Question

How do you find the exact values of$\sin \left(22.5°\right)$using the half-angle formula?

Answer 1

Compute the required value.

We know that, ${\sin }^2\frac{\theta }{2}=\frac{\left(1-\cos \theta \right)}{2}$

Here, $\theta =45°$

$\Rightarrow$ ${\sin }^2\left(\frac{45°}{2}\right)=\frac{\left[1-\cos \left(45°\right)\right]}{2}$

$\Rightarrow$ ${\sin }^2\left(22.5°\right)=\frac{\left[1-\cos 45°\right]}{2}$

$\Rightarrow$ ${\sin }^2\left(22.5°\right)=\frac{\left[1-{\displaystyle \frac{1}{\sqrt{2}}}\right]}{2}$

$\Rightarrow$ ${\sin }^2\left(22.5°\right)=\frac{\sqrt{2}-1}{2\sqrt{2}}$

$\Rightarrow$ ${\sin }^2\left(22.5°\right)=\frac{2-\sqrt{2}}{4}$

$\Rightarrow$ ${\sin }^2\left(22.5°\right)\approx \frac{2-1.414}{4}\left[\because \sqrt{2}\approx 1.414\right]$

$\Rightarrow$ ${\sin }^2\left(22.5°\right)\approx 0.15$

$\Rightarrow$ $\sin \left(22.5°\right)\approx \sqrt{0.15}\approx 0.38$

Hence, the value of $\sin \left(22.5°\right)$is approximately $0.38$.