proof left as an exercise
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07-02-2022, 11:44 PM
(This post was last modified: 07-04-2022 11:34 AM by Albert Chan.)
Post: #11
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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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