Question

The value of \( \sin^{-1}\left(\frac{3}{5}\right) + \sin^{-1}\left(\frac{8}{17}\right) \) is:

• A. $\sin^{-1} \frac{77}{85}$
• B. $\sin^{-1} \frac{13}{85}$
• C. $\sin^{-1} \frac{25}{85}$
• D. $\sin^{-1} \frac{7}{17}$

Hint:

Standard Pythagorean triplets like (3,4,5) and (8,15,17) make these calculations much faster.

Answer 1

The Correct Option is A

Step 1: Concept
Use the identity $\sin^{-1} x + \sin^{-1} y = \sin^{-1} (x\sqrt{1-y^2} + y\sqrt{1-x^2})$.
Step 2: Analysis
Let $x = \frac{3}{5}$ and $y = \frac{8}{17}$.
Result $= \sin^{-1} \left( \frac{3}{5} \sqrt{1 - (\frac{8}{17})^2} + \frac{8}{17} \sqrt{1 - (\frac{3}{5})^2} \right)$
$= \sin^{-1} \left( \frac{3}{5} \cdot \frac{15}{17} + \frac{8}{17} \cdot \frac{4}{5} \right) = \sin^{-1} \left( \frac{45 + 32}{85} \right)$.
Step 3: Conclusion
$= \sin^{-1} \frac{77}{85}$.
Final Answer: (A)