Chapter 14, Problem 24MCQ

To determine

The exact value of the expression.

Answer to Problem 24MCQ

The correct option is H.

Explanation of Solution

Given:

The expression is given as:

  $\cos \frac{5\pi }{12}$

The options are given as:

  $\begin{array}{l}F\sqrt{2}H\frac{\sqrt{6}-\sqrt{2}}{4}\\ G\frac{\sqrt{6}+\sqrt{2}}{2}J\frac{\sqrt{6}+\sqrt{2}}{4}\end{array}$

Consider the expression:

  $\cos \frac{5\pi }{12}$

Since,

  $\cos \left(\pi -\theta \right)=-\cos \theta$

Therefore,

  $\begin{array}{l}\cos \frac{5\pi }{12}=\cos \left(\pi -\frac{7\pi }{12}\right)\\ =-\cos \frac{7\pi }{12}\end{array}$

From sum identify:

  $\cos \left(A+B\right)=\cos A\cos B-\sin A\sin B$

Therefore,

  $\begin{array}{l}\cos \frac{7\pi }{12}=-\cos \left(\frac{\pi }{3}+\frac{\pi }{4}\right)\\ =-\left(\cos \frac{\pi }{3}\cos \frac{\pi }{4}-\sin \frac{\pi }{3}\sin \frac{\pi }{4}\right)\\ =-\left(\frac{1}{2}.\frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{2}.\frac{1}{\sqrt{2}}\right)\\ =\frac{\sqrt{3}-1}{2\sqrt{2}}\end{array}$

Multiply and divide the equation with $\sqrt{2}$ as follows:

  $\begin{array}{l}\cos \frac{7\pi }{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}.\frac{\sqrt{2}}{\sqrt{2}}\\ =\frac{\sqrt{6}-\sqrt{2}}{4}\end{array}$

Hence, the solution of the expression from the given options is H.