Question

Evaluate: $ \cos^{-1} \left( \cos \frac{35\pi}{18} \right) - \sin^{-1} \left( \sin \frac{35\pi}{18} \right) $

• A. 0
• B. \( \frac{\pi}{9} \)
• C. \( \frac{\pi}{18} \)
• D. \( \pi \)

Hint:

When dealing with inverse trigonometric functions, be mindful of their principal value ranges. Adjust the angle within the principal range by subtracting multiples of \( 2\pi \) or \( \pi \) as needed.

Answer 1

The Correct Option is B

We are tasked with evaluating the expression: \[ \cos^{-1} \left( \cos \frac{35\pi}{18} \right) - \sin^{-1} \left( \sin \frac{35\pi}{18} \right) \] First, let's simplify each part individually.
Step 1: Simplifying \( \cos^{-1} \left( \cos \frac{35\pi}{18} \right) \)
The principal value of the inverse cosine function, \( \cos^{-1} x \), lies between \( 0 \) and \( \pi \). Since \( \frac{35\pi}{18} \) is greater than \( \pi \), we need to find an equivalent angle within the range \( [0, \pi] \) by subtracting multiples of \( 2\pi \). \[ \frac{35\pi}{18} - 2\pi = \frac{35\pi}{18} - \frac{36\pi}{18} = -\frac{\pi}{18} \] Now, since \( \cos^{-1} (\cos \theta) = \theta \) for \( \theta \in [0, \pi] \), we have: \[ \cos^{-1} \left( \cos \frac{35\pi}{18} \right) = \cos^{-1} \left( \cos \left( -\frac{\pi}{18} \right) \right) = \frac{\pi}{18} \]
Step 2: Simplifying \( \sin^{-1} \left( \sin \frac{35\pi}{18} \right) \)
The principal value of the inverse sine function, \( \sin^{-1} x \), lies between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). Since \( \frac{35\pi}{18} \) is greater than \( \frac{\pi}{2} \), we again find an equivalent angle within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) by subtracting multiples of \( 2\pi \). \[ \frac{35\pi}{18} - 2\pi = -\frac{\pi}{18} \] Now, since \( \sin^{-1} (\sin \theta) = \theta \) for \( \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we have: \[ \sin^{-1} \left( \sin \frac{35\pi}{18} \right) = \sin^{-1} \left( \sin \left( -\frac{\pi}{18} \right) \right) = -\frac{\pi}{18} \]
Step 3: Subtracting the Results
Now, we subtract the two results: \[ \cos^{-1} \left( \cos \frac{35\pi}{18} \right) - \sin^{-1} \left( \sin \frac{35\pi}{18} \right) = \frac{\pi}{18} - \left( -\frac{\pi}{18} \right) = \frac{\pi}{9} \] Thus, the value of the given expression is \( \frac{\pi}{9} \).