Question

Calculate the value of $\sin \left(22.5\right)$. Show the steps.

Answer 1

Let us assume $\theta =22.5^0$

So, $2\theta$ is ${45}^0$.

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

On substituting the value of $\theta$ in equation $\left(1\right)$, we get

$\begin{array}{rcl}{\sin }^2\left(22.5^0\right)& =& \frac{\left[1-\cos \left(2\times 22.5^0\right)\right]}{2}\\ & =& \frac{\left[1-\cos \left(45°\right)\right]}{2}\\ & =& \frac{\left[1-\frac{1}{\sqrt{2}}\right]}{2}\\ & =& \frac{\left[\sqrt{2}-1\right]}{2\sqrt{2}}\\ & =& 0.146[\sqrt{2}=1.414]\end{array}$

So, $$\begin{gathered} \sin \left(22.5^0\right)=\sqrt{0.146} \\ =0.38 \end{gathered}$$

Hence, the value of $\sin \left(22.5^0\right)$ is $0.38$ .