Question

How do you solve $\sin \theta = - 0.5$

Answer 1

Hint: We will simplify the given equation and then by using the trigonometric table, we will find the value of $x$. Finally we get the required answer.

Complete step-by-step solution:
The given term is: $\sin \theta = - 0.5$
Now we know that the number $ - 0.5$ can be written in the form of a fraction as $ - \dfrac{1}{2}$ therefore, on substituting it in the right-hand side of the expression, we get:
$ \Rightarrow \sin \theta = - \dfrac{1}{2}$
Now from the trigonometric table, we know that:
$ \Rightarrow \sin \left( { - \dfrac{\pi }{6}} \right) = \left( {\sin \left( {\pi - \left( { - \dfrac{\pi }{6}} \right)} \right)} \right)$
This can be written as:
 $ \Rightarrow \sin \left( {\dfrac{{7\pi }}{6}} \right)$
This has the value: $\dfrac{1}{2}$
Therefore,
$ \Rightarrow x = {\left( {\dfrac{\pi }{6}} \right)^c}$which is \[{30^\circ }\] , $x = {\left( {\dfrac{{7\pi }}{6}} \right)^c}$which is ${210^\circ }$
Now we know that $\sin ( - \theta ) = - \sin \theta $

Therefore, on generalizing the answer, we get:
$ \Rightarrow x = \left( {2n\pi - \dfrac{\pi }{6}} \right)or\left( {2n\pi + \dfrac{{7\pi }}{6}} \right),n \to \varepsilon \to \mathbb{Z}$ , which is the required answer.

Note: This question can also be done by using the inverse trigonometric function as:
We have the given equation after simplification as: $\sin \theta = - \dfrac{1}{2}$
Now using the inverse trigonometric function, we get:
$ \Rightarrow \theta = {\sin ^{ - 1}}\left( { - \dfrac{1}{2}} \right)$
Therefore, the principal value of $\sin x$ is $\dfrac{\pi }{6}$
Now we know that $\sin ( - \theta ) = - \sin \theta $
Therefore, the principal value becomes $ - \dfrac{\pi }{6}$
Now since sine is positive in the first and second quadrant, we will subtract the principal value from $\pi $ to get the solution in the second quadrant.
Therefore,
$ \Rightarrow \pi - - \left( {\dfrac{\pi }{6}} \right)$
On simplifying we get:
 $ \Rightarrow \dfrac{{7\pi }}{6}$, which is the solution in the second quadrant.
Therefore, on generalizing the answer we get:
$ \Rightarrow x = \left( {2n\pi - \dfrac{\pi }{6}} \right)or\left( {2n\pi + \dfrac{{7\pi }}{6}} \right),n \to \varepsilon \to \mathbb{Z}$, which is the required answer.
It is to be remembered which trigonometric functions are positive and negative in what quadrants.
The formula used over here is for $\sin (n\pi + x)$ ,
It is to be remembered that $\sin (n\pi + x) = {( - 1)^n}\sin x$
Basic trigonometric formulas should be remembered to solve these types of sums.
The inverse trigonometric function of $\sin x$ which is ${\sin ^{ - 1}}x$ used in this sum
For example, if $\sin x = a$ then $x = {\sin ^{ - 1}}a$ .
And ${\sin ^{ - 1}}(\sin x) = x$ is a property of the inverse function.
There also exists inverse function for the other trigonometric relations such as tan and cos.
The inverse function is used to find the angle $x$ from the value of the trigonometric relation.