# How do you use a half-angle formula to find the exact value of cos22.5?

Aug 10, 2018

color(maroon)(cos 22.5^@ = + sqrt((sqrt2 -1)/(2 sqrt2)) ~~ + 0.3827

#### Explanation:

$\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

Let $\hat{\frac{\theta}{2}} = {22.5}^{\circ}$

$\hat{\theta} = 2 \cdot 22.5 = {45}^{\circ}$

cos (theta/2) = cos 22.5^@ = + sqrt((1 - cos 45) / 2)

We know $\cos 45 = \frac{1}{\sqrt{2}}$

:. color(maroon)(cos 22.5^@ = + sqrt((1 - 1/sqrt2)/2) = + sqrt((sqrt2 -1)/(2 sqrt2)) ~~ + 0.3827

Aug 10, 2018

$\cos {22.5}^{\circ} = \frac{1}{2} \left(\sqrt{2 + \sqrt{2}}\right)$

#### Explanation:

$\text{using the "color(blue)"half-angle formula for cos}$

•color(white)(x)cos(x/2)=+-sqrt((1+cosx)/2)#

$\cos {22.5}^{\circ} = + \sqrt{\frac{1 + \cos {45}^{\circ}}{2}}$

$\textcolor{w h i t e}{\times \times \times} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$

$\textcolor{w h i t e}{\times \times \times} = \sqrt{\frac{2 + \sqrt{2}}{4}}$

$\textcolor{w h i t e}{\times \times \times} = \frac{1}{2} \left(\sqrt{2 + \sqrt{2}}\right)$