A Shifted Sine Covering

★ 7/10trigonometryMonotonicityCasework

Problem

Let f(x)=2+3sin⁡xf(x)=2+3\sin x for x∈[0,π2]x\in\left[0,\dfrac{\pi}2\right]. Suppose θ\theta is such that for every x1∈[0,π2]x_1\in\left[0,\dfrac{\pi}2\right] there exists x2∈[0,π2]x_2\in\left[0,\dfrac{\pi}2\right] with

f(x1)+2f(x2+θ)=3.f(x_1)+2f(x_2+\theta)=3.

Find all such θ\theta in [0,2π)[0,2\pi).

Answer

Solution

Difficulty7/10
Topicstrigonometry, Monotonicity, Casework

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