Question

what is $csc(-\frac{15pi}{2})$?

Explanation:

Step1: Use the property of cosecant function

Since \(\csc(x)=\frac{1}{\sin(x)}\) and \(\sin(-x) = -\sin(x)\), then \(\csc(-\frac{15\pi}{2})=\frac{1}{\sin(-\frac{15\pi}{2})}=-\frac{1}{\sin(\frac{15\pi}{2})}\).

Step2: Simplify the angle using the period of sine function

The period of \(y = \sin(x)\) is \(2\pi\). We can write \(\frac{15\pi}{2}=6\pi+\frac{3\pi}{2}\). And \(\sin(x + 2k\pi)=\sin(x)\) for \(k\in\mathbb{Z}\). So \(\sin(\frac{15\pi}{2})=\sin(6\pi+\frac{3\pi}{2})=\sin(\frac{3\pi}{2})\).

Step3: Evaluate \(\sin(\frac{3\pi}{2})\)

We know that \(\sin(\frac{3\pi}{2})=- 1\).

Step4: Calculate \(\csc(-\frac{15\pi}{2})\)

Substitute \(\sin(\frac{3\pi}{2})=-1\) into \(-\frac{1}{\sin(\frac{15\pi}{2})}\), we get \(-\frac{1}{\sin(\frac{3\pi}{2})}=-\frac{1}{-1}=1\).

Answer:

\(1\)