Damped Sines

★ 8/10functions trigonometry Trigonometric Identities Cauchy-Schwarz inequality

Problem

For a positive integer $n$ and reals $a_0, \dots, a_n$ , let

f_n(x) = \sin(x + a_0) + \frac{\sin(x + a_1)}{2} + \cdots + \frac{\sin(x + a_n)}{2^n}.

1. If $a_0 = 0$ and $a_1 = \dfrac{\pi}{3}$ , find the range of $f_1$ . 2. For $n = 2020$ : do there exist $a_0, \dots, a_{2020}$ with $f_{2020}(1) = f_{2020}(2) = 0$ ? Exhibit a choice or prove none exists.

Answer

Solution

Difficulty	8/10
Topics	functions, trigonometry, Trigonometric Identities, Cauchy-Schwarz, inequality

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