Question

If \( a = \sin^{-1} (\sin(5)) \) and \( b = \cos^{-1} (\cos(5)) \), then \( a^2 + b^2 \) is equal to

• A. \( 4\pi^2 + 25 \)
• B. \( 8\pi^2 - 40\pi + 50 \)
• C. \( 4\pi^2 - 20\pi + 50 \)
• D. \( 25 \)

Answer 1

The Correct Option is B

Calculate \( a = \sin^{-1}(\sin(5)) \). To find \( a \), note that \( \sin^{-1}(\sin(x)) \) gives a result in the range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).

Since 5 is outside this range, we need to adjust it. We have:

\[ a = \sin^{-1}(\sin(5)) = 5 - 2\pi. \]

Thus,

\[ a = 5 - 2\pi. \]

Calculate \( b = \cos^{-1}(\cos(5)) \). To find \( b \), note that \( \cos^{-1}(\cos(x)) \) gives a result in the range \([0, \pi]\).

Since 5 is within this range, we can write:

\[ b = \cos^{-1}(\cos(5)) = 2\pi - 5. \]

Calculate \( a^2 + b^2 \). Now, substitute \( a = 5 - 2\pi \) and \( b = 2\pi - 5 \):

\[ a^2 + b^2 = (5 - 2\pi)^2 + (2\pi - 5)^2. \]

Expanding both terms:

\[ = (5 - 2\pi)^2 + (2\pi - 5)^2 = (25 - 20\pi + 4\pi^2) + (4\pi^2 - 20\pi + 25). \]

Combine like terms:

\[ = 8\pi^2 - 40\pi + 50. \]

Thus, the answer is:

\[ 8\pi^2 - 40\pi + 50 \]