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The value of ${\sin ^{ - 1}}\left( {\cos \dfrac{{53\pi }}{5}} \right)$ is
A.$\dfrac{{3\pi }}{5}$
B.$\dfrac{{ - 3\pi }}{5}$
C.$\dfrac{\pi }{{10}}$
D.$\dfrac{{ - \pi }}{{10}}$

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Last updated date: 16th Jul 2024
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Answer
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Hint : Try to break angles in general angles.

We know,
$
  \cos (2k\pi + \theta ) = \cos \theta \\
    \\
$
So, on comparing the above equation with question we get,
$\cos \dfrac{{53\pi }}{5} = $ $\cos \left( {10\pi + \dfrac{{3\pi }}{5}} \right) = \cos \dfrac{{3\pi }}{5}$ ……(i)
We know,
$\dfrac{{3\pi }}{5} = \dfrac{\pi }{2} + \dfrac{\pi }{{10}}$ ……(ii)
And we also know,
$\cos \left( {\dfrac{\pi }{2} + \dfrac{\pi }{{10}}} \right) = $$ - \sin \dfrac{\pi }{{10}}$= $\cos \dfrac{{3\pi }}{5}$ ……(iii) (From i & ii )
We have to find the value of
${\sin ^{ - 1}}\left( {\cos \dfrac{{53\pi }}{5}} \right)$
We know the value of $\left( {\cos \dfrac{{53\pi }}{5}} \right)$ is $ - \sin \dfrac{\pi }{{10}}$
So, ${\sin ^{ - 1}}\left( {\cos \dfrac{{53\pi }}{5}} \right)$=${\sin ^{ - 1}}\left( { - \sin \dfrac{\pi }{{10}}} \right)$=$ - \dfrac{\pi }{{10}}$
As we know(${\sin ^{ - 1}}\sin a = a$)
 Hence the correct option is (D).

Note : In these types of problems of finding value of trigonometry we have to use the quadrant rule of finding angle and also use some of the properties of inverse trigonometric functions as shown above.