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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:

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 lies in first quadrant $\Rightarrow \, \cos \theta = \frac{7}{25}$
$\cos \theta = 1 - 2 \sin^2 \frac{\theta}{2}$
$ 2 \sin^2 \frac{\theta}{2} = 1 - \cos \theta = 1 - \frac{7}{25}$
$2 \sin^2 \frac{\theta}{2} = \frac{18}{25}$
$\sin^2 \frac{\theta}{2} = \frac{9}{25} \, \Rightarrow \, \sin \frac{\theta}{2} = \pm \frac{3}{5}$
$\Rightarrow \, \sin \frac{\theta}{2} = \frac{3}{5}$ [Negative sign discarded since $\theta$ is in first quadrant]