$\sin \frac{x}{2}, \cos \frac{x}{2}$ અને $\tan \frac{x}{2}$ ની કિંમતો શોધો.: $\sin x=\frac{1}{4}, x$ એ ત્રીજા ચરણમાં છે.
Here, $x$ is in quadrant $II$.
i.e., $\frac{\pi}{2} < x < \pi$
$\Rightarrow \frac{\pi}{4}<\frac{x}{2}<\frac{\pi}{2}$
Therefore, $\sin \frac{x}{2}, \cos \frac{x}{2}, \tan \frac{x}{2}$ are all positive.
It is given that $\sin x=\frac{1}{4}$
$\cos ^{2} x=1-\sin ^{2} x=1-\left(\frac{1}{4}\right)^{2}=1-\frac{1}{16}=\frac{15}{16}$
$\Rightarrow \cos x=-\frac{\sqrt{15}}{4}[\cos x \text { is negative in quadrant II }]$
$\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}-\frac{1-\left(-\frac{\sqrt{15}}{4}\right)}{2}=\frac{4+\sqrt{15}}{8}$
$\Rightarrow \sin \frac{x}{2}=\sqrt{\frac{4+\sqrt{15}}{8}} \quad\left[\because \sin \frac{x}{2} \text { is negative }\right]$
$=\sqrt{\frac{4+\sqrt{15}}{8} \times \frac{2}{2}}$
$=\sqrt{\frac{8+2 \sqrt{15}}{16}}$
$=\frac{\sqrt{8+2 \sqrt{15}}}{4}$
$\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}=\frac{1+\left(-\frac{\sqrt{15}}{4}\right)}{2}=\frac{4-\sqrt{15}}{8}$
$\Rightarrow \cos \frac{x}{2}=\sqrt{\frac{4-\sqrt{15}}{8}} \quad\left[\because \cos \frac{x}{2} \text { is positve }\right]$
$=\sqrt{\frac{4+\sqrt{15}}{8} \times \frac{2}{2}}$
$=\sqrt{\frac{8-2 \sqrt{15}}{16}}$
$=\frac{\sqrt{8-2 \sqrt{15}}}{4}$
$\tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\left(\frac{\sqrt{8+2 \sqrt{15}}}{4}\right)}{\frac{\sqrt{8-2 \sqrt{15}}}{4}}=\frac{\sqrt{8+2 \sqrt{15}}}{\sqrt{8-2 \sqrt{15}}}$
$=\sqrt{\frac{8+2 \sqrt{15}}{8-2 \sqrt{15}}} \times \frac{8+2 \sqrt{15}}{8+2 \sqrt{15}}$
$=\sqrt{\frac{(8+2 \sqrt{15})^{2}}{64-60}}=\frac{8+2 \sqrt{15}}{2}=4+\sqrt{15}$
Thus, the respective values are $\sin \frac{x}{2}, \cos \frac{x}{2}$ and $\tan \frac{x}{2}$
$\operatorname{are} \frac{\sqrt{8+2 \sqrt{15}}}{4}, \frac{\sqrt{8-2 \sqrt{15}}}{4}$ and $4+\sqrt{15}$
$\cos \left(-1710^{\circ}\right)$ નું મૂલ્ય શોધો.
જો $x{\sin ^3}\alpha + y{\cos ^3}\alpha = \sin \alpha \cos \alpha $ અને $x\sin \alpha - y\cos \alpha = 0,$ તો ${x^2} + {y^2} = $
એક ચક્ર એક મિનિટમાં ${360^\circ }$ પરિભ્રમણ કરે છે, તો તે એક સેકન્ડમાં કેટલા રેડિયન માપ જેટલું ફરશે ?
જો $\cos (\alpha - \beta ) = 1$ અને $\cos (\alpha + \beta ) = \frac{1}{e}$, $ - \pi < \alpha ,\beta < \pi $, તો $(\alpha ,\beta )$ ની કુલ જોડની સંખ્યા મેળવો.
જો $\theta $ એ બીજા ચરણમાં હોય તો $\sqrt {\left( {\frac{{1 - \sin \theta }}{{1 + \sin \theta }}} \right)} + \sqrt {\left( {\frac{{1 + \sin \theta }}{{1 - \sin \theta }}} \right)} $ ની કિમત મેળવો.