8. given that \\( \\sin \\frac { 2 \\pi } { 5 } = \\cos x \\), first express \\( \\frac { 2 \\pi } { 5 } \\)…

8. given that \\( \\sin \\frac { 2 \\pi } { 5 } = \\cos x \\), first express \\( \\frac { 2 \\pi } { 5 } \\) as a difference between \\( \\frac { \\pi } { 2 } \\) and an angle, and then apply a cofunction identity to determine the measure of angle \\( x \\).

8. given that \\( \\sin \\frac { 2 \\pi } { 5 } = \\cos x \\), first express \\( \\frac { 2 \\pi } { 5 } \\) as a difference between \\( \\frac { \\pi } { 2 } \\) and an angle, and then apply a cofunction identity to determine the measure of angle \\( x \\).

Answer

Explanation:

Step1: Express (\frac{2\pi}{5}) as a difference

We know that (\frac{2\pi}{5}=\frac{\pi}{2}-\frac{\pi}{10}).

Step2: Apply co - function identity

The co - function identity is (\sin\theta=\cos(\frac{\pi}{2}-\theta)). If (\sin\frac{2\pi}{5}=\cos x) and (\frac{2\pi}{5}=\frac{\pi}{2}-\frac{\pi}{10}), then (\sin(\frac{\pi}{2}-\frac{\pi}{10})=\cos x). By the co - function identity (\sin A=\cos(\frac{\pi}{2}-A)), here (A = \frac{\pi}{2}-\frac{\pi}{10}), so (\cos(\frac{\pi}{10})=\cos x).

Answer:

(x=\frac{\pi}{10})