Chapter 12.3, Problem 52E

To determine

To calculate:

The exact value of $\cos 112.5°$ .

Answer to Problem 52E

The exact value of $\cos 112.5°$ is $-0.382$ .

Explanation of Solution

Calculation:

Consider the trigonometric function $\cos 112.5°$ .

We need to find the exact value of $\cos 112.5°$ .

Here, $\frac{a}{2}=112.5°$ since, the angle is in second quadrant.

Cos is negative number in second quadrant.

Thus, from the half angle formula,

  $\cos \left(\frac{\alpha }{2}\right)=\pm \sqrt{\frac{\left(1+\cos \alpha \right)}{2}}$

So,

  $\begin{array}{l}\cos \left(\frac{\alpha }{2}\right)=\pm \sqrt{\frac{\left(1+\cos \alpha \right)}{2}}\\ \cos \left(112.5\right)=-\sqrt{\frac{\left(1+\cos \left(225°\right)\right)}{2}}\\ \cos \left(112.5\right)=-\sqrt{\frac{\left(1-\frac{1}{\sqrt{2}}\right)}{2}}\\ \cos \left(112.5\right)=-\sqrt{\frac{\left(\sqrt{2}-1\right)}{2\sqrt{2}}}\\ \cos \left(112.5\right)=-\sqrt{\frac{0.414}{2.828}}\\ \cos \left(112.5\right)=-0.38268\end{array}$

Hence, the exact value of $\cos 112.5°$ is $-0.382$ .