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AP DSC School Assistant Math Model Test 3

Section: Subject Specific

Question : 103 of 160

Marks: +1, -0

\sin \left[ {\frac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \frac{1}{2}} \right)} \right]

is equal to :

Solution:

Concept:

The problem uses the property of inverse sine:

\sin^{-1}(-x) = -\sin^{-1}(x)

for

x

[-1,1]

, and the known value

\sin^{-1}\left(\frac12\right) = \frac{\pi}{6}

Explanation:

Step 1: Simplify

\sin^{-1}\left(-\frac12\right)

. Sine inverse of a negative angle gives a negative angle. So

\sin^{-1}\left(-\frac12\right) = -\sin^{-1}\left(\frac12\right) = -\frac{\pi}{6}

Step 2: Substitute into the expression:

\sin\left[\frac{\pi}{3} - \left(-\frac{\pi}{6}\right)\right] = \sin\left(\frac{\pi}{3} + \frac{\pi}{6}\right)

Step 3: Add the angles:

\frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}

Step 4: Evaluate

\sin\left(\frac{\pi}{2}\right) = 1

Answer:

The value is

1

, which corresponds to option D.

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