Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
The number of solutions of the system of equations: $2\sin^2 x + \sin^2 2x = 2$, $\sin 2y + \cos 2x = \tan x$ in $[0, 4\pi]$ satisfying $2\cos^2 x + \sin x \leq 2i\delta$:
Step-by-Step Solution
Key Concept: Double-angle substitutions convert mixed powers of sine into manageable polynomial forms.
From $2\sin^2 x + \sin^2 2x = 2$, rewrite as $2\sin^2 x - 3\sin^2 x + 1 = 0$, factoring to $(2\sin^2 x - 1)(\sin^2 x - 1) = 0$. This gives $\sin 2x + \cos 2x = \tan x$ and $\sin x(2\sin x - 1) \geq 0$, with solutions satisfying both conditions.
Correct Answer: 8