Trigonometry Solution – Problem 11: Find the domain of

Question

Domain of Trigonometric Function

Find the domain of $y = \sqrt{\frac{1}{2} - \cos x}$ .

For the function to be defined, we need:

$\frac{1}{2} - \cos x \geq 0$

$\cos x \leq \frac{1}{2}$

Since $\cos\frac{\pi}{3} = \frac{1}{2}$ , we have:

$\frac{\pi}{3} + 2k\pi \leq x \leq \frac{5\pi}{3} + 2k\pi, \quad k \in \mathbb{Z}$

$\left\{x \mid 2k\pi + \frac{\pi}{3} \leq x \leq 2k\pi + \frac{5\pi}{3}, k \in \mathbb{Z}\right\}$

Translate the question into conditions (2 pts): domain/range/period/symmetry/monotonicity constraints are written correctly.
Key transformation or rewrite (2 pts): rewrite the function into a standard form (e.g. amplitude/phase/period) or reduce using identities.
Correct interval/parameter reasoning (2 pts): derive the correct inequalities or argument interval.
Final answer (1 pt): state the final set / interval / period clearly.

Listing properties ("periodic", "even") without applying them to the given function.