Let S be the set of all R such that the equation 2x+ x = 2-7 has a solution.

Question

Let $S$ be the set of all $\alpha\in\mathbb{R}$ such that the equation $\cos2x+\alpha\sin x = 2\alpha-7$ has a solution. Then $S$ is equal to:

Accepted Answer

$$1-2\sin^2x+\alpha\sin x = 2\alpha-7 \implies 2\sin^2x-\alpha\sin x+(2\alpha-8)=0$$

$$\sin x = \frac{\alpha\pm\sqrt{\alpha^2-8(2\alpha-8)}}4 = \frac{\alpha\pm(\alpha-8)}4$$

Taking the root $\sin x = \dfrac{\alpha-4}4$ (the other root gives $\sin x=2$, rejected). For a solution to exist:

$$-1 \le \frac{\alpha-4}4 \le 1 \implies \alpha \in [2,6] \implies S = [2,6]$$

Solutions · 1