Question

What is the value of \( \sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{1}{2}\right) \)?

• A. \(0\)
• B. \(\pi/4\)
• C. \(\pi/2\)
• D. \(\pi\)

Hint:

Always remember the identity: \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \] This helps quickly solve many inverse trigonometry problems without detailed calculations.

Answer 1

The Correct Option is C

Concept: Inverse trigonometric functions have important identities. One of the most useful identities is: \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \] This identity holds for all values of \(x\) within the domain \([-1,1]\).

Step 1: Find the value of each term. We know that: \[ \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \] Therefore, \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \] Also, \[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \] Hence, \[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

Step 2: Add the values. \[ \frac{\pi}{6} + \frac{\pi}{3} \] Taking common denominator: \[ \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} \] \[ = \frac{\pi}{2} \]

Step 3: Final result. \[ \boxed{\frac{\pi}{2}} \]