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  4. If sin θ=(24/25) and 0°<θ<90° then what is the value of sin ((θ/2)) ?

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Q. If $\sin\, \theta=\frac{24}{25}$ and $0^{\circ}<\theta<90^{\circ}$ then what is the value of $\sin \left(\frac{\theta}{2}\right) ?$

Trigonometric Functions

Solution:

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We have, $\sin \,\theta=\frac{24}{25}, 0^{\circ}<\theta<90^{\circ}$
$\cos ^{2} \theta=1-\sin ^{2} \theta=1-\left(\frac{24}{25}\right)^{2}$
Since $\theta$ lies in first quadrant
$\Rightarrow \cos \theta=\frac{7}{25} \Rightarrow \cos \theta=1-2 \sin ^{2} \frac{\theta}{2}$
$\Rightarrow 2 \sin ^{2} \frac{\theta}{2}=1-\cos \theta=1-\frac{7}{25} \Rightarrow 2 \sin ^{2} \frac{\theta}{2}=\frac{18}{25}$
$\Rightarrow \sin ^{2} \frac{\theta}{2}=\frac{9}{25} \Rightarrow \sin \frac{\theta}{2}=\pm \frac{3}{5} \Rightarrow \sin \frac{\theta}{2}=\frac{3}{5}$
[Negative sign discarded since $\theta$ is in first quadrant]