Question

\(sin^{-1}(\frac{12}{13})+cos^{-1}(\frac{4}{5})+tan^{-1}(\frac{63}{16}) =\)

• A. \(\frac{π}{2}\)
• B. \(\frac{3π}{2}\)
• C. \(π\)
• D. \(2π\)

Hint:

Convert each term to an arctangent and add using the tangent formula.

Answer 1

The Correct Option is C

Step 1: Use degrees:
$\sin^{-1}\tfrac{12}{13} \approx 67.38^\circ$, $\cos^{-1}\tfrac45 \approx 36.87^\circ$.

Step 2: Third term:
$\tan^{-1}\tfrac{63}{16}\approx 75.75^\circ$ because $\tan75.75^\circ \approx 3.94$.

Step 3: Add:
$67.38 + 36.87 + 75.75 = 180.0$ degrees, which is $\pi$, option (C). An exact route is to note $\sin(A+B)=\tfrac{12}{13}\cdot\tfrac45+\tfrac5{13}\cdot\tfrac35=\tfrac{63}{65}$, which equals $\sin(\pi - C)$ for $C = \tan^{-1}\tfrac{63}{16}$.

Final Answer:
The total is pi. \[ \boxed{\text{(C) }\pi} \]