proof left as an exercise
06-08-2022, 01:50 AM
Post: #6
 Albert Chan Senior Member Posts: 2,629 Joined: Jul 2018
RE: proof left as an exercise
Another proof, letting t = tan(10°)

tan(30°) = 1/√3 = (3t-t³) / (1-3t²)                  // triple-angle formula
⇒ t³ = √3*t² + 3t - 1/√3
⇒ t4 = 6t² + 8/√3*t - 1

2*cos(30°) / (1 + 4*cos(20°)) ?=? tan(20°)

√3 / (1 + 4*(1-t²)/(1+t²)) ?=? 2t / (1-t²)        // tangent half-angle formula
√3*(1+t²) / (5-3t²) ?=? 2t / (1-t²)                  // cross multiply
√3*(1-t4) ?=? 10t - 6t³

√3*(1-t4) + 6t³ - 10t
= √3 - √3*(6t² + 8/√3*t - 1) + 6*(√3*t² + 3t - 1/√3) - 10t
= 0      QED
 « Next Oldest | Next Newest »

 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

User(s) browsing this thread: 1 Guest(s)