Proving 1+cos⁡x2+sin⁡x<43

Ahmed Stewart

Ahmed Stewart

Answered

2022-01-25

Proving 1+cosx2+sinx<43

Answer & Explanation

Johnny Cummings

Johnny Cummings

Expert

2022-01-26Added 7 answers

If cos(θ)=35 and sin(θ)=45 (which is possible since 32+42=52), the inequality (with rather than <) is equivalent to cos(xθ)1
utgyrnr0

utgyrnr0

Expert

2022-01-27Added 11 answers

We want to prove 1+cosx2+sinx<43. By multiplying by 3(2+sinx) (note that 2+sinx>0), we get that this is equivalent to 3(1+cosx)4(2+sinx) which is equivalent to prove that 3cosx4sinx5
This inequality comes from Cauchy Schwartz. Note that
(3cosx4sinx)2(32+(4)2)(cos2x+sin2x)=25

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