The equation 2cos^{^{-1}}x+sin^{-1}x=\frac{11\pi}{6}\, \, has

  • Option 1)

    No solution

  • Option 2)

    Only one solution

  • Option 3)

    two solutions

  • Option 4)

    three solutions

 

Answers (1)

 

Important Results of Inverse Trigonometric Functions -

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

- wherein

When \left | x \right |\leqslant 1

 

 

2 \cos ^{-1}x+ \sin ^{-1}x=\frac{11\pi }{6}

\Rightarrow \cos ^{-1}x+ \cos ^{-1}x + \sin ^{-1}x=\frac{11\pi }{6}

\Rightarrow \cos ^{-1}x +\frac{\pi }{2}=\frac{11\pi }{6}

\Rightarrow \cos ^{-1}x = \frac{11\pi }{6} - \frac{\pi}{2}=\frac{11\pi -3\pi }{6}=\frac{8\pi }{6} = \frac{4\pi }{3}

\Rightarrow but 0\simeq \cos ^{-1}x\leq \pi

and \Rightarrow \frac{4\pi }{3}> \Pi

 


Option 1)

No solution

Correct option

Option 2)

Only one solution

Incorrect option

Option 3)

two solutions

Incorrect option

Option 4)

three solutions

Incorrect option

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