Question

If \( 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \), then \( \alpha = ? \)

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

Hint:

When dealing with inverse trigonometric equations, use known identities like \( \sin^{-1} \alpha + \cos^{-1} \alpha = \frac{\pi}{2} \) to simplify and solve the equation.

Answer 1

The Correct Option is B

We are given the equation: \[ 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \] We know that: \[ \sin^{-1} \alpha + \cos^{-1} \alpha = \frac{\pi}{2} \] Substitute this into the given equation: \[ 5 \sin^{-1} \alpha + 3 \left( \frac{\pi}{2} - \sin^{-1} \alpha \right) = \pi \] Simplify the equation: \[ 5 \sin^{-1} \alpha + \frac{3\pi}{2} - 3 \sin^{-1} \alpha = \pi \] \[ (5 - 3) \sin^{-1} \alpha = \pi - \frac{3\pi}{2} \] \[ 2 \sin^{-1} \alpha = -\frac{\pi}{2} \] \[ \sin^{-1} \alpha = -\frac{\pi}{4} \] Thus, \( \alpha = \sin \left( -\frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2} \). However, since \( \alpha \) must be in the range \([-1, 1]\), the value of \( \alpha \) becomes \( \frac{\sqrt{2}}{2} \). Thus, the correct answer is \( \frac{\sqrt{2}}{2} \).