Question

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

Answer 1

Step:1 General case of $\sin \left(\mathrm{\theta }\right)$ :

Let's draw the figure :

We have a unit radius circle and $∆OAB$:

$∆OAB$ is a right-angle triangle.

For angle $\theta$ :

$\sin \left(\mathrm{\theta }\right)=\frac{AB}{OA}=\frac{AB}{1}$ [$OA$ is the radius of unit circle]

Step:2 Convert the angle from radians to degree.

$$\begin{gathered} \sin \left(\frac{\mathrm{\pi }}{2}\right)=\sin \left(\frac{\mathrm{\pi }\times 180}{2\times \mathrm{\pi }}\right) \\ \sin \left(\frac{\mathrm{\pi }}{2}\right)=\sin \left(90°\right) \end{gathered}$$

Step:3 Finding the required value.

As we can see, when the angle $\theta$ reaches to the$90°$, $AB\underset{}{\to }A\text{'}B$.

$$\begin{gathered} \sin \left(90°\right)=\frac{A\text{'}B}{OA} \\ \sin \left(90°\right)=\frac{1}{1} \\ \sin \left(90°\right)=1 \end{gathered}$$

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