proof left as an exercise
07-02-2022, 11:44 PM (This post was last modified: 07-04-2022 11:34 AM by Albert Chan.)
Post: #11
 Albert Chan Senior Member Posts: 1,911 Joined: Jul 2018
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^2-1) * (z^6-1)/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 - 06-06-2022, 11:41 PM RE: proof left as an exercise - Ángel Martin - 06-07-2022, 05:05 AM RE: proof left as an exercise - Thomas Klemm - 06-07-2022, 05:32 AM RE: proof left as an exercise - Albert Chan - 06-07-2022, 05:36 PM RE: proof left as an exercise - Albert Chan - 06-07-2022, 06:17 PM RE: proof left as an exercise - Albert Chan - 06-08-2022, 01:50 AM RE: proof left as an exercise - Albert Chan - 06-08-2022, 11:12 AM RE: proof left as an exercise - Thomas Klemm - 06-08-2022, 11:18 PM RE: proof left as an exercise - Albert Chan - 06-09-2022, 12:35 AM RE: proof left as an exercise - Albert Chan - 07-01-2022, 07:51 PM RE: proof left as an exercise - Albert Chan - 07-02-2022 11:44 PM

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