Question

In problems 11-26,use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sin \left(\frac{9\pi }{4}\right)$

Short answer

The exact value is$\frac{\sqrt{2}}{2}$.

Step-by-step solution

Step 1. Given information.  

We have:

$\sin \left(\frac{9\pi }{4}\right)$

Step 2. Find the exact value of trigonometric function.    

The function is $\sin \left(\frac{9\pi }{4}\right)$.

Formula used:

$\sin \left(\theta +2\pi \right)=\sin \theta$

Remove full rotations of $2\pi$until the angle is between $0$and $2\pi$.

We get:

$$\begin{gathered} \sin \left(\frac{9\pi }{4}\right)=\sin \left(\frac{\pi }{4}\right) \\ =\frac{\sqrt{2}}{2} \end{gathered}$$