Question

How do you find the exact values of $\cos 22.5°$ using the half-angle formula?

Answer 1

Compute the required value.

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

Here $A=45°$

$\Rightarrow$ $\cos A={\cos }^2\left(\frac{A}{2}\right)-\left(1-{\cos }^2\left(\frac{A}{2}\right)\right)$

$\Rightarrow$ $\cos A=2{\cos }^2\left(\frac{A}{2}\right)-1$

$\Rightarrow$ $\cos 45°=2{\cos }^{2}22.5°-1$

$\Rightarrow$ $\frac{1}{\sqrt{2}}+1=2{\cos }^{2}22.5°$ $\left[\cos 45°=\frac{1}{\sqrt{2}}\right]$

$\Rightarrow$ $\frac{\sqrt{2}}{2}+1=2{\cos }^{2}22.5°$

$\Rightarrow$ $\frac{\sqrt{2}+2}{4}={\cos }^{2}22.5°$

$\Rightarrow$ $\cos 22.5°=\frac{\sqrt{(\sqrt{2}+2)}}{2}$

$\Rightarrow$ $\cos 22.5°\approx 0.92$

Hence, the exact value of $\cos 22.5°$ is $\frac{\sqrt{(\sqrt{2}+2)}}{2}$ or $0.92$.