Question

If \( \cos 76^\circ = \cos \left( 90^\circ - \theta \right) \), then the general value of \( \theta \) is

• A. \( 76^\circ \)
• B. \( 90^\circ - 76^\circ \)
• C. \( 76^\circ \) and \( 180^\circ - 76^\circ \)
• D. \( 180^\circ - 76^\circ \)

Hint:

When solving trigonometric equations, consider both possible solutions for angles due to periodicity and symmetry of trigonometric functions.

Answer 1

The Correct Option is C

Step 1: Use the identity \( \cos \left( 90^\circ - \theta \right) = \sin \theta \).
This gives: \[ \cos 76^\circ = \sin \theta \] Therefore, \( \theta = 76^\circ \) or \( \theta = 180^\circ - 76^\circ = 104^\circ \).
Step 2: Conclusion.
The general value of \( \theta \) is \( 76^\circ \) and \( 180^\circ - 76^\circ \). Final Answer: \[ \boxed{76^\circ \text{ and } 180^\circ - 76^\circ} \]