If \frac{x}{\cos \theta} = \frac{y}{\cos(\frac{2\pi}{3} - \theta)} = \frac{z}{\cos(\frac{2\pi}{3} + \theta)} then what is x+y+z equal to?
- A. -1
- B. 0 ✓
- C. 1
- D. 3
Correct Answer: B. 0
Explanation
Let the given ratio be k. Then x+y+z = k[\cos\theta + \cos(\frac{2\pi}{3}-\theta) + \cos(\frac{2\pi}{3}+\theta)]. Using the formula \cos(A-B)+\cos(A+B) = 2\cos A\cos B, the sum becomes k[\cos\theta + 2\cos(\frac{2\pi}{3})\cos\theta] = k[\cos\theta + 2(-\frac{1}{2})\cos\theta] = k[\cos\theta - \cos\theta] = 0.