Question

How do you find the exact value of $ {\sin ^{ - 1}}0? $

Answer 1

Hint: In this question, you have been asked to find the inverse sine value of zero, in which you have to use inverse trigonometry. Also, since the question has not asked for any particular type of solution so better answer in the range of principle solution of the sine and if possible then also give the answer in general form.

Complete step-by-step answer:
In order to find the exact value of the given inverse trigonometric function $ {\sin ^{ - 1}}0 $ we will consider $ y = {\sin ^{ - 1}}0 $
Since the domain and range of the sine function is belongs as
Domain: $ [ - 1,\;1] $ and Range: $ \left[ { - \dfrac{\pi }{2},\;\dfrac{\pi }{2}} \right] $
Now we can see that the given function is defined because its argument is lying in the domain, therefore its value will lie in the range of the inverse sine function.
From the given function,
 $ y = {\sin ^{ - 1}}0 $
Taking sine function both the sides, we will get
 $ \Rightarrow \sin y = \sin \left( {{{\sin }^{ - 1}}0} \right) $
Here the we can see in the right hand side of the equation, that the inverse function is now become the argument of its parent function, so they will cancel out each other and we can write the equation further as
 $ \Rightarrow \sin y = 0 $
Now we know that the principal argument of $ \sin y = 0 $ is equals to $ 0 $
Therefore $ {\sin ^{ - 1}}0 = 0 $ is the required solution.
So, the correct answer is “0”.

Note: When solving questions related to trigonometry or inverse trigonometry then always find the solution in the principle range of the given trigonometric function until and unless you have been asked for a general solution or a particular solution.