cos 2 x + \cos^2 2 x} 4
| c = multiplying out
}}
{{eqn | r = \frac {1 + 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4
| c = [[Square of C... | Power Reduction Formulas/Cosine to 4th/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_4th/Proof_1 | [
"Power Reduction Formula for 4th Power of Cosine",
"Cosine Function"
] | [] | [
"Power Reduction Formulas/Cosine Squared",
"Power Reduction Formulas/Cosine Squared",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8141 | Power Reduction Formulas/Cosine to 4th | :$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cos ^4 x
| r = \paren {\frac {e^{i x} + e^{-i x} } 2}^4
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac {\paren {e^{i x} + e^{-i x} }^4} {16}
| c = rearranging
}}
{{eqn | r = \frac {\paren {e^{i x} }^4 + 4 \paren {e^{i x} }^3 \paren {e^{-i x} } + 6 \paren {e^{i x} }^2 \pa... | :$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cos ^4 x
| r = \paren {\frac {e^{i x} + e^{-i x} } 2}^4
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac {\paren {e^{i x} + e^{-i x} }^4} {16}
| c = rearranging
}}
{{eqn | r = \frac {\paren {e^{i x} }^4 + 4 \paren {e^{i x} }^3 \paren {e^{-i x} } + 6 \paren {e^{i x} }^2... | Power Reduction Formulas/Cosine to 4th/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_4th/Proof_2 | [
"Power Reduction Formula for 4th Power of Cosine",
"Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Euler's Cosine Identity"
] |
proofwiki-8142 | Power Reduction Formulas/Sine to 5th | :$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$ | {{begin-eqn}}
{{eqn | l = \sin 5 x
| r = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x
| c = Quintuple Angle Formula for Sine
}}
{{eqn | ll= \leadsto
| l = 16 \sin^5 x
| r = \sin 5 x + 20 \sin^3 x - 5 \sin x
| c = rearranging
}}
{{eqn | r = \sin 5 x + 20 \paren {\frac {3 \sin x - \sin 3 x} 4} - 5 \... | :$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$ | {{begin-eqn}}
{{eqn | l = \sin 5 x
| r = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x
| c = [[Quintuple Angle Formula for Sine]]
}}
{{eqn | ll= \leadsto
| l = 16 \sin^5 x
| r = \sin 5 x + 20 \sin^3 x - 5 \sin x
| c = rearranging
}}
{{eqn | r = \sin 5 x + 20 \paren {\frac {3 \sin x - \sin 3 x} 4} -... | Power Reduction Formulas/Sine to 5th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_to_5th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Sine_to_5th | [
"Sine Function"
] | [] | [
"Quintuple Angle Formulas/Sine",
"Power Reduction Formulas/Sine Cubed"
] |
proofwiki-8143 | Power Reduction Formulas/Cosine to 5th | :$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$ | {{begin-eqn}}
{{eqn | l = \cos 5 x
| r = 16 \cos^5 x - 20 \cos^3 x + 5 \cos x
| c = Quintuple Angle Formula for Cosine
}}
{{eqn | ll= \leadsto
| l = 16 \cos^5 x
| r = \cos 5 x + 20 \cos^3 x - 5 \cos x
| c = rearranging
}}
{{eqn | r = \cos 5 x + 20 \paren {\frac {3 \cos x + \cos 3 x} 4} - 5... | :$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$ | {{begin-eqn}}
{{eqn | l = \cos 5 x
| r = 16 \cos^5 x - 20 \cos^3 x + 5 \cos x
| c = [[Quintuple Angle Formula for Cosine]]
}}
{{eqn | ll= \leadsto
| l = 16 \cos^5 x
| r = \cos 5 x + 20 \cos^3 x - 5 \cos x
| c = rearranging
}}
{{eqn | r = \cos 5 x + 20 \paren {\frac {3 \cos x + \cos 3 x} 4}... | Power Reduction Formulas/Cosine to 5th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_5th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_5th | [
"Cosine Function"
] | [] | [
"Quintuple Angle Formulas/Cosine",
"Power Reduction Formulas/Cosine Cubed"
] |
proofwiki-8144 | Orthocomplement of Subset of Orthocomplement is Superset | Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $A, B \subseteq V$ be subsets of $V$ such that $B \subseteq A^\perp$, where $A^\perp$ is the orthocomplement of $A$.
Then:
:$A \subseteq B^\perp$. | Let $B \subseteq A^\perp$.
Then by Orthocomplement Reverses Subset:
:$A^{\perp\perp} \subseteq B^\perp$
By Double Orthocomplement is Closed Linear Span and the definition of closed linear span:
:$A \subseteq A^{\perp\perp}$
Hence, by Subset Relation is Transitive:
:$A \subseteq B^\perp$
{{qed}}
Category:Inner Product S... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]].
Let $A, B \subseteq V$ be [[Definition:Subset|subsets]] of $V$ such that $B \subseteq A^\perp$, where $A^\perp$ is the [[Definition:Orthocomplement|orthocomplement]] of $A$.
Then:
:$A \subseteq B^\perp$. | Let $B \subseteq A^\perp$.
Then by [[Orthocomplement Reverses Subset]]:
:$A^{\perp\perp} \subseteq B^\perp$
By [[Double Orthocomplement is Closed Linear Span]] and the definition of [[Definition:Closed Linear Span|closed linear span]]:
:$A \subseteq A^{\perp\perp}$
Hence, by [[Subset Relation is Transitive]]:
:$... | Orthocomplement of Subset of Orthocomplement is Superset | https://proofwiki.org/wiki/Orthocomplement_of_Subset_of_Orthocomplement_is_Superset | https://proofwiki.org/wiki/Orthocomplement_of_Subset_of_Orthocomplement_is_Superset | [
"Inner Product Spaces",
"Orthocomplements"
] | [
"Definition:Inner Product Space",
"Definition:Subset",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement"
] | [
"Orthocomplement Reverses Subset",
"Double Orthocomplement is Closed Linear Span",
"Definition:Closed Linear Span",
"Subset Relation is Transitive",
"Category:Inner Product Spaces",
"Category:Orthocomplements"
] |
proofwiki-8145 | Werner Formulas/Cosine by Cosine | :$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2
}}
{{eqn | r = \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} + \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2
| c = Cosine of Difference and Cosine of Sum
}}
{{eqn | r = \frac ... | :$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2
}}
{{eqn | r = \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} + \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2
| c = [[Cosine of Difference]] and [[Cosine of Sum]]
}}
{{eqn | r ... | Werner Formulas/Cosine by Cosine | https://proofwiki.org/wiki/Werner_Formulas/Cosine_by_Cosine | https://proofwiki.org/wiki/Werner_Formulas/Cosine_by_Cosine | [
"Werner Formula for Cosine by Cosine",
"Werner Formulas",
"Cosine Function"
] | [] | [
"Cosine of Difference",
"Cosine of Sum"
] |
proofwiki-8146 | Werner Formulas/Sine by Sine | :$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2
}}
{{eqn | r = \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} - \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2
| c = Cosine of Difference and Cosine of Sum
}}
{{eqn | r = \frac {... | :$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2
}}
{{eqn | r = \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} - \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2
| c = [[Cosine of Difference]] and [[Cosine of Sum]]
}}
{{eqn | r =... | Werner Formulas/Sine by Sine | https://proofwiki.org/wiki/Werner_Formulas/Sine_by_Sine | https://proofwiki.org/wiki/Werner_Formulas/Sine_by_Sine | [
"Werner Formula for Sine by Sine",
"Werner Formulas",
"Sine Function"
] | [] | [
"Cosine of Difference",
"Cosine of Sum"
] |
proofwiki-8147 | Werner Formulas/Cosine by Sine | :$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2
}}
{{eqn | r = \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} - \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2
| c = Sine of Sum and Sine of Difference
}}
{{eqn | r = \frac {2 \... | :$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2
}}
{{eqn | r = \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} - \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2
| c = [[Sine of Sum]] and [[Sine of Difference]]
}}
{{eqn | r = \f... | Werner Formulas/Cosine by Sine/Proof 1 | https://proofwiki.org/wiki/Werner_Formulas/Cosine_by_Sine | https://proofwiki.org/wiki/Werner_Formulas/Cosine_by_Sine/Proof_1 | [
"Werner Formula for Cosine by Sine",
"Werner Formulas",
"Sine Function",
"Cosine Function"
] | [] | [
"Sine of Sum",
"Sine of Difference"
] |
proofwiki-8148 | Werner Formulas/Cosine by Sine | :$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | l = \cos \alpha \sin \beta
| r = \sin \beta \cos \alpha
}}
{{eqn | r = \frac {\map \sin {\beta + \alpha} + \map \sin {\beta - \alpha} } 2
| c = Werner Formula for Sine by Cosine
}}
{{eqn | r = \frac {\map \sin {\alpha + \beta} + \map \sin {-\paren {\alpha - \beta} } } 2
| c =
}}... | :$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | l = \cos \alpha \sin \beta
| r = \sin \beta \cos \alpha
}}
{{eqn | r = \frac {\map \sin {\beta + \alpha} + \map \sin {\beta - \alpha} } 2
| c = [[Werner Formula for Sine by Cosine]]
}}
{{eqn | r = \frac {\map \sin {\alpha + \beta} + \map \sin {-\paren {\alpha - \beta} } } 2
| c =... | Werner Formulas/Cosine by Sine/Proof 2 | https://proofwiki.org/wiki/Werner_Formulas/Cosine_by_Sine | https://proofwiki.org/wiki/Werner_Formulas/Cosine_by_Sine/Proof_2 | [
"Werner Formula for Cosine by Sine",
"Werner Formulas",
"Sine Function",
"Cosine Function"
] | [] | [
"Werner Formulas/Sine by Cosine",
"Sine Function is Odd"
] |
proofwiki-8149 | Werner Formulas/Sine by Cosine | :$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\sin \paren {\alpha + \beta} + \sin \paren {\alpha - \beta} } 2
}}
{{eqn | r = \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} + \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2
| c = Sine of Sum and Sine of Difference
}}
{{eqn | r = \frac ... | :$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\sin \paren {\alpha + \beta} + \sin \paren {\alpha - \beta} } 2
}}
{{eqn | r = \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} + \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2
| c = [[Sine of Sum]] and [[Sine of Difference]]
}}
{{eqn | r ... | Werner Formulas/Sine by Cosine/Proof 1 | https://proofwiki.org/wiki/Werner_Formulas/Sine_by_Cosine | https://proofwiki.org/wiki/Werner_Formulas/Sine_by_Cosine/Proof_1 | [
"Werner Formula for Sine by Cosine",
"Werner Formulas",
"Sine Function",
"Cosine Function"
] | [] | [
"Sine of Sum",
"Sine of Difference"
] |
proofwiki-8150 | Werner Formulas/Sine by Cosine | :$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \sin \alpha \cos \beta
}}
{{eqn | r = 2 \paren {\dfrac {\map \exp {i \alpha} - \map \exp {-i \alpha} } {2 i} } \paren {\dfrac {\map \exp {i \beta} + \map \exp {-i \beta} } 2}
| c = Euler's Sine Identity and Euler's Cosine Identity
}}
{{eqn | r = \frac 1 {2 i} \paren {\map ... | :$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \sin \alpha \cos \beta
}}
{{eqn | r = 2 \paren {\dfrac {\map \exp {i \alpha} - \map \exp {-i \alpha} } {2 i} } \paren {\dfrac {\map \exp {i \beta} + \map \exp {-i \beta} } 2}
| c = [[Euler's Sine Identity]] and [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 1 {2 i} \pare... | Werner Formulas/Sine by Cosine/Proof 2 | https://proofwiki.org/wiki/Werner_Formulas/Sine_by_Cosine | https://proofwiki.org/wiki/Werner_Formulas/Sine_by_Cosine/Proof_2 | [
"Werner Formula for Sine by Cosine",
"Werner Formulas",
"Sine Function",
"Cosine Function"
] | [] | [
"Euler's Sine Identity",
"Euler's Cosine Identity"
] |
proofwiki-8151 | Sine of Integer Multiple of Argument/Formulation 1 | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \sin \theta \paren {\paren {2 \cos \theta}^{n - 1} - \dbinom {n - 2} 1 \paren {2 \cos \theta}^{n - 3} + \dbinom {n - 3} 2 \paren {2 \cos \theta}^{n - 5} - \cdots}
| c =
}}
{{eqn | r = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$ | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \sin \theta \paren {\paren {2 \cos \theta}^{n - 1} - \dbinom {n - 2} 1 \paren {2 \cos \theta}^{n - 3} + \dbinom {n - 3} 2 \paren {2 \cos \theta}^{n - 5} - \cdots}
| c =
}}
{{eqn | r = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sin n \theta = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }... | Sine of Integer Multiple of Argument/Formulation 1 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_1 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_1 | [
"Sine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8152 | Cosine to Power of Odd Integer | :$\ds \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \sum_{k \mathop = 0}^n \binom {2 n + 1} k \cos \paren {2 n - 2 k + 1} \theta$ | {{begin-eqn}}
{{eqn | l = \cos^{2 n + 1} \theta
| r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^{2 n + 1}
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac 1 {2^{2 n + 1} } \paren {e^{i \theta} + e^{-i \theta} }^{2 n + 1}
| c = Power of Product
}}
{{eqn | r = \frac 1 {2^{2 n + 1} } \sum^{2 n +... | :$\ds \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \sum_{k \mathop = 0}^n \binom {2 n + 1} k \cos \paren {2 n - 2 k + 1} \theta$ | {{begin-eqn}}
{{eqn | l = \cos^{2 n + 1} \theta
| r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^{2 n + 1}
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 1 {2^{2 n + 1} } \paren {e^{i \theta} + e^{-i \theta} }^{2 n + 1}
| c = [[Power of Product]]
}}
{{eqn | r = \frac 1 {2^{2 n + 1} } \su... | Cosine to Power of Odd Integer/Proof 1 | https://proofwiki.org/wiki/Cosine_to_Power_of_Odd_Integer | https://proofwiki.org/wiki/Cosine_to_Power_of_Odd_Integer/Proof_1 | [
"Cosine Function",
"Cosine to Power of Odd Integer"
] | [] | [
"Euler's Cosine Identity",
"Exponent Combination Laws/Power of Product",
"Binomial Theorem",
"Exponential of Sum",
"Symmetry Rule for Binomial Coefficients",
"Euler's Cosine Identity"
] |
proofwiki-8153 | Cosine to Power of Odd Integer | :$\ds \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \sum_{k \mathop = 0}^n \binom {2 n + 1} k \cos \paren {2 n - 2 k + 1} \theta$ | {{begin-eqn}}
{{eqn | l = \cos^n \theta
| r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n
| c = De Moivre's Theorem
}}
{{eqn | r = \frac {\paren {e^{i \theta} + e^{-i \theta} }^n} {2^n}
| c =
}}
{{eqn | r = \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - k} i \theta} e^{-k i \th... | :$\ds \cos^{2 n + 1} \theta = \frac 1 {2^{2 n} } \sum_{k \mathop = 0}^n \binom {2 n + 1} k \cos \paren {2 n - 2 k + 1} \theta$ | {{begin-eqn}}
{{eqn | l = \cos^n \theta
| r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n
| c = [[De Moivre's Theorem]]
}}
{{eqn | r = \frac {\paren {e^{i \theta} + e^{-i \theta} }^n} {2^n}
| c =
}}
{{eqn | r = \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - k} i \theta} e^{-k i... | Cosine to Power of Odd Integer/Proof 2 | https://proofwiki.org/wiki/Cosine_to_Power_of_Odd_Integer | https://proofwiki.org/wiki/Cosine_to_Power_of_Odd_Integer/Proof_2 | [
"Cosine Function",
"Cosine to Power of Odd Integer"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient"
] |
proofwiki-8154 | Sum of Arctangent and Arccotangent | Let $x \in \R$ be a real number.
Then:
:$\arctan x + \arccot x = \dfrac \pi 2$
where $\arctan$ and $\arccot$ denote arctangent and arccotangent respectively. | Let $y \in \R$ such that:
:$\exists x \in \R: x = \map \cot {y + \dfrac \pi 2}$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \map \cot {y + \frac \pi 2}
| c =
}}
{{eqn | r = -\tan y
| c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = \map \tan {-y}
| c = Tangent Function is Odd
}}
{{end-eqn}}
S... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\arctan x + \arccot x = \dfrac \pi 2$
where $\arctan$ and $\arccot$ denote [[Definition:Real Arctangent|arctangent]] and [[Definition:Real Arccotangent|arccotangent]] respectively. | Let $y \in \R$ such that:
:$\exists x \in \R: x = \map \cot {y + \dfrac \pi 2}$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \map \cot {y + \frac \pi 2}
| c =
}}
{{eqn | r = -\tan y
| c = [[Cotangent of Angle plus Right Angle]]
}}
{{eqn | r = \map \tan {-y}
| c = [[Tangent Function is Odd]]
}}
{{en... | Sum of Arctangent and Arccotangent | https://proofwiki.org/wiki/Sum_of_Arctangent_and_Arccotangent | https://proofwiki.org/wiki/Sum_of_Arctangent_and_Arccotangent | [
"Arctangent Function",
"Arccotangent Function"
] | [
"Definition:Real Number",
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Inverse Cotangent/Real/Arccotangent"
] | [
"Cotangent of Angle plus Right Angle",
"Tangent Function is Odd"
] |
proofwiki-8155 | Arcsine of Reciprocal equals Arccosecant | :$\map \arcsin {\dfrac 1 x} = \arccsc x$ | {{begin-eqn}}
{{eqn | l = \map \arcsin {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \sin y
| c = {{Defof|Real Arcsine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \csc y
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | ll= \leadstoandfr... | :$\map \arcsin {\dfrac 1 x} = \arccsc x$ | {{begin-eqn}}
{{eqn | l = \map \arcsin {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \sin y
| c = {{Defof|Real Arcsine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \csc y
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | ll= \leadstoa... | Arcsine of Reciprocal equals Arccosecant | https://proofwiki.org/wiki/Arcsine_of_Reciprocal_equals_Arccosecant | https://proofwiki.org/wiki/Arcsine_of_Reciprocal_equals_Arccosecant | [
"Arcsine Function",
"Arccosecant Function",
"Reciprocals"
] | [] | [
"Cosecant is Reciprocal of Sine"
] |
proofwiki-8156 | Arccosine of Reciprocal equals Arcsecant | :$\map \arccos {\dfrac 1 x} = \arcsec x$ | {{begin-eqn}}
{{eqn | l = \map \arccos {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \cos y
| c = {{Defof|Real Arccosine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sec y
| c = Secant is Reciprocal of Cosine
}}
{{eqn | ll= \leadstoand... | :$\map \arccos {\dfrac 1 x} = \arcsec x$ | {{begin-eqn}}
{{eqn | l = \map \arccos {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \cos y
| c = {{Defof|Real Arccosine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sec y
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | ll= \leadst... | Arccosine of Reciprocal equals Arcsecant | https://proofwiki.org/wiki/Arccosine_of_Reciprocal_equals_Arcsecant | https://proofwiki.org/wiki/Arccosine_of_Reciprocal_equals_Arcsecant | [
"Arccosine Function",
"Arcsecant Function",
"Reciprocals"
] | [] | [
"Secant is Reciprocal of Cosine"
] |
proofwiki-8157 | Arctangent of Reciprocal equals Arccotangent | :$\map \arctan {\dfrac 1 x} = \arccot x$ | {{begin-eqn}}
{{eqn | l = \map \arctan {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \tan y
| c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cot y
| c = Cotangent is Reciprocal of Tangent
}}
{{eqn | ll= \leads... | :$\map \arctan {\dfrac 1 x} = \arccot x$ | {{begin-eqn}}
{{eqn | l = \map \arctan {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \tan y
| c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cot y
| c = [[Cotangent is Reciprocal of Tangent]]
}}
{{eqn | ll= \l... | Arctangent of Reciprocal equals Arccotangent | https://proofwiki.org/wiki/Arctangent_of_Reciprocal_equals_Arccotangent | https://proofwiki.org/wiki/Arctangent_of_Reciprocal_equals_Arccotangent | [
"Arctangent Function",
"Arccotangent Function",
"Reciprocals"
] | [] | [
"Cotangent is Reciprocal of Tangent"
] |
proofwiki-8158 | Inverse Sine is Odd Function | Everywhere that the function is defined:
:$\map \arcsin {-x} = -\arcsin x$ | {{begin-eqn}}
{{eqn | l = \map \arcsin {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \sin y:
| rr = -\frac \pi 2 \le y \le \frac \pi 2
| c = {{Defof|Real Arcsine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\sin y:
| rr = -\frac \pi 2 \le y \l... | Everywhere that the function is defined:
:$\map \arcsin {-x} = -\arcsin x$ | {{begin-eqn}}
{{eqn | l = \map \arcsin {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \sin y:
| rr = -\frac \pi 2 \le y \le \frac \pi 2
| c = {{Defof|Real Arcsine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\sin y:
| rr = -\frac \pi 2 \le y \l... | Inverse Sine is Odd Function | https://proofwiki.org/wiki/Inverse_Sine_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Sine_is_Odd_Function | [
"Arcsine Function",
"Examples of Odd Functions"
] | [] | [
"Sine Function is Odd"
] |
proofwiki-8159 | Arccosine of Negative Argument | Everywhere that the function is defined:
:$\map \arccos {-x} = \pi - \arccos x$ | {{begin-eqn}}
{{eqn | l = \map \arccos {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \cos y:
| rr= 0 \le y \le \pi
| c = {{Defof|Real Arccosine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\cos y:
| rr= -\pi \le y \le 0
| c =
}}
{{eqn |... | Everywhere that the function is defined:
:$\map \arccos {-x} = \pi - \arccos x$ | {{begin-eqn}}
{{eqn | l = \map \arccos {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \cos y:
| rr= 0 \le y \le \pi
| c = {{Defof|Real Arccosine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\cos y:
| rr= -\pi \le y \le 0
| c =
}}
{{eqn |... | Arccosine of Negative Argument | https://proofwiki.org/wiki/Arccosine_of_Negative_Argument | https://proofwiki.org/wiki/Arccosine_of_Negative_Argument | [
"Arccosine Function"
] | [] | [
"Cosine of Supplementary Angle"
] |
proofwiki-8160 | Inverse Tangent is Odd Function | Everywhere that the function is defined:
:$\map \arctan {-x} = -\arctan x$ | {{begin-eqn}}
{{eqn | l = \map \arctan {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \tan y:
| rr= -\frac \pi 2 \le y \le \frac \pi 2
| c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\tan y:
| rr= -\frac \pi 2 \le y \... | Everywhere that the function is defined:
:$\map \arctan {-x} = -\arctan x$ | {{begin-eqn}}
{{eqn | l = \map \arctan {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \tan y:
| rr= -\frac \pi 2 \le y \le \frac \pi 2
| c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\tan y:
| rr= -\frac \pi 2 \le y \... | Inverse Tangent is Odd Function | https://proofwiki.org/wiki/Inverse_Tangent_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Tangent_is_Odd_Function | [
"Arctangent Function",
"Examples of Odd Functions"
] | [] | [
"Tangent Function is Odd"
] |
proofwiki-8161 | Arccotangent of Negative Argument | Everywhere that the function is defined:
:$\map \arccot {-x} = \pi - \arccot x$ | {{begin-eqn}}
{{eqn | l = \map \arccot {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \cot y:
| rr= 0 \le y \le \pi
| c = {{Defof|Arccotangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\cot y:
| rr= -\pi \le y \le 0
| c =
}}
{{eqn | l... | Everywhere that the function is defined:
:$\map \arccot {-x} = \pi - \arccot x$ | {{begin-eqn}}
{{eqn | l = \map \arccot {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \cot y:
| rr= 0 \le y \le \pi
| c = {{Defof|Arccotangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\cot y:
| rr= -\pi \le y \le 0
| c =
}}
{{eqn | l... | Arccotangent of Negative Argument | https://proofwiki.org/wiki/Arccotangent_of_Negative_Argument | https://proofwiki.org/wiki/Arccotangent_of_Negative_Argument | [
"Arccotangent Function"
] | [] | [
"Cotangent of Supplementary Angle"
] |
proofwiki-8162 | Arcsecant of Negative Argument | Everywhere that the function is defined:
:$\map \arcsec {-x} = \pi - \arcsec x$ | {{begin-eqn}}
{{eqn | l = \map \arcsec {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \sec y:
| rr= 0 \le y \le \pi
| c = {{Defof|Arcsecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\sec y:
| rr= -\pi \le y \le 0
| c =
}}
{{eqn | ll= ... | Everywhere that the function is defined:
:$\map \arcsec {-x} = \pi - \arcsec x$ | {{begin-eqn}}
{{eqn | l = \map \arcsec {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \sec y:
| rr= 0 \le y \le \pi
| c = {{Defof|Arcsecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\sec y:
| rr= -\pi \le y \le 0
| c =
}}
{{eqn | ll= ... | Arcsecant of Negative Argument | https://proofwiki.org/wiki/Arcsecant_of_Negative_Argument | https://proofwiki.org/wiki/Arcsecant_of_Negative_Argument | [
"Arcsecant Function"
] | [] | [
"Cosecant of Supplementary Angle"
] |
proofwiki-8163 | Inverse Cosecant is Odd Function | Everywhere that the function is defined:
:$\map \arccsc {-x} = -\arccsc x$ | {{begin-eqn}}
{{eqn | l = \map \arccsc {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \csc y:
| rr= -\frac \pi 2 \le y \le \frac \pi 2
| c = {{Defof|Arccosecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\csc y:
| rr= -\frac \pi 2 \le y \le \... | Everywhere that the function is defined:
:$\map \arccsc {-x} = -\arccsc x$ | {{begin-eqn}}
{{eqn | l = \map \arccsc {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \csc y:
| rr= -\frac \pi 2 \le y \le \frac \pi 2
| c = {{Defof|Arccosecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\csc y:
| rr= -\frac \pi 2 \le y \le \... | Inverse Cosecant is Odd Function | https://proofwiki.org/wiki/Inverse_Cosecant_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Cosecant_is_Odd_Function | [
"Arccosecant Function",
"Examples of Odd Functions"
] | [] | [
"Cosecant Function is Odd"
] |
proofwiki-8164 | Law of Tangents | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$ | Let $d = \dfrac a {\sin A}$.
From the Law of Sines, let:
:$d = \dfrac a {\sin A} = \dfrac b {\sin B}$
so that:
{{begin-eqn}}
{{eqn | l = a
| r = d \sin A
| c =
}}
{{eqn | l = b
| r = d \sin B
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {a + b} {a - b}
| r = \frac {d \sin A + d \sin B... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$ | Let $d = \dfrac a {\sin A}$.
From the [[Law of Sines]], let:
:$d = \dfrac a {\sin A} = \dfrac b {\sin B}$
so that:
{{begin-eqn}}
{{eqn | l = a
| r = d \sin A
| c =
}}
{{eqn | l = b
| r = d \sin B
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {a + b} {a - b}
| r = \frac {d \sin A + d ... | Law of Tangents | https://proofwiki.org/wiki/Law_of_Tangents | https://proofwiki.org/wiki/Law_of_Tangents | [
"Law of Tangents",
"Triangles",
"Tangent Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)"
] | [
"Law of Sines",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Tangent is Sine divided by Cosine"
] |
proofwiki-8165 | Law of Tangents | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$ | {{begin-eqn}}
{{eqn | l = \dfrac {a + b} {a - b}
| r = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }
| c = Law of Tangents
}}
{{eqn | ll= \leadsto
| l = \tan \frac {A - B} 2
| r = \dfrac {a - b} {a + b} \tan \frac {A + B} 2
| c = algebraic manipulation
}}
{{eqn |... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$ | {{begin-eqn}}
{{eqn | l = \dfrac {a + b} {a - b}
| r = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }
| c = [[Law of Tangents]]
}}
{{eqn | ll= \leadsto
| l = \tan \frac {A - B} 2
| r = \dfrac {a - b} {a + b} \tan \frac {A + B} 2
| c = algebraic manipulation
}}
{{e... | Law of Tangents/Corollary/Proof 1 | https://proofwiki.org/wiki/Law_of_Tangents | https://proofwiki.org/wiki/Law_of_Tangents/Corollary/Proof_1 | [
"Law of Tangents",
"Triangles",
"Tangent Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)"
] | [
"Law of Tangents",
"Sum of Angles of Triangle equals Two Right Angles",
"Tangent of Complement equals Cotangent"
] |
proofwiki-8166 | Law of Tangents | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$ | {{begin-eqn}}
{{eqn | l = \dfrac {a - b} {a + b}
| r = \dfrac {2 R \sin A - 2 R \sin B} {2 R \sin A + 2 R \sin B}
| c = Law of Sines
}}
{{eqn | r = \dfrac {2 \cos \frac {A + B} 2 \sin \frac {A - B} 2} {2 \sin \frac {A + B} 2 \cos \frac {A - B} 2}
| c = Sine minus Sine, Sine plus Sine
}}
{{eqn | r = \d... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$ | {{begin-eqn}}
{{eqn | l = \dfrac {a - b} {a + b}
| r = \dfrac {2 R \sin A - 2 R \sin B} {2 R \sin A + 2 R \sin B}
| c = [[Law of Sines]]
}}
{{eqn | r = \dfrac {2 \cos \frac {A + B} 2 \sin \frac {A - B} 2} {2 \sin \frac {A + B} 2 \cos \frac {A - B} 2}
| c = [[Sine minus Sine]], [[Sine plus Sine]]
}}
{{... | Law of Tangents/Corollary/Proof 2 | https://proofwiki.org/wiki/Law_of_Tangents | https://proofwiki.org/wiki/Law_of_Tangents/Corollary/Proof_2 | [
"Law of Tangents",
"Triangles",
"Tangent Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)"
] | [
"Law of Sines",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Tangent is Sine divided by Cosine",
"Sum of Angles of Triangle equals Two Right Angles",
"Tangent of Complement equals Cotangent",
"Tangent of Complement equals Cotangent"
] |
proofwiki-8167 | Sine of Angle of Triangle by Semiperimeter | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
: $\sin A = \dfrac 2 {b c} \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $\sin$ denotes sine and $s$ is the semiperimeter: $s = \dfrac {a + b + c} 2$. | Let $Q$ be the area of $\triangle ABC$.
From Area of Triangle in Terms of Two Sides and Angle:
:$Q = \dfrac {b c \sin A} 2$
From Heron's Formula:
:$Q = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
Equating the two:
:$\dfrac {b c \sin A} 2 = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
from which... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
: $\sin A = \dfrac 2 {b c} \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $\sin$ denotes [[Definition:Sine|sine]] and $s$ is t... | Let $Q$ be the [[Definition:Area|area]] of $\triangle ABC$.
From [[Area of Triangle in Terms of Two Sides and Angle]]:
:$Q = \dfrac {b c \sin A} 2$
From [[Heron's Formula]]:
:$Q = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
Equating the two:
:$\dfrac {b c \sin A} 2 = \sqrt {s \paren {s - a} \paren {s - ... | Sine of Angle of Triangle by Semiperimeter | https://proofwiki.org/wiki/Sine_of_Angle_of_Triangle_by_Semiperimeter | https://proofwiki.org/wiki/Sine_of_Angle_of_Triangle_by_Semiperimeter | [
"Triangles",
"Sine Function"
] | [
"Definition:Triangle (Geometry)",
"Definition:Sine",
"Definition:Semiperimeter"
] | [
"Definition:Area",
"Area of Triangle in Terms of Two Sides and Angle",
"Heron's Formula"
] |
proofwiki-8168 | Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Conditional | : $\set {\neg, \land, \lor, \implies}$: Not, And, Or and Implies | From Functional Completeness over Finite Number of Arguments, it suffices to consider binary truth functions.
From Count of Truth Functions, there are $16$ of these.
These are enumerated in Binary Truth Functions, and are analysed in turn as follows. | : $\set {\neg, \land, \lor, \implies}$: [[Definition:Logical Not|Not]], [[Definition:Conjunction|And]], [[Definition:Disjunction|Or]] and [[Definition:Conditional|Implies]] | From [[Functional Completeness over Finite Number of Arguments]], it suffices to consider binary [[Definition:Truth Function|truth functions]].
From [[Count of Truth Functions]], there are $16$ of these.
These are enumerated in [[Binary Truth Functions]], and are analysed in turn as follows. | Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Conditional | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation,_Conjunction,_Disjunction_and_Conditional | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation,_Conjunction,_Disjunction_and_Conditional | [
"Functional Completeness"
] | [
"Definition:Logical Not",
"Definition:Conjunction",
"Definition:Disjunction",
"Definition:Conditional"
] | [
"Functional Completeness over Finite Number of Arguments",
"Definition:Truth Function",
"Count of Truth Functions",
"Binary Truth Functions",
"Binary Truth Functions"
] |
proofwiki-8169 | Functionally Complete Logical Connectives/Negation and Disjunction | :$\set {\neg, \lor}$: Not and Or | From Functionally Complete Logical Connectives: Negation and Conjunction, $\set {\neg, \land}$ is functionally complete.
That is: any expression can be expressed in terms of $\neg$ and $\land$.
From De Morgan's laws: Conjunction, we have that:
:$p \land q \dashv \vdash \neg \paren {\neg p \lor \neg q}$
Thus all occurre... | :$\set {\neg, \lor}$: [[Definition:Logical Not|Not]] and [[Definition:Disjunction|Or]] | From [[Functionally Complete Logical Connectives/Negation and Conjunction|Functionally Complete Logical Connectives: Negation and Conjunction]], $\set {\neg, \land}$ is [[Definition:Functionally Complete|functionally complete]].
That is: any expression can be expressed in terms of $\neg$ and $\land$.
From [[De Morgan... | Functionally Complete Logical Connectives/Negation and Disjunction | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation_and_Disjunction | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation_and_Disjunction | [
"Functional Completeness"
] | [
"Definition:Logical Not",
"Definition:Disjunction"
] | [
"Functionally Complete Logical Connectives/Negation and Conjunction",
"Definition:Functionally Complete",
"De Morgan's Laws (Logic)/Conjunction",
"Definition:Functionally Complete"
] |
proofwiki-8170 | Functionally Complete Logical Connectives/Negation and Conjunction | :$\set {\neg, \land}$: Not and And | From Functionally Complete Logical Connectives: Negation, Conjunction, Disjunction and Conditional, all sixteen of the binary truth functions can be expressed in terms of $\neg, \land, \lor, \implies$.
From Conjunction and Conditional, we have that:
:$p \implies q \dashv \vdash \neg \paren {p \land \neg q}$
From De Mor... | :$\set {\neg, \land}$: [[Definition:Logical Not|Not]] and [[Definition:Conjunction|And]] | From [[Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Conditional|Functionally Complete Logical Connectives: Negation, Conjunction, Disjunction and Conditional]], all sixteen of the [[Binary Truth Functions|binary truth functions]] can be expressed in terms of $\neg, \land, \lor, \impl... | Functionally Complete Logical Connectives/Negation and Conjunction | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation_and_Conjunction | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation_and_Conjunction | [
"Functional Completeness"
] | [
"Definition:Logical Not",
"Definition:Conjunction"
] | [
"Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Conditional",
"Binary Truth Functions",
"Conjunction and Conditional",
"De Morgan's Laws (Logic)/Disjunction",
"Definition:Functionally Complete"
] |
proofwiki-8171 | Functionally Complete Logical Connectives/Negation and Conditional | :$\set {\neg, \implies}$: Not and Implies | From Functionally Complete Logical Connectives: Negation and Conjunction, we can represent any boolean expression in terms of $\land$ and $\neg$.
From Conjunction and Conditional, we have that:
:$p \land q \dashv \vdash \neg \paren {p \implies \neg q}$
So it follows that we can replace all occurrences of $\land$ by $\i... | :$\set {\neg, \implies}$: [[Definition:Logical Not|Not]] and [[Definition:Conditional|Implies]] | From [[Functionally Complete Logical Connectives/Negation and Conjunction|Functionally Complete Logical Connectives: Negation and Conjunction]], we can represent any boolean expression in terms of $\land$ and $\neg$.
From [[Conjunction and Conditional]], we have that:
:$p \land q \dashv \vdash \neg \paren {p \implies ... | Functionally Complete Logical Connectives/Negation and Conditional | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation_and_Conditional | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Negation_and_Conditional | [
"Functional Completeness"
] | [
"Definition:Logical Not",
"Definition:Conditional"
] | [
"Functionally Complete Logical Connectives/Negation and Conjunction",
"Conjunction and Conditional",
"Definition:Functionally Complete"
] |
proofwiki-8172 | Functionally Complete Logical Connectives/Conjunction, Negation and Disjunction | :$\set {\neg, \land, \lor}$: Not, And and Or | From the stronger results:
:Functionally Complete Logical Connectives: Negation and Disjunction:
::the set of logical connectives: $\set {\neg, \lor}$ is functionally complete
:Functionally Complete Logical Connectives: Negation and Conjunction:
::the set of logical connectives: $\set {\neg, \land}$ is functionally com... | :$\set {\neg, \land, \lor}$: [[Definition:Logical Not|Not]], [[Definition:Conjunction|And]] and [[Definition:Disjunction|Or]] | From the [[Definition:Stronger Statement|stronger]] results:
:[[Functionally Complete Logical Connectives/Negation and Disjunction|Functionally Complete Logical Connectives: Negation and Disjunction]]:
::the [[Definition:Set|set]] of [[Definition:Logical Connective|logical connectives]]: $\set {\neg, \lor}$ is [[Defini... | Functionally Complete Logical Connectives/Conjunction, Negation and Disjunction | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Conjunction,_Negation_and_Disjunction | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/Conjunction,_Negation_and_Disjunction | [
"Functional Completeness"
] | [
"Definition:Logical Not",
"Definition:Conjunction",
"Definition:Disjunction"
] | [
"Definition:Conditional/Language of Conditional/Strong",
"Functionally Complete Logical Connectives/Negation and Disjunction",
"Definition:Set",
"Definition:Logical Connective",
"Definition:Functionally Complete",
"Functionally Complete Logical Connectives/Negation and Conjunction",
"Definition:Set",
... |
proofwiki-8173 | Functionally Complete Logical Connectives/NAND | :$\set \uparrow$: NAND | From Functionally Complete Logical Connectives: Negation and Conjunction, any boolean expression can be expressed in terms of $\land$ and $\neg$.
From NAND with Equal Arguments:
:$\neg p \dashv \vdash p \uparrow p$
From Conjunction in terms of NAND:
:$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \up... | :$\set \uparrow$: [[Definition:Logical NAND|NAND]] | From [[Functionally Complete Logical Connectives/Negation and Conjunction|Functionally Complete Logical Connectives: Negation and Conjunction]], any boolean expression can be expressed in terms of $\land$ and $\neg$.
From [[NAND with Equal Arguments]]:
:$\neg p \dashv \vdash p \uparrow p$
From [[Conjunction in term... | Functionally Complete Logical Connectives/NAND | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/NAND | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/NAND | [
"Functional Completeness",
"Logical NAND"
] | [
"Definition:Logical NAND"
] | [
"Functionally Complete Logical Connectives/Negation and Conjunction",
"NAND with Equal Arguments",
"Conjunction in terms of NAND",
"Definition:Functionally Complete"
] |
proofwiki-8174 | Functionally Complete Logical Connectives/NOR | :$\set \downarrow$: NOR | From Functionally Complete Logical Connectives: Negation and Disjunction, any boolean expression can be expressed in terms of $\lor$ and $\neg$.
From NOR with Equal Arguments:
:$\neg p \dashv \vdash p \downarrow p$
From Disjunction in terms of NOR:
:$p \lor q \dashv \vdash \paren {p \downarrow q} \downarrow \paren {p \... | :$\set \downarrow$: [[Definition:Logical NOR|NOR]] | From [[Functionally Complete Logical Connectives/Negation and Disjunction|Functionally Complete Logical Connectives: Negation and Disjunction]], any boolean expression can be expressed in terms of $\lor$ and $\neg$.
From [[NOR with Equal Arguments]]:
:$\neg p \dashv \vdash p \downarrow p$
From [[Disjunction in term... | Functionally Complete Logical Connectives/NOR | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/NOR | https://proofwiki.org/wiki/Functionally_Complete_Logical_Connectives/NOR | [
"Functional Completeness",
"Logical NOR"
] | [
"Definition:Logical NOR"
] | [
"Functionally Complete Logical Connectives/Negation and Disjunction",
"NOR with Equal Arguments",
"Disjunction in terms of NOR",
"Definition:Functionally Complete"
] |
proofwiki-8175 | Functionally Complete Singleton Sets | The only binary logical connectives that form singleton sets which are functionally complete are NAND and NOR. | Let $\uparrow$ and $\downarrow$ denote NAND and NOR respectively.
From:
: NAND is Functionally Complete
and
: NOR is Functionally Complete
the singleton sets $\left\{{\uparrow}\right\}$ and $\left\{{\downarrow}\right\}$ are functionally complete.
Suppose $\circ$ is a binary logical connective such that $\left\{{\circ}\... | The only [[Definition:Binary Logical Connective|binary logical connectives]] that form [[Definition:Singleton|singleton sets]] which are [[Definition:Functionally Complete|functionally complete]] are [[Definition:Logical NAND|NAND]] and [[Definition:Logical NOR|NOR]]. | Let $\uparrow$ and $\downarrow$ denote [[Definition:Logical NAND|NAND]] and [[Definition:Logical NOR|NOR]] respectively.
From:
: [[NAND is Functionally Complete]]
and
: [[NOR is Functionally Complete]]
the [[Definition:Singleton|singleton sets]] $\left\{{\uparrow}\right\}$ and $\left\{{\downarrow}\right\}$ are [[Defin... | Functionally Complete Singleton Sets | https://proofwiki.org/wiki/Functionally_Complete_Singleton_Sets | https://proofwiki.org/wiki/Functionally_Complete_Singleton_Sets | [
"Functional Completeness"
] | [
"Definition:Logical Connective/Binary",
"Definition:Singleton",
"Definition:Functionally Complete",
"Definition:Logical NAND",
"Definition:Logical NOR"
] | [
"Definition:Logical NAND",
"Definition:Logical NOR",
"Functionally Complete Logical Connectives/NAND",
"Functionally Complete Logical Connectives/NOR",
"Definition:Singleton",
"Definition:Functionally Complete",
"Definition:Logical Connective/Binary",
"Definition:Functionally Complete",
"Definition:... |
proofwiki-8176 | Conditional and Converse are not Equivalent | A conditional statement:
:$p \implies q$
is not logically equivalent to its converse:
:$q \implies p$ | We apply the Method of Truth Tables to the proposition:
:$\paren {p \implies q} \iff \paren {q \implies p}$
$\begin{array}{|ccc|c|ccc|} \hline
p & \implies & q) & \iff & (q & \implies & p) \\
\hline
\F & \T & \F & \T & \F & \T & \F \\
\F & \T & \T & \F & \T & \F & \F \\
\T & \F & \F & \F & \F & \T & \T \\
\T & \T & \T ... | A [[Definition:Conditional|conditional statement]]:
:$p \implies q$
is not [[Definition:Logical Equivalence|logically equivalent]] to its [[Definition:Converse Statement|converse]]:
:$q \implies p$ | We apply the [[Method of Truth Tables]] to the proposition:
:$\paren {p \implies q} \iff \paren {q \implies p}$
$\begin{array}{|ccc|c|ccc|} \hline
p & \implies & q) & \iff & (q & \implies & p) \\
\hline
\F & \T & \F & \T & \F & \T & \F \\
\F & \T & \T & \F & \T & \F & \F \\
\T & \F & \F & \F & \F & \T & \T \\
\T & \T ... | Conditional and Converse are not Equivalent | https://proofwiki.org/wiki/Conditional_and_Converse_are_not_Equivalent | https://proofwiki.org/wiki/Conditional_and_Converse_are_not_Equivalent | [
"Conditional"
] | [
"Definition:Conditional",
"Definition:Logical Equivalence",
"Definition:Converse Statement"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-8177 | De Morgan's Laws (Predicate Logic)/Assertion of Universality | :$\forall x: \map P x \dashv \vdash \neg \paren {\exists x: \neg \map P x}$
::''If everything '''is''', there exists nothing that '''is not'''.'' | {{BeginTableau|\forall x: \map P x \vdash \neg \paren {\exists x: \neg \map P x} }}
{{Premise|1|\forall x: \map P x}}
{{Assumption|2|\exists x: \neg \map P x}}
{{TableauLine|n = 3|pool = 2|f = \neg \map P {\mathbf a}|rlnk = Existential Instantiation|rtxt = Existential Instantiation|dep = 2|c = for some arbitrary $\math... | :$\forall x: \map P x \dashv \vdash \neg \paren {\exists x: \neg \map P x}$
::''If everything '''is''', there exists nothing that '''is not'''.'' | {{BeginTableau|\forall x: \map P x \vdash \neg \paren {\exists x: \neg \map P x} }}
{{Premise|1|\forall x: \map P x}}
{{Assumption|2|\exists x: \neg \map P x}}
{{TableauLine|n = 3|pool = 2|f = \neg \map P {\mathbf a}|rlnk = Existential Instantiation|rtxt = Existential Instantiation|dep = 2|c = for some arbitrary $\math... | De Morgan's Laws (Predicate Logic)/Assertion of Universality | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Assertion_of_Universality | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Assertion_of_Universality | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)"
] | [] | [
"De Morgan's Laws (Predicate Logic)/Denial of Universality"
] |
proofwiki-8178 | De Morgan's Laws (Predicate Logic)/Denial of Existence | :$\forall x: \neg \map P x \dashv \vdash \neg \paren {\exists x: \map P x}$
::''If everything '''is not''', there exists nothing that '''is'''.'' | {{BeginTableau|\forall x: \neg \map P x \vdash \neg \paren {\exists x: \map P x} }}
{{Premise|1|\forall x: \neg \map P x}}
{{Assumption|2|\exists x: \map P x}}
{{TableauLine|n = 3|pool = 2|f = \map P {\mathbf a}|rlnk = Existential Instantiation|rtxt = Existential Instantiation|dep = 2|c = for an arbitrary $\mathbf a$}}... | :$\forall x: \neg \map P x \dashv \vdash \neg \paren {\exists x: \map P x}$
::''If everything '''is not''', there exists nothing that '''is'''.'' | {{BeginTableau|\forall x: \neg \map P x \vdash \neg \paren {\exists x: \map P x} }}
{{Premise|1|\forall x: \neg \map P x}}
{{Assumption|2|\exists x: \map P x}}
{{TableauLine|n = 3|pool = 2|f = \map P {\mathbf a}|rlnk = Existential Instantiation|rtxt = Existential Instantiation|dep = 2|c = for an arbitrary $\mathbf a$}}... | De Morgan's Laws (Predicate Logic)/Denial of Existence | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Existence | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Denial_of_Existence | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)"
] | [] | [] |
proofwiki-8179 | De Morgan's Laws (Predicate Logic)/Assertion of Existence | :$\neg \paren {\forall x: \neg \map P x} \dashv \vdash \exists x: \map P x$
::''If not everything '''is not''', there exists something that '''is'''.'' | {{BeginTableau|\neg \paren {\forall x: \neg \map P x} \vdash \exists x: \map P x}}
{{Premise|1|\neg \paren {\forall x: \neg \map P x} }}
{{Assumption|2|\neg \paren {\exists x: \map P x} }}
{{SequentIntro|3|2|\forall x: \neg \map P x|2|Denial of Existence}}
{{NonContradiction|4|1, 2|1|3}}
{{Reductio|5|1|\exists x: \map ... | :$\neg \paren {\forall x: \neg \map P x} \dashv \vdash \exists x: \map P x$
::''If not everything '''is not''', there exists something that '''is'''.'' | {{BeginTableau|\neg \paren {\forall x: \neg \map P x} \vdash \exists x: \map P x}}
{{Premise|1|\neg \paren {\forall x: \neg \map P x} }}
{{Assumption|2|\neg \paren {\exists x: \map P x} }}
{{SequentIntro|3|2|\forall x: \neg \map P x|2|[[Denial of Existence]]}}
{{NonContradiction|4|1, 2|1|3}}
{{Reductio|5|1|\exists x: \... | De Morgan's Laws (Predicate Logic)/Assertion of Existence | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Assertion_of_Existence | https://proofwiki.org/wiki/De_Morgan's_Laws_(Predicate_Logic)/Assertion_of_Existence | [
"Universal Quantifier",
"Existential Quantifier",
"De Morgan's Laws (Logic)"
] | [] | [
"De Morgan's Laws (Predicate Logic)/Denial of Existence"
] |
proofwiki-8180 | Conjunction in terms of NAND | :$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \uparrow q}$
where $\land$ denotes logical conjunction and $\uparrow$ denotes logical NAND. | {{begin-eqn}}
{{eqn | l = p \land q
| o = \dashv \vdash
| r = \neg \neg \paren {p \land q}
| c = Double Negation
}}
{{eqn | o = \dashv \vdash
| r = \neg \paren {p \uparrow q}
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \uparrow q} \uparrow \paren {p \upa... | :$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \uparrow q}$
where $\land$ denotes [[Definition:Conjunction|logical conjunction]] and $\uparrow$ denotes [[Definition:Logical NAND|logical NAND]]. | {{begin-eqn}}
{{eqn | l = p \land q
| o = \dashv \vdash
| r = \neg \neg \paren {p \land q}
| c = [[Double Negation]]
}}
{{eqn | o = \dashv \vdash
| r = \neg \paren {p \uparrow q}
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \uparrow q} \uparrow \paren {p ... | Conjunction in terms of NAND | https://proofwiki.org/wiki/Conjunction_in_terms_of_NAND | https://proofwiki.org/wiki/Conjunction_in_terms_of_NAND | [
"Logical NAND",
"Conjunction"
] | [
"Definition:Conjunction",
"Definition:Logical NAND"
] | [
"Double Negation",
"NAND with Equal Arguments"
] |
proofwiki-8181 | Disjunction in terms of NOR | :$p \lor q \dashv \vdash \paren {p \downarrow q} \downarrow \paren {p \downarrow q}$
where $\lor$ denotes logical disjunction and $\downarrow$ denotes logical NOR. | {{begin-eqn}}
{{eqn | l = p \lor q
| o = \dashv \vdash
| r = \neg \neg \paren {p \lor q}
| c = Double Negation
}}
{{eqn | o = \dashv \vdash
| r = \neg \paren {p \downarrow q}
| c = {{Defof|Logical NOR}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \downarrow q} \downarrow \paren {p \... | :$p \lor q \dashv \vdash \paren {p \downarrow q} \downarrow \paren {p \downarrow q}$
where $\lor$ denotes [[Definition:Disjunction|logical disjunction]] and $\downarrow$ denotes [[Definition:Logical NOR|logical NOR]]. | {{begin-eqn}}
{{eqn | l = p \lor q
| o = \dashv \vdash
| r = \neg \neg \paren {p \lor q}
| c = [[Double Negation]]
}}
{{eqn | o = \dashv \vdash
| r = \neg \paren {p \downarrow q}
| c = {{Defof|Logical NOR}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \downarrow q} \downarrow \paren ... | Disjunction in terms of NOR | https://proofwiki.org/wiki/Disjunction_in_terms_of_NOR | https://proofwiki.org/wiki/Disjunction_in_terms_of_NOR | [
"Logical NOR",
"Disjunction"
] | [
"Definition:Disjunction",
"Definition:Logical NOR"
] | [
"Double Negation",
"NOR with Equal Arguments",
"Category:Logical NOR",
"Category:Disjunction"
] |
proofwiki-8182 | Disjunction in terms of NAND | :$p \lor q \dashv \vdash \paren {p \uparrow p} \uparrow \paren {q \uparrow q}$
where $\lor$ denotes logical disjunction and $\uparrow$ denotes logical NAND. | {{begin-eqn}}
{{eqn | l = p \lor q
| o = \dashv \vdash
| r = \neg \paren {\neg p \land \neg q}
| c = De Morgan's Laws (Logic): Disjunction
}}
{{eqn | o = \dashv \vdash
| r = \neg p \uparrow \neg q
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \uparrow p} \... | :$p \lor q \dashv \vdash \paren {p \uparrow p} \uparrow \paren {q \uparrow q}$
where $\lor$ denotes [[Definition:Disjunction|logical disjunction]] and $\uparrow$ denotes [[Definition:Logical NAND|logical NAND]]. | {{begin-eqn}}
{{eqn | l = p \lor q
| o = \dashv \vdash
| r = \neg \paren {\neg p \land \neg q}
| c = [[De Morgan's Laws (Logic)/Disjunction|De Morgan's Laws (Logic): Disjunction]]
}}
{{eqn | o = \dashv \vdash
| r = \neg p \uparrow \neg q
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \... | Disjunction in terms of NAND | https://proofwiki.org/wiki/Disjunction_in_terms_of_NAND | https://proofwiki.org/wiki/Disjunction_in_terms_of_NAND | [
"Logical NAND",
"Disjunction"
] | [
"Definition:Disjunction",
"Definition:Logical NAND"
] | [
"De Morgan's Laws (Logic)/Disjunction",
"NAND with Equal Arguments"
] |
proofwiki-8183 | Conditional in terms of NAND | :$p \implies q \dashv \vdash p \uparrow \paren {q \uparrow q}$ | {{begin-eqn}}
{{eqn | l = p \implies q
| o = \dashv \vdash
| r = \neg \paren {p \land \neg q}
| c = Conditional is Equivalent to Negation of Conjunction with Negative
}}
{{eqn | o = \dashv \vdash
| r = p \uparrow \neg q
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| r = ... | :$p \implies q \dashv \vdash p \uparrow \paren {q \uparrow q}$ | {{begin-eqn}}
{{eqn | l = p \implies q
| o = \dashv \vdash
| r = \neg \paren {p \land \neg q}
| c = [[Conditional is Equivalent to Negation of Conjunction with Negative]]
}}
{{eqn | o = \dashv \vdash
| r = p \uparrow \neg q
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| ... | Conditional in terms of NAND/Proof 1 | https://proofwiki.org/wiki/Conditional_in_terms_of_NAND | https://proofwiki.org/wiki/Conditional_in_terms_of_NAND/Proof_1 | [
"Conditional in terms of NAND",
"Logical NAND",
"Conditional"
] | [] | [
"Conditional is Equivalent to Negation of Conjunction with Negative",
"NAND with Equal Arguments"
] |
proofwiki-8184 | Conditional in terms of NAND | :$p \implies q \dashv \vdash p \uparrow \paren {q \uparrow q}$ | {{begin-eqn}}
{{eqn | l = p \implies q
| o = \dashv \vdash
| r = \neg p \lor q
| c = Rule of Material Implication
}}
{{eqn | o = \dashv \vdash
| r = \neg p \lor \neg \neg q
| c = Double Negation Introduction
}}
{{eqn | o = \dashv \vdash
| r = p \uparrow \neg q
| c = NAND as Dis... | :$p \implies q \dashv \vdash p \uparrow \paren {q \uparrow q}$ | {{begin-eqn}}
{{eqn | l = p \implies q
| o = \dashv \vdash
| r = \neg p \lor q
| c = [[Rule of Material Implication]]
}}
{{eqn | o = \dashv \vdash
| r = \neg p \lor \neg \neg q
| c = [[Double Negation Introduction]]
}}
{{eqn | o = \dashv \vdash
| r = p \uparrow \neg q
| c = [[N... | Conditional in terms of NAND/Proof 2 | https://proofwiki.org/wiki/Conditional_in_terms_of_NAND | https://proofwiki.org/wiki/Conditional_in_terms_of_NAND/Proof_2 | [
"Conditional in terms of NAND",
"Logical NAND",
"Conditional"
] | [] | [
"Rule of Material Implication",
"Double Negation/Double Negation Introduction",
"NAND as Disjunction of Negations",
"NAND with Equal Arguments"
] |
proofwiki-8185 | Biconditional in terms of NAND | :$p \iff q \dashv \vdash \paren {\paren {p \uparrow p} \uparrow \paren {q \uparrow q} } \uparrow \paren {p \uparrow q}$
where $\iff$ denotes logical biconditional and $\uparrow$ denotes logical NAND. | {{begin-eqn}}
{{eqn | l = p \iff q
| o = \dashv \vdash
| r = \neg \paren {p \oplus q}
| c = Exclusive Or is Negation of Biconditional
}}
{{eqn | o = \dashv \vdash
| r = \neg \paren {\paren {p \lor q} \land \neg \paren {p \land q} }
| c = {{Defof|Exclusive Or}}
}}
{{eqn | o = \dashv \vdash
... | :$p \iff q \dashv \vdash \paren {\paren {p \uparrow p} \uparrow \paren {q \uparrow q} } \uparrow \paren {p \uparrow q}$
where $\iff$ denotes [[Definition:Biconditional|logical biconditional]] and $\uparrow$ denotes [[Definition:Logical NAND|logical NAND]]. | {{begin-eqn}}
{{eqn | l = p \iff q
| o = \dashv \vdash
| r = \neg \paren {p \oplus q}
| c = [[Exclusive Or is Negation of Biconditional]]
}}
{{eqn | o = \dashv \vdash
| r = \neg \paren {\paren {p \lor q} \land \neg \paren {p \land q} }
| c = {{Defof|Exclusive Or}}
}}
{{eqn | o = \dashv \vd... | Biconditional in terms of NAND | https://proofwiki.org/wiki/Biconditional_in_terms_of_NAND | https://proofwiki.org/wiki/Biconditional_in_terms_of_NAND | [
"Logical NAND",
"Biconditional"
] | [
"Definition:Biconditional",
"Definition:Logical NAND"
] | [
"Exclusive Or is Negation of Biconditional",
"Disjunction in terms of NAND"
] |
proofwiki-8186 | Mapping is Injection and Surjection iff Inverse is Mapping | Let $f: S \to T$ be a mapping.
Then:
: $f: S \to T$ can be defined as a bijection in the sense that:
::$(1): \quad f$ is an injection
::$(2): \quad f$ is a surjection
{{iff}}:
:the inverse $f^{-1}$ of $f$ is itself a mapping. | === Necessary Condition ===
{{:Inverse of Injective and Surjective Mapping is Mapping}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
: $f: S \to T$ can be defined as a [[Definition:Bijection/Definition 1|bijection]] in the sense that:
::$(1): \quad f$ is an [[Definition:Injection|injection]]
::$(2): \quad f$ is a [[Definition:Surjection|surjection]]
{{iff}}:
:the [[Definition:Inverse of ... | === [[Inverse of Injective and Surjective Mapping is Mapping|Necessary Condition]] ===
{{:Inverse of Injective and Surjective Mapping is Mapping}} | Mapping is Injection and Surjection iff Inverse is Mapping | https://proofwiki.org/wiki/Mapping_is_Injection_and_Surjection_iff_Inverse_is_Mapping | https://proofwiki.org/wiki/Mapping_is_Injection_and_Surjection_iff_Inverse_is_Mapping | [
"Mapping is Injection and Surjection iff Inverse is Mapping",
"Equivalence of Definitions of Bijection",
"Injections",
"Surjections",
"Inverse Mappings"
] | [
"Definition:Mapping",
"Definition:Bijection/Definition 1",
"Definition:Injection",
"Definition:Surjection",
"Definition:Inverse of Mapping",
"Definition:Mapping"
] | [
"Inverse of Injective and Surjective Mapping is Mapping"
] |
proofwiki-8187 | Mapping is Injection and Surjection iff Inverse is Mapping | Let $f: S \to T$ be a mapping.
Then:
: $f: S \to T$ can be defined as a bijection in the sense that:
::$(1): \quad f$ is an injection
::$(2): \quad f$ is a surjection
{{iff}}:
:the inverse $f^{-1}$ of $f$ is itself a mapping. | === Necessary Condition ===
{{:Inverse of Injective and Surjective Mapping is Mapping/Proof 2}}{{qed|lemma}}
=== Sufficient Condition ===
Let $f^{-1}: T \to S$ be a mapping.
By Inverse Mapping is Bijection, both $f$ and $f^{-1}$ are bijections.
Hence, in particular, $f$ is a bijection.
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
: $f: S \to T$ can be defined as a [[Definition:Bijection/Definition 1|bijection]] in the sense that:
::$(1): \quad f$ is an [[Definition:Injection|injection]]
::$(2): \quad f$ is a [[Definition:Surjection|surjection]]
{{iff}}:
:the [[Definition:Inverse of ... | === [[Inverse of Injective and Surjective Mapping is Mapping/Proof 2|Necessary Condition]] ===
{{:Inverse of Injective and Surjective Mapping is Mapping/Proof 2}}{{qed|lemma}}
=== Sufficient Condition ===
Let $f^{-1}: T \to S$ be a [[Definition:Mapping|mapping]].
By [[Inverse Mapping is Bijection]], both $f$ and $f... | Mapping is Injection and Surjection iff Inverse is Mapping/Proof 2 | https://proofwiki.org/wiki/Mapping_is_Injection_and_Surjection_iff_Inverse_is_Mapping | https://proofwiki.org/wiki/Mapping_is_Injection_and_Surjection_iff_Inverse_is_Mapping/Proof_2 | [
"Mapping is Injection and Surjection iff Inverse is Mapping",
"Equivalence of Definitions of Bijection",
"Injections",
"Surjections",
"Inverse Mappings"
] | [
"Definition:Mapping",
"Definition:Bijection/Definition 1",
"Definition:Injection",
"Definition:Surjection",
"Definition:Inverse of Mapping",
"Definition:Mapping"
] | [
"Inverse of Injective and Surjective Mapping is Mapping/Proof 2",
"Definition:Mapping",
"Inverse Mapping is Bijection",
"Definition:Bijection",
"Definition:Bijection"
] |
proofwiki-8188 | Condition for Composite Mapping to be Identity | Let $S$ and $T$ be sets.
Let $f: S \to T$ and $g: T \to S$ be mappings such that:
: $g \circ f = I_S$
where $I_S$ is the identity mapping on $S$.
Then $f$ is an injection and $g$ is a surjection. | From Identity Mapping is Bijection, $I_S$ is a bijection.
From Identity Mapping is Injection, $I_S$ is an injection.
From Injection if Composite is Injection it follows that $f$ is an injection.
From Identity Mapping is Surjection, $I_S$ is a surjection.
From Surjection if Composite is Surjection it follows that $g$ is... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ and $g: T \to S$ be [[Definition:Mapping|mappings]] such that:
: $g \circ f = I_S$
where $I_S$ is the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then $f$ is an [[Definition:Injection|injection]] and $g$ is a [[Definition:Surjection|surjection... | From [[Identity Mapping is Bijection]], $I_S$ is a [[Definition:Bijection|bijection]].
From [[Identity Mapping is Injection]], $I_S$ is an [[Definition:Injection|injection]].
From [[Injection if Composite is Injection]] it follows that $f$ is an [[Definition:Injection|injection]].
From [[Identity Mapping is Surject... | Condition for Composite Mapping to be Identity | https://proofwiki.org/wiki/Condition_for_Composite_Mapping_to_be_Identity | https://proofwiki.org/wiki/Condition_for_Composite_Mapping_to_be_Identity | [
"Identity Mappings",
"Injections",
"Surjections"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Identity Mapping",
"Definition:Injection",
"Definition:Surjection"
] | [
"Identity Mapping is Bijection",
"Definition:Bijection",
"Identity Mapping is Injection",
"Definition:Injection",
"Injection if Composite is Injection",
"Definition:Injection",
"Identity Mapping is Surjection",
"Definition:Surjection",
"Surjection if Composite is Surjection",
"Definition:Surjectio... |
proofwiki-8189 | Bijection iff exists Mapping which is Left and Right Inverse | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Then $f$ is a bijection {{iff}}:
:there exists a mapping $g: T \to S$ such that:
::$g \circ f = I_S$
::$f \circ g = I_T$
:where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$ respectively. | === Necessary Condition ===
Let $f$ be a bijection.
Then for each $y \in T$ there exists one and only one $x \in S$ such that $\map f x = y$.
That is, that there exists a mapping $g: T \to S$ with the property that:
: $\forall y \in T: \exists x \in S: \map g y = x$
Let $y \in T$.
Let $x = g \map g y$.
Then:
{{begin-eq... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then $f$ is a [[Definition:Bijection|bijection]] {{iff}}:
:there exists a [[Definition:Mapping|mapping]] $g: T \to S$ such that:
::$g \circ f = I_S$
::$f \circ g = I_T$
:where $I_S$ and $I_T$ are the [[Definition:Identi... | === Necessary Condition ===
Let $f$ be a [[Definition:Bijection/Definition 4|bijection]].
Then [[Definition:Universal Quantifier|for each]] $y \in T$ there exists [[Definition:Unique|one and only one]] $x \in S$ such that $\map f x = y$.
That is, that there exists a [[Definition:Mapping|mapping]] $g: T \to S$ with t... | Bijection iff exists Mapping which is Left and Right Inverse | https://proofwiki.org/wiki/Bijection_iff_exists_Mapping_which_is_Left_and_Right_Inverse | https://proofwiki.org/wiki/Bijection_iff_exists_Mapping_which_is_Left_and_Right_Inverse | [
"Bijections"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Mapping",
"Definition:Identity Mapping"
] | [
"Definition:Bijection/Definition 4",
"Definition:Universal Quantifier",
"Definition:Unique",
"Definition:Mapping"
] |
proofwiki-8190 | Eigenvalues of G-Representation are Roots of Unity | Let $G$ be a finite group.
Let $\left({K, +, \cdot}\right)$ be a field.
Let $V$ be a $G$-module over $K$ (i.e. $V$ is a $K \left[{G}\right]$-module).
Then $\forall g \in G$, the eigenvalues of the action by the vector $g \in K \left[{G}\right]$ on $V$ are roots of unity. | Fix an arbitrary $g \in G$ and consider the corresponding vector $g \in K \left[{G}\right]$.
Let $\lambda$ be an eigenvalue of $g$, that is $\lambda$ is an eigenvalue of the map in $\operatorname{Aut} \left({V}\right): \vec v \mapsto g \vec v$.
Then by definition of an eigenvalue we have:
:$\exists \vec v_\lambda \in V... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $\left({K, +, \cdot}\right)$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:G-Module|$G$-module]] over $K$ (i.e. $V$ is a $K \left[{G}\right]$-module).
Then $\forall g \in G$, the [[Definition:Eigenvalue|eigenvalues]] of the a... | Fix an arbitrary $g \in G$ and consider the corresponding vector $g \in K \left[{G}\right]$.
Let $\lambda$ be an eigenvalue of $g$, that is $\lambda$ is an eigenvalue of the map in $\operatorname{Aut} \left({V}\right): \vec v \mapsto g \vec v$.
Then by definition of an eigenvalue we have:
:$\exists \vec v_\lambda \i... | Eigenvalues of G-Representation are Roots of Unity | https://proofwiki.org/wiki/Eigenvalues_of_G-Representation_are_Roots_of_Unity | https://proofwiki.org/wiki/Eigenvalues_of_G-Representation_are_Roots_of_Unity | [] | [
"Definition:Finite Group",
"Definition:Field (Abstract Algebra)",
"Definition:Module over Group",
"Definition:Eigenvalue",
"Definition:Root of Unity"
] | [
"Definition:Linear Group Action",
"Definition:Linear Group Action"
] |
proofwiki-8191 | Character of Representations over C are Algebraic Integers | Let $G$ be a finite group.
Let $\chi$ be the character of any $\C \sqbrk G$-module $\struct {V, \rho}$.
Then for all $g \in G$, it follows that $\map \chi g$ is an algebraic integer. | By the definition of character:
:$\map \chi g = \map \tr {\rho_g}$
where $\map \tr {\rho_g}$ is the trace of $\rho_g$.
{{explain|$\rho_g$}}
where:
:$\rho \in \map \hom {\C \sqbrk G, \Aut V}: \vec {e_g} \mapsto \rho_g$
by definition.
{{explain|The above definition is not stated in that form on {{ProofWiki}}. Link to $\m... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $\chi$ be the [[Definition:Character (Representation Theory)|character]] of any [[Definition:G-Module|$\C \sqbrk G$-module]] $\struct {V, \rho}$.
Then for all $g \in G$, it follows that $\map \chi g$ is an [[Definition:Algebraic Integer|algebraic integer]]. | By the definition of [[Definition:Character (Representation Theory)|character]]:
:$\map \chi g = \map \tr {\rho_g}$
where $\map \tr {\rho_g}$ is the [[Definition:Trace (Linear Algebra)|trace]] of $\rho_g$.
{{explain|$\rho_g$}}
where:
:$\rho \in \map \hom {\C \sqbrk G, \Aut V}: \vec {e_g} \mapsto \rho_g$
by definition... | Character of Representations over C are Algebraic Integers | https://proofwiki.org/wiki/Character_of_Representations_over_C_are_Algebraic_Integers | https://proofwiki.org/wiki/Character_of_Representations_over_C_are_Algebraic_Integers | [
"Module Theory",
"Group Theory",
"Complex Analysis"
] | [
"Definition:Finite Group",
"Definition:Character (Representation Theory)",
"Definition:Module over Group",
"Definition:Algebraic Integer"
] | [
"Definition:Character (Representation Theory)",
"Definition:Trace (Linear Algebra)",
"Definition:Order of Group Element",
"Definition:Trace (Linear Algebra)",
"Definition:Eigenvalue",
"Eigenvalues of G-Representation are Roots of Unity",
"Definition:Eigenvalue",
"Definition:Root of Unity/Complex",
"... |
proofwiki-8192 | Injection from Finite Set to Itself is Surjection | Let $S$ be a finite set.
Let $f: S \to S$ be an injection.
Then $f$ is also a surjection. | Let $a \in S$.
We need to show that there exists $b \in S$ such that $a = \map f b$.
Consider what happens when $f$ is applied repeatedly on $S$.
Let $f^2$ denote $f \circ f$ and, generally, $f^n := f \circ f^{n-1}$.
Consider the sequence of elements of $S$:
:$a, \map f a, \map {f^2} a, \ldots$
Because $S$ is a finite ... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to S$ be an [[Definition:Injection|injection]].
Then $f$ is also a [[Definition:Surjection|surjection]]. | Let $a \in S$.
We need to show that there exists $b \in S$ such that $a = \map f b$.
Consider what happens when $f$ is applied repeatedly on $S$.
Let $f^2$ denote $f \circ f$ and, generally, $f^n := f \circ f^{n-1}$.
Consider the [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$:
:$a, \map... | Injection from Finite Set to Itself is Surjection | https://proofwiki.org/wiki/Injection_from_Finite_Set_to_Itself_is_Surjection | https://proofwiki.org/wiki/Injection_from_Finite_Set_to_Itself_is_Surjection | [
"Injections",
"Surjections",
"Finite Sets"
] | [
"Definition:Finite Set",
"Definition:Injection",
"Definition:Surjection"
] | [
"Definition:Sequence",
"Definition:Element",
"Definition:Finite Set",
"Definition:Injection",
"Composite of Injections is Injection",
"Definition:Injection",
"Injection iff Left Cancellable",
"Definition:Left Cancellable Mapping"
] |
proofwiki-8193 | Codomain of Bijection is Domain of Inverse | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a bijection.
Let $f^{-1}: T \to S$ be the inverse of $f$.
Then the domain of $f^{-1}$ equals the codomain of $f$. | Follows directly from the definition of domain and codomain:
:$\Dom f = S$ and $\Cdm f = T$
:$\Dom {f^{-1} } = T$ and $\Cdm {f^{-1} } = S$
That is:
:$\Dom {f^{-1} } = T = \Cdm f$
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Bijection|bijection]].
Let $f^{-1}: T \to S$ be the [[Definition:Inverse Mapping|inverse]] of $f$.
Then the [[Definition:Domain of Mapping|domain]] of $f^{-1}$ equals the [[Definition:Codomain of Mapping|codomain]] of $f$. | Follows directly from the definition of [[Definition:Domain of Mapping|domain]] and [[Definition:Codomain of Mapping|codomain]]:
:$\Dom f = S$ and $\Cdm f = T$
:$\Dom {f^{-1} } = T$ and $\Cdm {f^{-1} } = S$
That is:
:$\Dom {f^{-1} } = T = \Cdm f$
{{qed}} | Codomain of Bijection is Domain of Inverse | https://proofwiki.org/wiki/Codomain_of_Bijection_is_Domain_of_Inverse | https://proofwiki.org/wiki/Codomain_of_Bijection_is_Domain_of_Inverse | [
"Bijections"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping"
] |
proofwiki-8194 | Domain of Bijection is Codomain of Inverse | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a bijection.
Let $f^{-1}: T \to S$ be the inverse of $f$.
Then the codomain of $f^{-1}$ equals the domain of $f$. | Follows directly from the definition of domain and codomain:
:$\Dom f = S$ and $\Cdm f = T$
:$\Dom {f^{-1} } = T$ and $\Cdm {f^{-1} } = S$
That is:
:$\Cdm {f^{-1} } = S = \Dom f$
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Bijection|bijection]].
Let $f^{-1}: T \to S$ be the [[Definition:Inverse Mapping|inverse]] of $f$.
Then the [[Definition:Codomain of Mapping|codomain]] of $f^{-1}$ equals the [[Definition:Domain of Mapping|domain]] of $f$. | Follows directly from the definition of [[Definition:Domain of Mapping|domain]] and [[Definition:Codomain of Mapping|codomain]]:
:$\Dom f = S$ and $\Cdm f = T$
:$\Dom {f^{-1} } = T$ and $\Cdm {f^{-1} } = S$
That is:
:$\Cdm {f^{-1} } = S = \Dom f$
{{qed}} | Domain of Bijection is Codomain of Inverse | https://proofwiki.org/wiki/Domain_of_Bijection_is_Codomain_of_Inverse | https://proofwiki.org/wiki/Domain_of_Bijection_is_Codomain_of_Inverse | [
"Bijections"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping"
] |
proofwiki-8195 | Conditions for Uniqueness of Left Inverse Mapping | Let $S$ and $T$ be sets such that $S \ne \O$.
Let $f: S \to T$ be an injection.
Then a left inverse mapping of $f$ is in general not unique.
Uniqueness occurs under either of two circumstances:
:$(1): \quad S$ is a singleton
:$(2): \quad f$ is a bijection. | If $f$ is a bijection, then by definition $f$ is also a surjection.
Then:
:$T \setminus \Img f = \O$
:and we have that $g = f^{-1}$.
As $f^{-1}$ is uniquely defined $g$ is itself unique.
{{qed|lemma}}
If $S$ is a singleton then there can only be one mapping $g: T \to S$:
:$\forall t \in T: \map g t = s$
{{qed|lemma}}
I... | Let $S$ and $T$ be [[Definition:Set|sets]] such that $S \ne \O$.
Let $f: S \to T$ be an [[Definition:Injection|injection]].
Then a [[Definition:Left Inverse Mapping|left inverse mapping]] of $f$ is in general not [[Definition:Unique|unique]].
[[Definition:Unique|Uniqueness]] occurs under either of two circumstance... | If $f$ is a [[Definition:Bijection|bijection]], then by definition $f$ is also a [[Definition:Surjection|surjection]].
Then:
:$T \setminus \Img f = \O$
:and we have that $g = f^{-1}$.
As $f^{-1}$ is uniquely defined $g$ is itself unique.
{{qed|lemma}}
If $S$ is a [[Definition:Singleton|singleton]] then there can on... | Conditions for Uniqueness of Left Inverse Mapping | https://proofwiki.org/wiki/Conditions_for_Uniqueness_of_Left_Inverse_Mapping | https://proofwiki.org/wiki/Conditions_for_Uniqueness_of_Left_Inverse_Mapping | [
"Left Inverse Mappings",
"Conditions for Uniqueness of Left Inverse Mapping"
] | [
"Definition:Set",
"Definition:Injection",
"Definition:Left Inverse Mapping",
"Definition:Unique",
"Definition:Unique",
"Definition:Singleton",
"Definition:Bijection"
] | [
"Definition:Bijection",
"Definition:Surjection",
"Definition:Singleton",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Singleton",
"Definition:Unique"
] |
proofwiki-8196 | Surjection iff Right Inverse/Non-Uniqueness | A right inverse of $f$ is in general not unique.
Uniqueness occurs {{iff}} $f$ is a bijection. | If $f$ is not an injection then:
:$\exists y \in T: \exists x_1, x_2 \in S: \map f {x_1} = y = \map f {x_2}$
Hence we have more than one choice in $\map {f^{-1} } {\set y}$ for how to map $\map g y$.
That is, $\map g y$ is not unique.
This does not happen {{iff}} $f$ is an injection.
Hence the result.
{{qed}} | A [[Definition:Right Inverse Mapping|right inverse]] of $f$ is in general not [[Definition:Unique|unique]].
Uniqueness occurs {{iff}} $f$ is a [[Definition:Bijection|bijection]]. | If $f$ is not an [[Definition:Injection|injection]] then:
:$\exists y \in T: \exists x_1, x_2 \in S: \map f {x_1} = y = \map f {x_2}$
Hence we have more than one choice in $\map {f^{-1} } {\set y}$ for how to map $\map g y$.
That is, $\map g y$ is not [[Definition:Unique|unique]].
This does not happen {{iff}} $f$ i... | Surjection iff Right Inverse/Non-Uniqueness | https://proofwiki.org/wiki/Surjection_iff_Right_Inverse/Non-Uniqueness | https://proofwiki.org/wiki/Surjection_iff_Right_Inverse/Non-Uniqueness | [
"Surjection iff Right Inverse"
] | [
"Definition:Right Inverse Mapping",
"Definition:Unique",
"Definition:Bijection"
] | [
"Definition:Injection",
"Definition:Unique",
"Definition:Injection"
] |
proofwiki-8197 | Mapping reflects Preordering | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let ${\precsim} \subseteq T \times T$ be a preordering on $T$.
Let $\RR$ be the relation defined on $S$ by the rule:
:$x \mathrel \RR y \iff \map f x \precsim \map f y$
Then $\RR$ is a preordering on $S$. | === Reflexivity ===
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadsto
| l = \map f x
| o = \in
| r = T
| c = Definition of $f$
}}
{{eqn | ll= \leadsto
| l = \map f x
| o = \precsim
| r = \map f x
| c = as $\precsim$ is a preord... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let ${\precsim} \subseteq T \times T$ be a [[Definition:Preordering|preordering]] on $T$.
Let $\RR$ be the [[Definition:Relation|relation]] defined on $S$ by the rule:
:$x \mathrel \RR y \iff \map f x \precsim \map f y... | === Reflexivity ===
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadsto
| l = \map f x
| o = \in
| r = T
| c = Definition of $f$
}}
{{eqn | ll= \leadsto
| l = \map f x
| o = \precsim
| r = \map f x
| c = as $\precsim$ is a [[Def... | Mapping reflects Preordering | https://proofwiki.org/wiki/Mapping_reflects_Preordering | https://proofwiki.org/wiki/Mapping_reflects_Preordering | [
"Preorder Theory",
"Mapping Theory"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Preordering",
"Definition:Relation",
"Definition:Preordering"
] | [
"Definition:Preordering",
"Definition:Reflexive Relation",
"Definition:Preordering",
"Definition:Preordering"
] |
proofwiki-8198 | Bernoulli's Inequality | Let $x \in \R$ be a real number such that $x > -1$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\paren {1 + x}^n \ge 1 + n x$ | Let $0 < x < 1$.
Let $y = -x$.
Then $y > -1$ and by Bernoulli's Inequality:
:$\paren {1 + y}^n \ge 1 + n y$
Thus:
:$\paren {1 + \paren {-x} }^n \ge 1 + n \paren {-x}$
Hence the result.
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > -1$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\paren {1 + x}^n \ge 1 + n x$ | Let $0 < x < 1$.
Let $y = -x$.
Then $y > -1$ and by [[Bernoulli's Inequality]]:
:$\paren {1 + y}^n \ge 1 + n y$
Thus:
:$\paren {1 + \paren {-x} }^n \ge 1 + n \paren {-x}$
Hence the result.
{{qed}} | Bernoulli's Inequality/Corollary/Proof 1 | https://proofwiki.org/wiki/Bernoulli's_Inequality | https://proofwiki.org/wiki/Bernoulli's_Inequality/Corollary/Proof_1 | [
"Bernoulli's Inequality",
"Inequalities",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Positive/Integer"
] | [
"Bernoulli's Inequality"
] |
proofwiki-8199 | Bernoulli's Inequality | Let $x \in \R$ be a real number such that $x > -1$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\paren {1 + x}^n \ge 1 + n x$ | Proof by induction:
Let $0 < x < 1$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\paren {1 - x}^n \ge 1 - n x$
=== Basis for the Induction ===
$\map P 0$ is the case:
:$\paren {1 - x}^0 = 1 \ge 1 - 0 x = 1$
so $\map P 0$ holds.
This is our basis for the induction.
=== Induction Hypothesis ===
Now ... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > -1$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\paren {1 + x}^n \ge 1 + n x$ | Proof by [[Principle of Mathematical Induction|induction]]:
Let $0 < x < 1$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {1 - x}^n \ge 1 - n x$
=== Basis for the Induction ===
$\map P 0$ is the case:
:$\paren {1 - x}^0 = 1 \ge 1 - 0 x = 1$
so $\map P 0$ holds... | Bernoulli's Inequality/Corollary/Proof 2 | https://proofwiki.org/wiki/Bernoulli's_Inequality | https://proofwiki.org/wiki/Bernoulli's_Inequality/Corollary/Proof_2 | [
"Bernoulli's Inequality",
"Inequalities",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Bernoulli's Inequality/Corollary/Proof 2",
"Principle of Mathematical Induction"
] |
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