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If \sin ^{-1}\left ( \frac{x}{5} \right )+cosec^{-1}\left ( \frac{5}{4} \right )= \frac{\pi }{2}, then the values of x is

  • Option 1)

    4

  • Option 2)

    5

  • Option 3)

    1

  • Option 4)

    3

 

Answers (1)

best_answer

As we learnt in 

Important Results of Inverse Trigonometric Functions -

\sin ^{-1}x + \cos ^{-1}x = \frac{\pi }{2}

- wherein

When \left | x \right |\leqslant 1

 

 \sin ^{-1}\frac{x}{5}+\cos ^{-1}\frac{5}{4} = \frac{\pi }{2}    

\Rightarrow \cos ^{-1}\frac{5}{4}= \frac{\pi }{2} -\sin ^{-1}\frac{x}{5}= \cos ^{-1} \frac{x}{5}

taking cos on both sides,

\cos \sec^{-1}\frac{5}{4}=\frac{x}{5}      \Rightarrow x=5\ cos\ cosec^{-1}\left(\frac{5}{4} \right )=5\times \sqrt{\left(\frac{5^{2}-4^{2}}{5} \right )}=3

x=3


Option 1)

4

Incorrect

Option 2)

5

Incorrect

Option 3)

1

Incorrect

Option 4)

3

Correct

Posted by

prateek

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