Question

How do you use a half-angle formula to find $\cos \left(\frac{\mathrm{\pi }}{8}\right)$?

Answer 1

Use double angle identities to obtain half angle relations

We know that,

$\cos 2A={\cos }^{2}A-{\sin }^{2}A$

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

$\Rightarrow$ $\cos 2A=2{\cos }^{2}A-1$

Now replace $2A$ by $A$ and $A$ by $\frac{A}{2}$

$\Rightarrow$ $\cos A=2{\cos }^2\frac{A}{2}-1$

Substitute value of $A=\frac{\mathrm{\pi }}{4}$ we get,

$\Rightarrow$ $\cos \frac{\mathrm{\pi }}{4}=2{\cos }^2\frac{{\displaystyle \frac{\mathrm{\pi }}{4}}}{2}-1$

$\Rightarrow$ $\frac{1}{\sqrt{2}}=2{\cos }^2\frac{{\displaystyle \mathrm{\pi }}}{8}-1$ $...\left[cos\frac{\pi }{4}=\frac{1}{\sqrt{2}}\right]$

$\Rightarrow$ $2{\cos }^2\frac{{\displaystyle \mathrm{\pi }}}{8}=\frac{1+\sqrt{2}}{\sqrt{2}}$

$\Rightarrow$ ${\cos }^2\frac{{\displaystyle \mathrm{\pi }}}{8}=\frac{1+\sqrt{2}}{2\sqrt{2}}$

$\Rightarrow$ $\cos \frac{{\displaystyle \mathrm{\pi }}}{8}=\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}=0.923$

Hence, the value of $\cos \left(\frac{\mathrm{\pi }}{8}\right)$ by half angle formula is $\sqrt{\frac{1+\sqrt{2}}{2\sqrt{2}}}=0.923$