Question

The value of \( \sin \theta + \cos(90^\circ + \theta) + \sin(180^\circ - \theta) + \sin(180^\circ + \theta) \) is:

• A. 0
• B. -1
• C. 1
• D. \( \frac{1}{2} \)

Hint:

For simplifying trigonometric expressions, always apply standard trigonometric identities first. Look for opportunities to cancel out terms.

Answer 1

The Correct Option is A

We begin by simplifying the given trigonometric expression. Using the following trigonometric identities: \[ \cos(90^\circ + \theta) = -\sin(\theta) \] \[ \sin(180^\circ - \theta) = \sin(\theta) \] \[ \sin(180^\circ + \theta) = -\sin(\theta) \] Substitute these identities into the expression: \[ \sin \theta + (-\sin \theta) + \sin \theta + (-\sin \theta) \] Simplifying: \[ \sin \theta - \sin \theta + \sin \theta - \sin \theta = 0 \] Therefore, the correct answer is 0.