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

The trigonometric expression is simplified using the following identities: \[\n\cos(90^\circ + \theta) = -\sin(\theta)\n\] \[\n\sin(180^\circ - \theta) = \sin(\theta)\n\] \[\n\sin(180^\circ + \theta) = -\sin(\theta)\n\] Substitution yields: \[\n\sin \theta + (-\sin \theta) + \sin \theta + (-\sin \theta)\n\] Simplifying the expression: \[\n\sin \theta - \sin \theta + \sin \theta - \sin \theta = 0\n\] The answer is 0.