Question

$\sin \frac{5\mathrm{\pi }}{4}=\_\_\_\_\_\_\_\_.$

Short answer

The value of$\sin \frac{5\mathrm{\pi }}{4}=-\frac{1}{\sqrt{2}}.$

Step-by-step solution

Step 1. Given Information 

The sine function is a periodic function. The ratio of the perpendicular and the hypotenuse is called the sine function.

Step 2. Filling the blank 

We can write $\sin \frac{5\mathrm{\pi }}{4}$as $\sin \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right).$

As we know sine function is negative in the third quadrant.

Thus,

$$\begin{gathered} \sin \left(\frac{5\mathrm{\pi }}{4}\right)=\sin \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right) \\ \sin \left(\frac{5\mathrm{\pi }}{4}\right)=-\sin \left(\frac{\mathrm{\pi }}{4}\right) \\ \sin \left(\frac{5\mathrm{\pi }}{4}\right)=-\frac{1}{\sqrt{2}} \end{gathered}$$