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Question: The value of \(\sin \left( {\dfrac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \rig...

The value of sin⁡(π3−sin⁡−1(−32))\sin \left( {\dfrac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right)} \right) is?
A. 32\dfrac{{\sqrt 3 }}{2}
B. −32 - \dfrac{{\sqrt 3 }}{2}
C. 12\dfrac{1}{2}
D. −12 - \dfrac{1}{2}

Explanation

Solution

As we can see that this question is related to trigonometry. We have been given a trigonometric expression and we have to solve it. So we will apply the trigonometric formulas and identities to solve this question. We can see that in the equation we have the sine function, which is one of the basic trigonometric ratios. So we will try to apply the formula which includes the inverse of the sine function.

Formula used:
sin⁡−1(−x)=−sin⁡−1x{\sin ^{ - 1}}( - x) = - {\sin ^{ - 1}}x

Complete answer:
According to the question here we have:
sin⁡(π3−sin⁡−1(−32))\sin \left( {\dfrac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right)} \right)
In this question, let us take one part and solve it. We have
sin⁡−1(−32){\sin ^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right)
So by comparing with the formula, we can write, can write that
⇒sin⁡−1(−32)=(−)(−)sin⁡−1(−32)\Rightarrow {\sin ^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right) = ( - )( - ){\sin ^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right)
On simplifying this expression we have:
sin⁡−1(−32){\sin ^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right)
We will now substitute this value in the equation and we have:
=sin⁡(π3+sin⁡−1(−32))= \sin \left( {\dfrac{\pi }{3} + {{\sin }^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right)} \right)
Now we know the value of
⇒sin⁡−1(−32)=π3\Rightarrow {\sin ^{ - 1}}\left( { - \dfrac{{\sqrt 3 }}{2}} \right) = \dfrac{\pi }{3}
Putting this value in the equation, we can write
sin⁡(π3+π3)\sin \left( {\dfrac{\pi }{3} + \dfrac{\pi }{3}} \right)
We have sin⁡(2π3)\sin \left( {\dfrac{{2\pi }}{3}} \right)
Now again we know the value that
sin⁡(2π3)=32\sin \left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{{\sqrt 3 }}{2}
Therefore this gives us the required value.
Hence the correct option is (A) 32\dfrac{{\sqrt 3 }}{2} .

Therefore, the correct option is A

Note: We should note that we can also express the above equation i.e.
sin⁡(2π3)=sin⁡(π−2π3)\sin \left( {\dfrac{{2\pi }}{3}} \right) = \sin \left( {\pi - \dfrac{{2\pi }}{3}} \right)
We will now simplify this value by taking the LCM and it gives us
sin⁡(3π−2π3)\sin \left( {\dfrac{{3\pi - 2\pi }}{3}} \right)
It gives us a new expression which is sin⁡π3\sin \dfrac{\pi }{3}
It can further be simplified as
⇒sin⁡1803=sin⁡60∘\Rightarrow \sin \dfrac{{180}}{3} = \sin 60^\circ
Now we will apply the basic trigonometric formula of singe angles and we know that
sin⁡60∘=32\sin 60^\circ = \dfrac{{\sqrt 3 }}{2} .
We should always know the formulas and identities before solving this kind of question . We should also take care of the sign associated with the trigonometric functions and solve carefully to avoid calculation mistakes.