given that \\( \\cos 2 x = \\frac { 4 } { 5 } \\), with \\( \\frac { \\pi } { 2 } < x < \\pi \\), determine…

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

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

Answer

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})