Question

How do you simplify the value of \[\arcsin \left( { - \dfrac{1}{2}} \right)\]?

Answer 1

Hint: In the given question, we have been given a trigonometric function. This trigonometric function is raised to some negative power. Also, this trigonometric function has a constant as its argument. We just need to know what this negative power means. Any trigonometric function raised to this negative power means that it is the inverse of that trigonometric function. So, we have to find the inverse of the given trigonometric function. By finding the inverse it means that we have to find the angle which when put into the given trigonometric function, yields the same value as is given in the inverse of the trigonometric function.

Complete step by step answer:
The given trigonometric function is \[\arcsin \left( { - \dfrac{1}{2}} \right)\]. This expression can be written as \[si{n^{ - 1}}\left( { - \dfrac{1}{2}} \right)\]. Now, when a trigonometric function is raised to a negative power, it means that we have to find the inverse of the given trigonometric function. By finding the inverse it means that we have to calculate the angle which gives the value which is inside the given inverse of the trigonometric function.
We know, \[\sin \left( { - \dfrac{\pi }{6}} \right) = - \dfrac{1}{2}\]
So, \[si{n^{ - 1}}\left( { - \dfrac{1}{2}} \right) = - \dfrac{\pi }{6}\]

Hence, the value of the given trigonometric function is \[ - \dfrac{\pi }{6}\].

Note:
As we know that inverse of any trigonometric function means the inverse of that function. Here, we noticed that the inverse of any function is written in two different ways. For example, the inverse of sin is written as arcsin or ${\sin ^{ - 1}}$.