MathLabs

Problem 4

Let f(θ)=1−acos⁡θ−bsin⁡θ−Acos⁡2θ−Bsin⁡2θf(\theta)=1-a\cos\theta-b\sin\theta-A\cos2\theta-B\sin2\theta, where a,b,A,Ba,b,A,B are real. Prove that if f(θ)≥0f(\theta)\ge0 for all real θ\theta, then a2+b2≤2a^2+b^2\le2 and A2+B2≤1A^2+B^2\le1.
Step 3 of 3: Conclude both bounds
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A2+B2≤1,a2+b2≤2A^2+B^2\le1,\qquad a^2+b^2\le2
Detailed analysis

The two shifts separately bound the amplitudes of the second and first harmonics. Together they prove both required inequalities.