Chapter 13.8, Problem 12WE

To determine

To find:The trigonometric functionin a single angle.

Answer to Problem 12WE

The trigonometric function of a single angle is $\cos \alpha$ .

Explanation of Solution

Given information:

The function is $\sqrt{\frac{1+\cos 2\alpha }{2}}$ .

Calculation:

Consider the formula.

  $\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \beta \sin \alpha$

Thus,

  $\begin{array}{c}\sqrt{\frac{1+\cos 2\alpha }{2}}=\sqrt{\frac{1+\cos \left(\alpha +\alpha \right)}{2}}\\ =\sqrt{\frac{1+\cos \alpha \cos \alpha -\sin \alpha \sin \alpha }{2}}\\ =\sqrt{\frac{1+{\cos }^2\alpha -{\sin }^2\alpha }{2}}\\ =\sqrt{\frac{{\cos }^2\alpha +{\cos }^2\alpha }{2}}\\ =\sqrt{\frac{2{\cos }^2\alpha }{2}}\\ =\sqrt{{\cos }^2\alpha }\\ =\cos \alpha \end{array}$