How many values does (x+y) have?
For the following two (02) items: Let \cos(2x+3y) = \frac{1}{2} and \cos(3x+2y) = \frac{\sqrt{3}}{2}, where -\pi \lt (2x+3y) \lt \pi and -\pi \lt (3x+2y) \lt \pi.
- A. Two
- B. Three
- C. Four ✓
- D. More than four
Correct Answer: C. Four
Explanation
Let u = 2x+3y and v = 3x+2y. In the domain (-\pi, \pi), u \in \{-\frac{\pi}{3}, \frac{\pi}{3}\} and v \in \{-\frac{\pi}{6}, \frac{\pi}{6}\}. Since u+v = 5(x+y), we evaluate all 4 combinations of u+v. All 4 combinations yield distinct sums, meaning x+y has 4 values.