Question

Evaluate the expression: \[ \cos^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \tan^{-1} \frac{16}{63} \]

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)

Hint:

Use trigonometric identities and inverse function properties to simplify complex expressions.

Answer 1

The Correct Option is A

Step 1: Evaluating \( \cos^{-1} \frac{3}{5} \).
Let \( \theta = \cos^{-1} \frac{3}{5} \). Then, \[ \cos \theta = \frac{3}{5}, \quad \text{so using the Pythagorean theorem,} \quad \sin \theta = \frac{4}{5} \] Step 2: Evaluating \( \sin^{-1} \frac{5}{13} \).
Let \( \phi = \sin^{-1} \frac{5}{13} \). Then, \[ \sin \phi = \frac{5}{13}, \quad \text{so using the Pythagorean theorem,} \quad \cos \phi = \frac{12}{13} \] Step 3: Evaluating \( \tan^{-1} \frac{16}{63} \).
Let \( \psi = \tan^{-1} \frac{16}{63} \). Then, \[ \tan \psi = \frac{16}{63}, \quad \text{so using trigonometric identity,} \quad \sin \psi = \frac{16}{65}, \quad \cos \psi = \frac{63}{65} \] Step 4: Adding the angles.
\[ \theta + \phi + \psi = \cos^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \tan^{-1} \frac{16}{63} \] Using the identity: \[ \cos^{-1} a + \sin^{-1} a = \frac{\pi}{2} \] we simplify: \[ \theta + \phi + \psi = \frac{\pi}{2} \] Thus, the correct answer is: \[ \mathbf{\frac{\pi}{2}} \]