Question

Find the value of cos 22°30'

Answer 1

$\textbf{To find:}$

$cos\,22.5^{\circ}$

$\text{We know that the following trigonometry identity}$

$\boxed{\bf\;cosA=2\,cos^2{\frac{A}{2}}-1}$

$\implies\,cos^2{\frac{A}{2}}=\frac{1+cosA}{2}$

$\text{Put A=45^{\circ} }$

$cos^2({\frac{45^{\circ}}{2}})=\frac{1+cos45^{\circ}}{2}$

$\implies\,cos^2{22.5^{\circ}}=\frac{1+\frac{1}{\sqrt2}}{2}$

$\implies\,cos^2{22.5^{\circ}}=\frac{\sqrt{2}+1}{2\sqrt2}}$

$\implies\bf\,cos{22.5^{\circ}}=\sqrt{\frac{\sqrt{2}+1}{2\sqrt2}}$

$\therefore\textbf{The value of \bf\,cos\,22^{\circ}5' is \bf\sqrt{\frac{\sqrt{2}+1}{2\sqrt2}} }$