Question

Whow do you solve \[\sin \theta = 0?\]

Answer 1

Hint: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 \[0\] .