What is a value of \sin 3x+\sin 3y?
Consider the following for the next two (02) items that follow : Let \sin x+\sin y=\sqrt{3}(\cos y-\cos x); x+y=\frac{\pi}{2}, 0 \lt x, y \lt \frac{\pi}{2}.
- A. -1
- B. 0 ✓
- C. 1
- D. 3
Correct Answer: B. 0
Explanation
The given relation simplifies to 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) = 2\sqrt{3}\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right). This yields \tan\left(\frac{x-y}{2}\right) = \frac{1}{\sqrt{3}} \implies x-y = \frac{\pi}{3}. Combined with the condition x+y = \frac{\pi}{2}, we solve to find x = 75^\circ and y = 15^\circ. Evaluating the required expression: \sin(3 \times 75^\circ) + \sin(3 \times 15^\circ) = \sin 225^\circ + \sin 45^\circ = -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = 0.