Question

given that \\( \cos 2 x = \frac { 3 } { 5 } \\), with \\( \frac { \pi } { 2 } < x < \pi \\), determine the exact value \\( \cos x \\).

\\( \cos x = \\)
(simplify your answer. type an exact answer, using radicals as needed. rationalize the de

Explanation:

Step1: Use the double - angle formula

The double - angle formula for cosine is $\cos2x = 2\cos^{2}x - 1$. Given $\cos2x=\frac{3}{5}$, we substitute it into the formula:
$$\frac{3}{5}=2\cos^{2}x - 1$$

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

Add 1 to both sides of the equation:
$$\frac{3}{5}+1 = 2\cos^{2}x$$
$$\frac{3 + 5}{5}=2\cos^{2}x$$
$$\frac{8}{5}=2\cos^{2}x$$
Divide both sides by 2:
$$\cos^{2}x=\frac{8}{5}\times\frac{1}{2}=\frac{4}{5}$$

Step3: Determine the sign of $\cos x$

Since $\frac{\pi}{2}Take the square root of $\cos^{2}x=\frac{4}{5}$:
$$\cos x=-\sqrt{\frac{4}{5}}=-\frac{2}{\sqrt{5}}$$
Rationalize the denominator:
$$\cos x=-\frac{2\sqrt{5}}{5}$$

Answer:

$-\frac{2\sqrt{5}}{5}$