Question

The value of \(sin^{-1}\frac{12}{13} +cos^{-1}\frac45+tan^{-1}\frac{63}{16}\) is :

• A. \(\frac{\pi}{2}\)
• B. \(\pi\)
• C. \(\frac{\pi}{3}\)
• D. \(\frac{\pi}{4}\)

Answer 1

The Correct Option is B

To solve the problem of determining the value of \( \sin^{-1}\frac{12}{13} + \cos^{-1}\frac{4}{5} + \tan^{-1}\frac{63}{16} \), we will evaluate each term individually and then add them together.

1. First, calculate \(\sin^{-1}\frac{12}{13}\):
In a right triangle where the opposite side is \(12\) and the hypotenuse is \(13\), the adjacent side, calculated by the Pythagorean theorem, is \(5\) (because \(5^2+12^2=13^2\)). Thus, \(\cos(\theta) = \frac{5}{13}\), where \(\theta = \sin^{-1}\frac{12}{13}\).

2. Next, calculate \(\cos^{-1}\frac{4}{5}\):
In a right triangle where the adjacent side is \(4\) and the hypotenuse is \(5\), the opposite side is \(3\) (since \(3^2+4^2=5^2\)). Thus, \(\sin(\phi) = \frac{3}{5}\), where \(\phi = \cos^{-1}\frac{4}{5}\).

3. Lastly, calculate \(\tan^{-1}\frac{63}{16}\):
This represents an angle \(\psi\) where the opposite side is \(63\) and the adjacent side is \(16\), so the tangent is \(\frac{63}{16}\).

Now sum the angles. Using the identities:
\(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\)
We find:
\(\theta + \phi = \frac{\pi}{2}\)

The angle \(\psi\) such that \(\tan(\psi)=\frac{63}{16}\) corresponds to the angle whose tangent equals the division of opposite by adjacent sides in a triangle, completing the \( \frac{\pi}{2} \) gap to reach \(\pi\).

So, the sum of the three angles provides:
\(\theta + \phi + \psi = \frac{\pi}{2} + \psi = \pi\)
Thus, the correct answer is \(\pi\).