Question

\(cos(cos^{-1}(-\frac{1}{2})+\frac{π}{3}) =\)

• A. \(0\)
• B. \(-1\)
• C. \(1\)
• D. \(\frac{1}{2}\)

Hint:

arccos(-1/2) is 2pi/3, so the angle becomes pi.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The principal value of \(\cos^{-1}\) lies in \([0, \pi]\).

Step 2: Evaluate the inner term:
\[ \cos^{-1}\left(-\frac12\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

Step 3: Add and evaluate:
\[ \cos\left(\frac{2\pi}{3} + \frac{\pi}{3}\right) = \cos\pi = -1 \]

Step 4: Check the options:
Option (A) 0 would need an angle of \(\pi/2\), (C) 1 would need 0 or \(2\pi\), and (D) \(\tfrac12\) would need \(\pi/3\). Only \(-1\) matches the angle \(\pi\).

Final Answer:
The angle is pi, so the cosine is -1. \[ \boxed{\text{(B) }-1} \]