Ucale

# 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

1. $\displaystyle \sin { 2A } =2\sin { A } \cos { A } =\frac { 2tanA }{ 1+{ tan }^{ 2 }A }$
2. $\displaystyle \cos { 2A } ={ cos }^{ 2 }A-{ sin }^{ 2 }A=1-2{ sin }^{ 2 }A=2{ cos }^{ 2 }A-1=\frac { 1-{ tan }^{ 2 }A }{ 1+{ tan }^{ 2 }A }$
3. $\displaystyle 1+\cos { 2A } =2{ cos }^{ 2 }A$ ; $\displaystyle 1-\cos { 2A } =2{ sin }^{ 2 }A$ ; $\displaystyle or\frac { 1+\cos { 2A } }{ 2 } ={ cos }^{ 2 }A$ ; $\displaystyle \frac { 1-\cos { 2A } }{ 2 } ={ sin }^{ 2 }A$
4. $\displaystyle \tan { 2A } =\frac { 2\tan { A } }{ 1-{ tan }^{ 2 }A } ,\quad where\quad A\neq \left( 2n+1 \right) \frac { \pi }{ 4 }$
5. $\displaystyle \frac { 1-\cos { A } }{ \sin { A } } =\tan { \left( \frac { A }{ 2 } \right) } ,\quad where\quad A\neq 2n\pi$
6. $\displaystyle \frac { 1+\cos { A } }{ \sin { A } } =\cot { \left( \frac { A }{ 2 } \right) } ,\quad where\quad A\neq \left( 2n+1 \right) \pi$
7. $\displaystyle \frac { 1-\cos { A } }{ 1+\cos { A } } ={ tan }^{ 2 }\left( \frac { A }{ 2 } \right) ,\quad where\quad A\neq \left( 2n+1 \right) \pi$
8. $\displaystyle \frac { 1+\cos { A } }{ 1-\cos { A } } ={ cot }^{ 2 }\left( \frac { A }{ 2 } \right) ,\quad where\quad A\neq 2n\pi$
9. $\displaystyle \sin { 3A } =3\sin { A } -4{ sin }^{ 3 }A$
10. $\displaystyle \cos { 3A } =4{ cos }^{ 3 }A-3\cos { A }$
11. $\displaystyle \tan { 3A } =\frac { 3\tan { A } -{ tan }^{ 3 }A }{ 1-3{ tan }^{ 2 }A }$
12. $\displaystyle \cos { A } \cos { 2A } .\cos { { 2 }^{ 2 }A } .\cos { { 2 }^{ 3 }A } .....\cos { { 2 }^{ n-1 } } A=\frac { \sin { { 2 }^{ n }A } }{ { 2 }^{ n }\sin { A } }$
February 22, 2019
Which class you are presently in?
Choose an option. You can change your option at any time.
You will be solving questions and growing your critical thinking skills.   