Ucale

Trigonometrical Ratios of Submultiple of an Angle

An angle of the form A/n where n is an integer is called submultiple angle of A

1. $\displaystyle \left| \sin { \frac { A }{ 2 } } +\cos { \frac { A }{ 2 } } \right|$=$\displaystyle\sqrt { 1+\sin { A } }$ or$\displaystyle \sin { \frac { A }{ 2 } } +\cos { \frac { A }{ 2 } }$=$\displaystyle \pm \sqrt { 1+\sin { A } }$ $\displaystyle \left[ +ve, if\quad 2n\pi -\frac { \pi }{ 4 } \le \frac { A }{ 2 } \le 2n\pi +\frac { 3 }{ 4 } \\ otherwise,-ve\right]$
2. $\displaystyle \left| \sin { \frac { A }{ 2 } } -\cos { \frac { A }{ 2 } } \right|$=$\displaystyle \sqrt { 1-\sin { A } }$ or $\displaystyle \sin { \frac { A }{ 2 } } -\cos { \frac { A }{ 2 } }$=$\displaystyle \pm \sqrt { 1-\sin { A } }$ $\displaystyle \left[ +ve,if\quad 2n\pi +\frac { \pi }{ 4 } \le \frac { A }{ 2 } \le 2n\pi +\frac { 5\pi }{ 4 } \\ otherwise,-ve\right]$
3. $\displaystyle \tan { \frac { A }{ 2 } }$=$\displaystyle \frac { \pm \sqrt { { tan }^{ 2 }A+1 } -1 }{ \tan { A } }$

February 22, 2019
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