Question

If \( \sec^2 \theta = \frac{4}{3} \), then the general value of \( \theta \) is:

• A. \( 2n\pi \pm \frac{\pi}{6} \)
• B. \( n\pi \pm \frac{\pi}{6} \)
• C. \( 2n\pi \pm \frac{\pi}{3} \)
• D. \( n\pi \pm \frac{\pi}{3} \)

Hint:

For trigonometric equations, use fundamental identities and general formulas to derive the solution. The solution for \( \tan \theta = \pm \tan A \) is \( n\pi \pm A \).

Answer 1

The Correct Option is B

Given: \[ \sec^2 \theta = \frac{4}{3} \] Using the identity: \[ \sec^2 \theta = 1 + \tan^2 \theta \] Substituting: \[ 1 + \tan^2 \theta = \frac{4}{3} \] \[ \tan^2 \theta = \frac{4}{3} - 1 = \frac{1}{3} \] \[ \tan \theta = \pm \frac{1}{\sqrt{3}} \] The general solution for \( \tan \theta = \pm \frac{1}{\sqrt{3}} \) is: \[ \theta = n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \] Thus, the correct answer is \( n\pi \pm \frac{\pi}{6} \).