9. given that \\( \\csc \\frac { 3 \\pi } { 10 } = \\sec y \\), first express \\( \\frac { 3 \\pi } { 10 }…

9. 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 \\).

9. 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 \\).

Answer

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})