Question

What is the exact value of cos112.5?

Answer 1

The answer is -0.382683432 if 112.5 is in degrees

Answer 2

The exact value of cos⁡ 112.5 is $\bold{-\frac{\sqrt{(2-\sqrt{2})}}{2}= 0.3827531}$

Solution:

Use half angle formula for Cos,

$\begin{array}{l}{\cos \left(\frac{x}{2}\right)=\pm \sqrt{\frac{1+\cos x}{2}}} \\ {\cos 112.5=\cos \frac{225}{2}=-\sqrt{\frac{1+\cos 225}{2}} \quad(\text { equation } 1)}\end{array}$

(Since cos 112.5 is in II quadrant ,negative sign is used)            

cos⁡  225 = cos⁡ (45+180)

cos ⁡(a+b) = cos a cos b+sin a sin⁡ b

cos ⁡(45+180) = cos 45 cos⁡ 180+ sin⁡ 45 sin ⁡180

$\begin{array}{l}{\cos 45=\frac{\sqrt{2}}{2}} \\ \\ {\sin 45=\frac{\sqrt{2}}{2}}\end{array}$

cos⁡ 180 = -1  

sin⁡ 180 = 0

$\begin{array}{l}{\cos (45+180)=\frac{\sqrt{2}}{2}(-1)+\frac{\sqrt{2}}{2}(0)} \\\\ {\cos (45+180)=-\frac{\sqrt{2}}{2}+0} \\\\ {\cos (45+180)=-\frac{\sqrt{2}}{2}(\text { equation } 2)}\end{array}$

apply equation 2 in equation 1

$\begin{array}{l}{\cos \frac{225}{2}=-\sqrt{\frac{1+\left(-\frac{\sqrt{2}}{2}\right)}{2}}=-\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}=-\sqrt{\frac{2-\sqrt{2}}{2}}=-\sqrt{\frac{2-\sqrt{2}}{4}}} \\ {\cos \frac{225}{2}=-\frac{\sqrt{2-\sqrt{2}}}{2}} \\ {\cos 112.5=-\frac{\sqrt{2-\sqrt{2}}}{2}}=0.3827531\end{array}$