Question

The value of \[ \sin^{-1} \left[ \cos \left( \frac{39\pi}{5} \right) \right] \] is

• A. \( \frac{\pi}{2} \)
• B. \( \frac{3\pi}{10} \)
• C. \( \frac{3\pi}{5} \)
• D. \( -\frac{3\pi}{10} \)

Hint:

When dealing with inverse trigonometric functions and periodic functions like cosine, always simplify the angle to lie within the appropriate range using the function's periodicity.

Answer 1

The Correct Option is B

We are asked to find the value of: \[ \sin^{-1} \left[ \cos \left( \frac{39\pi}{5} \right) \right] \] Step 1: Simplify the angle \( \frac{39\pi}{5} \) First, simplify the angle \( \frac{39\pi}{5} \). We know that the cosine function has a period of \( 2\pi \), so we can subtract multiples of \( 2\pi \) to bring the angle within a more manageable range: \[ \frac{39\pi}{5} = 2\pi \times 3 + \frac{3\pi}{5} \] Thus, we have: \[ \cos \left( \frac{39\pi}{5} \right) = \cos \left( \frac{3\pi}{5} \right) \] Step 2: Evaluate the inverse sine function Now, we need to find: \[ \sin^{-1} \left( \cos \left( \frac{3\pi}{5} \right) \right) \] We know that \( \cos \left( \frac{3\pi}{5} \right) \) is positive and lies within the range of the inverse sine function, so we can directly apply the result: \[ \sin^{-1} \left( \cos \left( \frac{3\pi}{5} \right) \right) = \frac{3\pi}{10} \] Thus, the correct answer is \( \boxed{\frac{3\pi}{10}} \).