Question

1. given that \\( \csc \frac { 3 \pi } { 10 } = \sec y \\), first express \\( \frac { 3 \pi } { 10 } \\) as a difference between \\( \frac { \pi } { 2 } \\) and an angle, and then apply a cofunction identity to determine the measure of angle \\( y \\).

Explanation:

Step1: Express \(\frac{3\pi}{10}\) as a difference

We know that \(\frac{3\pi}{10}=\frac{\pi}{2}-\frac{\pi}{5}\). So, \(\csc\frac{3\pi}{10}=\csc(\frac{\pi}{2}-\frac{\pi}{5})\).

Step2: Apply co - function identity

The co - function identity for cosecant is \(\csc(A - B)=\sec B\) when \(A=\frac{\pi}{2}\). Using the co - function identity \(\csc(\frac{\pi}{2}-\theta)=\sec\theta\), if \(A = \frac{\pi}{2}\) and \(\theta=\frac{\pi}{5}\), then \(\csc(\frac{\pi}{2}-\frac{\pi}{5})=\sec\frac{\pi}{5}\).
Since \(\csc\frac{3\pi}{10}=\sec y\) and \(\csc\frac{3\pi}{10}=\sec\frac{\pi}{5}\), by the equality of secant functions.

Answer:

\(y = \frac{\pi}{5}\)