Question

1. 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 \\).

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:

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