Question

Whow do you solve $\sin \theta = 0?$

Answer 1

We need to know the trigonometric table values and basic definitions $\sin \theta = 0$ . To to solve the given problem we need to find the value of $\theta$ . Also, we need to know the process of calculating ${\tan ^{ - 1}}$ value in the calculator. Also, we need to know when we have the same answer $\sin \theta$ for different $\theta$ values.

Complete step by step solution:
The given equation is shown below,
$\sin \theta = 0$

We need to find the value $\theta$ from the above equation. Before that, we need to know the basic definition $\sin \theta$ .

The above figure represents a triangle marked with the opposite side, hypotenuse side, and adjacent side according to the position of $\theta$ .

We know that,
$\sin \theta = \dfrac{{opposite}}{{hypotenuse}}$ $\to \left( 1 \right)$

The given equation is,
$\sin \theta = 0$

The above equation can also be written as,
$\sin \theta = \dfrac{0}{1}$ $\to \left( 2 \right)$

By comparing the equation $\left( 1 \right)$ and $\left( 2 \right)$ , we get the value of the opposite side is $0$ , and the value of the hypotenuse side is $1$ .

When the term” $\sin$ ” is the move from the left side to the right side of the equation, it converts into $\arcsin$ .

So, the value of $\theta$ is given below,

$\sin \theta = 0$
$\theta = \arcsin \left( 0 \right)$

The above equation can also be written as,
$\arcsin \left( 0 \right) = \theta$

By using the trigonometric table values, we find that the value of $\sin \theta$ is $0$ when the value of $\theta$ is $0,\pi ,2\pi ,3\pi ,......$

So, we get
$\arcsin \left( 0 \right) = {0^ \circ },\arcsin \left( 0 \right) = \pi ,\arcsin \left( 0 \right) = 2\pi .....$

**So, the final answer is,
$\theta = {0^ \circ },\pi ,2\pi ,3\pi ,......$ **

Note: In these types of questions we would find the value of $\theta$ from the given equation. In this question, we use trigonometric table values for finding the value of $\theta$ . We can also use a scientific calculator to find the value $\theta$ . On finding the $\theta$ value in the calculator We can use either radian mode or degree mode. It $0(or)\pi$ is involved in the value of $\theta$ ,
then the value of $\sin \theta$ is always $0$ .