How do I prove tan⁡1<π2? Prove that the equation sin⁡xsin⁡(sin⁡x)=π2cos⁡(sin⁡x) Let t=sin⁡x,−1≤t≤1. Then the expression...

Alisha Pitts

Alisha Pitts

Answered

2022-01-23

How do I prove tan1<π2?
Prove that the equation
sinxsin(sinx)=π2cos(sinx)
Let t=sinx,1t1. Then the expression above is equvalent to tsint=π2cost. As the function f(t)=tsintπ2cost is even, and t=0 is not a solution, I have to prove that f(t) has no positive roots (t>0). So, for the left side 0<t1 and 0<sintsin1, then tsintsin1. For the right side costcos1, so π2costπ2cos1. The objective is to prove that sin1<π2cos1, or, equivalently, tan1<π2

Answer & Explanation

Tapanuiwp

Tapanuiwp

Expert

2022-01-24Added 13 answers

We can use the Taylor series and alternating series theorem to say
sin1<113!+15!=101120
cos1>112!+14!16!=112+1241720=389720
tan1=sin1cos1<606389<1.56<π2

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