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Trigonometric ratios of multiples of angles

An angle of the form nA, where n is an integer is called a multiple angle, for example 2A , 3A , 4A ,…. etc are multiples angle of A

$\displaystyle \sin { 2A } =2\sin { A } \cos { A } =\frac { 2tanA }{ 1+{ tan }^{ 2 }A }$
$\displaystyle \cos { 2A } ={ cos }^{ 2 }A-{ sin }^{ 2 }A=1-2{ sin }^{ 2 }A=2{ cos }^{ 2 }A-1=\frac { 1-{ tan }^{ 2 }A }{ 1+{ tan }^{ 2 }A }$
$\displaystyle 1+\cos { 2A } =2{ cos }^{ 2 }A$ ; $\displaystyle 1-\cos { 2A } =2{ sin }^{ 2 }A$ ; $\displaystyle or\frac { 1+\cos { 2A } }{ 2 } ={ cos }^{ 2 }A$ ; $\displaystyle \frac { 1-\cos { 2A } }{ 2 } ={ sin }^{ 2 }A$
$\displaystyle \tan { 2A } =\frac { 2\tan { A } }{ 1-{ tan }^{ 2 }A } ,\quad where\quad A\neq \left( 2n+1 \right) \frac { \pi }{ 4 }$
$\displaystyle \frac { 1-\cos { A } }{ \sin { A } } =\tan { \left( \frac { A }{ 2 } \right) } ,\quad where\quad A\neq 2n\pi$
$\displaystyle \frac { 1+\cos { A } }{ \sin { A } } =\cot { \left( \frac { A }{ 2 } \right) } ,\quad where\quad A\neq \left( 2n+1 \right) \pi$
$\displaystyle \frac { 1-\cos { A } }{ 1+\cos { A } } ={ tan }^{ 2 }\left( \frac { A }{ 2 } \right) ,\quad where\quad A\neq \left( 2n+1 \right) \pi$
$\displaystyle \frac { 1+\cos { A } }{ 1-\cos { A } } ={ cot }^{ 2 }\left( \frac { A }{ 2 } \right) ,\quad where\quad A\neq 2n\pi$
$\displaystyle \sin { 3A } =3\sin { A } -4{ sin }^{ 3 }A$
$\displaystyle \cos { 3A } =4{ cos }^{ 3 }A-3\cos { A }$
$\displaystyle \tan { 3A } =\frac { 3\tan { A } -{ tan }^{ 3 }A }{ 1-3{ tan }^{ 2 }A }$
$\displaystyle \cos { A } \cos { 2A } .\cos { { 2 }^{ 2 }A } .\cos { { 2 }^{ 3 }A } .....\cos { { 2 }^{ n-1 } } A=\frac { \sin { { 2 }^{ n }A } }{ { 2 }^{ n }\sin { A } }$

February 22, 2019

Trigonometry

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Nishant Agrawal

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