Absolute Sines in a Row

★ 8/10trigonometryEstimationCauchy-Schwarzinequality

Problem

Let f(x)=∣sin⁡x∣f(x) = |\sin x|.

1. Prove that

sin⁡1≤f(x)+f(x+1)≤2cos⁡12.\sin 1 \le f(x) + f(x+1) \le 2\cos\frac12.
  1. Prove that for every positive integer nn,
f(n)n+f(n+1)n+1+⋯+f(3n−1)3n−1>sin⁡12.\frac{f(n)}{n} + \frac{f(n+1)}{n+1} + \cdots + \frac{f(3n-1)}{3n-1} > \frac{\sin 1}{2}.

Answer

Solution

Difficulty8/10
Topicstrigonometry, Estimation, Cauchy-Schwarz, inequality

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