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: Understanding the Concept:
Convert to a common form: \(\sin^{-1}\tfrac{12}{13} = \tan^{-1}\tfrac{12}{5}\) (3-4-5 style triangle 5, 12, 13) and \(\cos^{-1}\tfrac45 = \tan^{-1}\tfrac34\).

Step 2: Add the first two:
Since \(\tfrac{12}{5}\cdot\tfrac34 = \tfrac95 > 1\), the formula for \(\tan^{-1}p + \tan^{-1}q\) gains a \(\pi\):
\[ \tan^{-1}\tfrac{12}{5} + \tan^{-1}\tfrac34 = \pi + \tan^{-1}\frac{\tfrac{12}{5} + \tfrac34}{1 - \tfrac95} = \pi + \tan^{-1}\left(-\frac{63}{16}\right) = \pi - \tan^{-1}\frac{63}{16} \]

Step 3: Add the third:
\[ \left(\pi - \tan^{-1}\tfrac{63}{16}\right) + \tan^{-1}\tfrac{63}{16} = \pi \]

Step 4: Check:
Numerically: \(67.38^\circ + 36.87^\circ + 75.75^\circ = 180^\circ\). The value is \(\pi\), option (C).

Final Answer:
The three angles add up to pi. \[ \boxed{\text{(C) }\pi} \]