Question

If \( \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 3 \), then what is the value of \( \cos \theta_1 + \cos \theta_2 + \cos \theta_3 \)?

• A. 1
• B. 0
• C. 2
• D. 3

Hint:

In trigonometric problems, consider angles where sine or cosine reaches their extreme values (like 0° and 90°) to simplify calculations.

Answer 1

The Correct Option is B

We are given the equation: \[ \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 3 \] The sum of sines reaches its maximum value when each sine term is equal to 1, which occurs when each angle is \( 90^\circ \). Hence, we assume: \[ \theta_1 = \theta_2 = \theta_3 = 90^\circ \] For \( \theta_1 = \theta_2 = \theta_3 = 90^\circ \), we have: \[ \sin 90^\circ = 1 \] Thus: \[ \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 1 + 1 + 1 = 3 \] Now, for the cosine terms: \[ \cos 90^\circ = 0 \] So: \[ \cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 0 + 0 + 0 = 0 \]