Question

If $\sin^{-1} \frac{5}{x} + \sin^{-1} \frac{12}{x} = \frac{\pi}{2}$ then $x =$

• A. 12
• B. 7
• C. 13
• D. 15

Hint:

If the sum of two inverse sines is $\pi/2$, the arguments follow the Pythagorean triplet rule: $a^2 + b^2 = x^2$.

Answer 1

The Correct Option is C

Step 1: Concept
Use the identity $\sin^{-1} \alpha + \cos^{-1} \alpha = \pi/2$.

Step 2: Meaning
If $\sin^{-1} \frac{5}{x} + \sin^{-1} \frac{12}{x} = \pi/2$, then it implies $\sin^{-1} \frac{12}{x} = \cos^{-1} \frac{5}{x}$.

Step 3: Analysis
For $\sin^{-1} \frac{12}{x}$ to equal $\cos^{-1} \frac{5}{x}$, we consider a right-angled triangle where the hypotenuse is $x$. By Pythagoras' theorem: $5^2 + 12^2 = x^2 \implies 25 + 144 = x^2 \implies 169 = x^2$.

Step 4: Conclusion
Therefore, $x = 13$.
Final Answer: (C)