The value of ^{-1}(-{3})- ^{-1}≤ft({1}{{2}})+ ^{-1}≤ft(-1/2) is

Correct Answer is : π12π/1212π

Correct Answer is : π12π/1212π

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MHTCET 16 May 2024 Shift 2 Paper

Question : 141 of 150

Marks: +1, -0

The value of

\tan ^{-1}(-\sqrt{3})-\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)+\cos ^{-1}\left(\frac{-1}{2}\right)

Solution:

Concept:

Use principal values of inverse trigonometric functions and simplify the expression.

Explanation:

Evaluate each term using standard principal values.

\tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3}

\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}

\cos^{-1}\left(-\frac{1}{2}\right) = \pi - \cos^{-1}\left(\frac{1}{2}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}

Now substitute these values into the given expression.

-\frac{\pi}{3} - \frac{\pi}{4} + \frac{2\pi}{3}

Combine the terms involving

\pi/3

\left(-\frac{\pi}{3} + \frac{2\pi}{3}\right) - \frac{\pi}{4} = \frac{\pi}{3} - \frac{\pi}{4}

Take LCM of

3

and

4

, which is

12

\frac{4\pi - 3\pi}{12} = \frac{\pi}{12}

Answer:

The value is

\frac{\pi}{12}

, which matches option C.

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