Question

Find \( \sin \left( \sin^{-1} \frac{2}{3} \right) + \tan^{-1} \left( \tan \frac{3\pi}{4} \right) \).

• A. \( \frac{17\pi}{12} \)
• B. \( \frac{5\pi}{12} \)
• C. \( \frac{\pi}{12} \)
• D. \( -\frac{\pi}{12} \)

Hint:

When working with trigonometric inverses and functions, remember that \( \sin(\sin^{-1} x) = x \) and use the periodicity of the tangent function to simplify expressions.

Answer 1

The Correct Option is C

We are asked to find: \[ \sin \left( \sin^{-1} \frac{2}{3} \right) + \tan^{-1} \left( \tan \frac{3\pi}{4} \right) \] We know that: \[ \sin \left( \sin^{-1} \frac{2}{3} \right) = \frac{2}{3} \] For \( \tan^{-1} \left( \tan \frac{3\pi}{4} \right) \), since \( \tan \frac{3\pi}{4} = -1 \), we have: \[ \tan^{-1} (-1) = -\frac{\pi}{4} \] Thus, the expression becomes: \[ \frac{2}{3} + \left( -\frac{\pi}{4} \right) \] Now simplify: \[ \frac{2}{3} - \frac{\pi}{4} = \frac{8}{12} - \frac{3\pi}{12} = \frac{8 - 3\pi}{12} \] Thus, the correct answer is \( \frac{\pi}{12} \).