Question

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

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

Hint:

Always use \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \) to reduce equations.

Answer 1

The Correct Option is C

Concept: \[ \sin^{-1}\alpha + \cos^{-1}\alpha = \frac{\pi}{2} \]

Step 1: Substitute identity.
\[ \cos^{-1}\alpha = \frac{\pi}{2} - \sin^{-1}\alpha \]

Step 2: Substitute into given equation.
\[ 5\sin^{-1}\alpha + 3\left(\frac{\pi}{2} - \sin^{-1}\alpha\right) = \pi \]

Step 3: Simplify.
\[ 5\sin^{-1}\alpha + \frac{3\pi}{2} - 3\sin^{-1}\alpha = \pi \] \[ 2\sin^{-1}\alpha + \frac{3\pi}{2} = \pi \] \[ 2\sin^{-1}\alpha = -\frac{\pi}{2} \Rightarrow \sin^{-1}\alpha = -\frac{\pi}{4} \]

Step 4: Find \( \alpha \).
\[ \alpha = \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]