Question

The principal solutions of the equation \( \cos\theta = \frac{1}{2} \) are _________.

• A. \( \frac{\pi}{6}, \frac{5\pi}{6} \)
• B. \( \frac{\pi}{3}, \frac{5\pi}{3} \)
• C. \( \frac{\pi}{6}, \frac{7\pi}{6} \)
• D. \( \frac{\pi}{3}, \frac{2\pi}{3} \)

Hint:

For \( \cos\theta = a \), the principal solutions are \( \theta = \pm\cos^{-1}(a) \) within \( [0, 2\pi] \).

Answer 1

The Correct Option is B

Step 1: Finding the General Solutions
The equation \( \cos\theta = \frac{1}{2} \) holds at: \[ \theta = \pm \frac{\pi}{3} + 2k\pi, \quad k \in \mathbb{Z} \] Step 2: Principal Solutions
The general solutions in the principal range \( [0, 2\pi] \) are: \[ \theta = \frac{\pi}{3}, \frac{5\pi}{3} \]