given that \\( \\cos 2 x = \\frac { 2 } { 3 } \\), with \\( \\frac { \\pi } { 2 } < x < \\pi \\), determine…

given that \\( \\cos 2 x = \\frac { 2 } { 3 } \\), with \\( \\frac { \\pi } { 2 } < x < \\pi \\), determine the exact value \\( \\cos x \\).\n\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 { 2 } { 3 } \\), with \\( \\frac { \\pi } { 2 } < x < \\pi \\), determine the exact value \\( \\cos x \\).\n\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{2}{3}), we substitute it into the formula: (\frac{2}{3}=2\cos^{2}x - 1).

Step2: Solve for (\cos^{2}x)

Add (1) to both sides of the equation: (2\cos^{2}x=\frac{2}{3}+ 1=\frac{2 + 3}{3}=\frac{5}{3}). Then divide both sides by (2): (\cos^{2}x=\frac{5}{6}).

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{5}{6}): (\cos x=-\sqrt{\frac{5}{6}}=-\frac{\sqrt{5}}{\sqrt{6}}). Rationalize the denominator: (\cos x=-\frac{\sqrt{5}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=-\frac{\sqrt{30}}{6}).

Answer:

(-\frac{\sqrt{30}}{6})