Sum of \(\displaystyle{1}-{{\cos{\theta}}_{{i}}}\) is bounded For each

tambustqaze

tambustqaze

Answered question

2022-03-31

Sum of 1cosθi is bounded
For each i=1,,N let θi0 be angles such that i=1Nθiπ. Prove that i=1N(1cosθi)2

Answer & Explanation

Nunnaxf18

Nunnaxf18

Beginner2022-04-01Added 18 answers

Prove the sharper inequality
i=1k(1cosθi)1cosσk
for 1kN
σk=i=1kθi
The base case is clear, and for the induction step, one must show that
1cosθcosσcos(σ+θ)
for θ,σ0 and σ+θπ. Geometrically that is clear by looking at the circle, the projection of an arc of fixed length in the upper semicircle to the x-axis is shortest when the arc is at the ends of the semicircle. Analytically, the assertion is
2sin2θ22sinσ+θ2sinθ2,
i.e. sinθ2sinσ+θ2 which follows since θ2σ+θ2π2

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