Question

How do you evaluate $\sin \left(\frac{7\pi }{2}\right)$?

Answer 1

Evaluating the value of $\sin \left(\frac{7\pi }{2}\right)$

$\sin \left(\frac{7\pi }{2}\right)$ can be written as $\sin \left(\frac{6\mathrm{\pi }+\mathrm{\pi }}{2}\right)$.

We know that sine is a periodic function with period equal to $\mathrm{\pi }$ and the value of sine function is negative in the third quadrant.

Therefore,

$\begin{array}{rcl}\sin (3\pi +\theta )& =& -\sin \theta \\ & \Rightarrow & \sin \left(3\mathrm{\pi }+\frac{\mathrm{\pi }}{2}\right)=-\sin \left(\frac{\pi }{2}\right)\\ & \Rightarrow & \sin \left(\frac{7\mathrm{\pi }}{2}\right)=-1\end{array}$

Hence, the value of $\sin \left(\frac{7\pi }{2}\right)$ is$-1$