Question

given that \\( \cos 2 x = \frac { 4 } { 5 } \\), with \\( \frac { \pi } { 2 } < x < \pi \\), determine the exact value \\( \cos x \\).
\\( \cos x = \square \\)
(simplify your answer. type an exact answer, using radicals as needed. rationalize the denominator.)

Explanation:

Step1: Use the double - angle formula

The double - angle formula for cosine is $\cos2x = 2\cos^{2}x - 1$.
Given $\cos2x=\frac{4}{5}$, we substitute it into the formula:
$$\frac{4}{5}=2\cos^{2}x - 1$$

Step2: Solve for $\cos^{2}x$

Add 1 to both sides of the equation:
$$\frac{4}{5}+1 = 2\cos^{2}x$$
$$\frac{4 + 5}{5}=2\cos^{2}x$$
$$\frac{9}{5}=2\cos^{2}x$$
Then divide both sides by 2:
$$\cos^{2}x=\frac{9}{10}$$

Step3: Determine the sign of $\cos x$

Since $\frac{\pi}{2}\lt x\lt\pi$, \(x\) is in the second quadrant. In the second quadrant, \(\cos x\lt0\).
Take the square root of \(\cos^{2}x=\frac{9}{10}\):
$$\cos x=-\sqrt{\frac{9}{10}}=-\frac{3}{\sqrt{10}}$$
Rationalize the denominator:
$$\cos x =-\frac{3\sqrt{10}}{10}$$

Answer:

\(-\frac{3\sqrt{10}}{10}\)