Question

How do you find the exact value of \[\sin \left( {{{\sin }^{ - 1}}\left( {0.3} \right)} \right)\] ?

Answer 1

Hint: We will use the question itself to solve the question. We will use substitution here. The inverse function will be submitted as a variable. Then we will take the sine function value on both the sides. That will directly give the answer to us.

Complete step-by-step answer:
Given that,
 \[\sin \left( {{{\sin }^{ - 1}}\left( {0.3} \right)} \right)\]
Let,
 \[{\sin ^{ - 1}}\left( {0.3} \right) = \alpha \]
Then taking sin function on both the sides,
 \[0.3 = \sin \alpha \]
Now we can clearly see that,
 \[\sin \left( {{{\sin }^{ - 1}}\left( {0.3} \right)} \right) = \sin \alpha \]
And value of \[\sin \alpha \] is 0.3. so we conclude that,
 \[\sin \left( {{{\sin }^{ - 1}}\left( {0.3} \right)} \right) = 0.3\]
This is our final answer.
So, the correct answer is “0.3”.

Note: Here note that we can directly find the answer as \[\sin \left( {{{\sin }^{ - 1}}\theta } \right) = \theta \] but the solution needs to be elaborated. If sometimes the combination of function is used then we need to find the solution the way we found here.
Sometimes the inverse functions are written as arc functions like
 \[{\sin ^{ - 1}}\theta \Rightarrow \arcsin \theta \] . So do read the question first and don’t get confused. There are some identities in trigonometry related to this inverse functionality.