JEE Main & Advanced Mathematics Trigonometrical Ratios and Identities Trigonometric Ratio of Sub-multiple of an Angle

Trigonometric Ratio of Sub-multiple of an Angle

Category : JEE Main & Advanced

(1) $\left| \,\sin \frac{A}{2}+\cos \frac{A}{2}\, \right|=\sqrt{1+\sin A}$

or $\sin \frac{A}{2}+\cos \frac{A}{2}=\pm \sqrt{1+\sin A}$

i.e., $\left\{ \begin{matrix} +,\,\text{If }2n\pi -\pi /4\le A/2\le 2n\pi +\frac{3\pi }{4} \\ -,\,\text{otherwise}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{matrix} \right.$

(2) $\left| \,\sin \frac{A}{2}-\cos \frac{A}{2}\, \right|=\sqrt{1-\sin A}$

or  $(\sin \frac{A}{2}-\cos \frac{A}{2})=\pm \sqrt{1-\sin A}$

i.e., $\left\{ \begin{matrix} +,\,\text{If }2n\pi +\pi /4\le A/2\le 2n\pi +\frac{5\pi }{4} \\ -,\,\text{otherwise}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{matrix} \right.$

(3) (i) $\tan \frac{A}{2}=\frac{\pm \sqrt{{{\tan }^{2}}A+1}-1}{\tan A}=\pm \,\sqrt{\frac{1-\cos A}{1+\cos A}}=\frac{1-\cos A}{\sin A}$,

where $A\ne (2n+1)\pi$

(ii) $\cot \frac{A}{2}=\pm \,\sqrt{\frac{1+\cos A}{1-\cos A}}=\frac{1+\cos A}{\sin A}$, where $A\ne 2n\pi$

The ambiguities of signs are removed by locating the quadrants in which $\frac{A}{2}$ lies or you can follow the following figure,

(4) ${{\tan }^{2}}\frac{A}{2}=\frac{1-\cos A}{1+\cos A}$ ; where $A\ne (2n+1)\pi$   (5) ${{\cot }^{2}}\frac{A}{2}=\frac{1+\cos A}{1-\cos A}$; where $A\ne 2n\pi$

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