Chapter 13.6, Problem 23PPS

To determine

To find the exact value of $\sin \left(\frac{11\pi }{4}\right)$ .

Answer to Problem 23PPS

  $\frac{1}{\sqrt{2}}$

Explanation of Solution

Given:

  $\sin \left(\frac{11\pi }{4}\right)$

Calculation:

Given angle $\sin \left(\frac{11\pi }{4}\right),$

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

  $\Rightarrow \sin \left(\frac{11\pi }{4}\right)=\sin \left(3\pi -\frac{\pi }{4}\right)$

Here $\sin$ ratio lies in second quadrant, so it is positive.

  $\sin \left(\frac{11\pi }{4}\right)=\sin \left(3\pi -\frac{\pi }{4}\right)$

  $\begin{array}{l}=\sin \left(\frac{\pi }{4}\right)\\ =\frac{1}{\sqrt{2}}\end{array}$

Conclusion:

Therefore, the exact value of $\sin \left(\frac{11\pi }{4}\right)$ is $\frac{1}{\sqrt{2}}$ .