proof left as an exercise

07022022, 11:44 PM
(This post was last modified: 07042022 11:34 AM by Albert Chan.)
Post: #11




RE: proof left as an exercise
Using complex numbers, proof turns out very simple !
Let z = cis(20°), 1° = pi/180 cos(60°) = (z^3+1/z^3)/2 = 1/2 → (z^3+1/z^3) = 1 1 + 4*cos(20°) = (z^3+1/z^3) + 2*(z+1/z) = (z^2+1)*(z^4+z^2+1) / z^3 = (z^2+1)/(z^21) * (z^61)/z^3 = (2*cos(20°)) / (2i*sin(20°)) * (2i*sin(60°)) = 2*sin(60°) / tan(20°) > 2*cos(30°) / (1 + 4*sin(70°)) = tan(20°) 

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Messages In This Thread 
proof left as an exercise  Thomas Klemm  06062022, 11:41 PM
RE: proof left as an exercise  Ángel Martin  06072022, 05:05 AM
RE: proof left as an exercise  Thomas Klemm  06072022, 05:32 AM
RE: proof left as an exercise  Albert Chan  06072022, 05:36 PM
RE: proof left as an exercise  Albert Chan  06072022, 06:17 PM
RE: proof left as an exercise  Albert Chan  06082022, 01:50 AM
RE: proof left as an exercise  Albert Chan  06082022, 11:12 AM
RE: proof left as an exercise  Thomas Klemm  06082022, 11:18 PM
RE: proof left as an exercise  Albert Chan  06092022, 12:35 AM
RE: proof left as an exercise  Albert Chan  07012022, 07:51 PM
RE: proof left as an exercise  Albert Chan  07022022 11:44 PM

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