Question

Find the value of \[ \cos\left(\sin^{-1}\left(\frac{1}{2}\right)\right). \]

• A. \( \frac{1}{2} \)
• B. \( \frac{\sqrt{3}}{2} \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

For expressions like \(\cos(\sin^{-1}x)\), first find the angle represented by \(\sin^{-1}x\), then apply the outer trigonometric function.

Answer 1

The Correct Option is B

Concept: Inverse trigonometric functions convert a trigonometric value into its corresponding angle. If \[ \theta=\sin^{-1}\left(\frac12\right), \] then \(\theta\) is the angle whose sine is \(\frac12\). The standard value is: \[ \sin 30^\circ=\frac12 \] Therefore, \[ \sin^{-1}\left(\frac12\right)=30^\circ=\frac{\pi}{6} \]

Step 1: Find the angle represented by \(\sin^{-1}\left(\frac12\right)\) Let \[ \theta=\sin^{-1}\left(\frac12\right) \] Then \[ \sin\theta=\frac12 \] The principal value satisfying this condition is \[ \theta=\frac{\pi}{6} \]

Step 2: Evaluate the cosine of the angle Substituting \(\theta=\frac{\pi}{6}\), \[ \cos\left(\sin^{-1}\left(\frac12\right)\right) = \cos\frac{\pi}{6} \] Using the standard trigonometric value, \[ \cos\frac{\pi}{6} = \frac{\sqrt3}{2} \]

Step 3: Verify using trigonometric identity Let \[ \theta=\sin^{-1}\left(\frac12\right) \] Then \[ \sin\theta=\frac12 \] Using \[ \sin^2\theta+\cos^2\theta=1 \] we get \[ \cos\theta = \sqrt{1-\sin^2\theta} \] \[ = \sqrt{1-\left(\frac12\right)^2} \] \[ = \sqrt{1-\frac14} \] \[ = \sqrt{\frac34} \] \[ = \frac{\sqrt3}{2} \] which confirms the answer. Final Answer: \[ \boxed{\frac{\sqrt3}{2}} \]