Question

find the exact value. do not use a calculator.

\\( \csc \frac { 3 \pi } { 2 } \\)

\\( \csc \frac { 3 \pi } { 2 } = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )

Explanation:

Step1: Recall the reciprocal identity

The cosecant function is the reciprocal of the sine function, i.e., \(\csc\theta=\frac{1}{\sin\theta}\). So, \(\csc\frac{3\pi}{2}=\frac{1}{\sin\frac{3\pi}{2}}\).

Step2: Find the value of \(\sin\frac{3\pi}{2}\)

On the unit - circle, for \(\theta = \frac{3\pi}{2}\), the \(y\) - coordinate (which gives the value of \(\sin\theta\)) is \(- 1\). So, \(\sin\frac{3\pi}{2}=-1\).

Step3: Calculate the value of \(\csc\frac{3\pi}{2}\)

Substitute \(\sin\frac{3\pi}{2}=-1\) into the formula \(\csc\frac{3\pi}{2}=\frac{1}{\sin\frac{3\pi}{2}}\). Then \(\csc\frac{3\pi}{2}=\frac{1}{-1}\).

Answer:

\(-1\)