Question

If \( \cos 2\theta = \sin \alpha \), then \( \theta = \)

• A. \( 2\pi n \pm \left( \frac{\pi}{2} - \alpha \right), n \in \mathbb{Z} \)
• B. \( \pi n \pm \left( \frac{\pi}{4} + \frac{\alpha}{2} \right), n \in \mathbb{Z} \)
• C. \( \frac{1}{2} \left[ n\pi + (-1)^n \alpha \right], n \in \mathbb{Z} \)
• D. \( \pi n \pm \left( \frac{\pi}{4} - \frac{\alpha}{2} \right), n \in \mathbb{Z} \)

Hint:

For equations involving trigonometric functions, always consider multiple angle identities and the periodic nature of trigonometric functions to derive general solutions.

Answer 1

The Correct Option is D

Step 1: Understanding the equation.
We are given \( \cos 2\theta = \sin \alpha \). Using the identity \( \cos 2\theta = \cos(\pi - 2\theta) \), we equate it to \( \sin \alpha \). This leads to multiple possible angles for \( \theta \), and we derive the general solution.

Step 2: Analyzing the options.
Option (D) correctly matches the derived formula for \( \theta \), which is \( \pi n \pm \left( \frac{\pi}{4} - \frac{\alpha}{2} \right), n \in \mathbb{Z} \). This formula reflects the periodic nature of trigonometric functions.

Step 3: Conclusion.
Thus, the correct answer is option (D).