Odd Angles Formula Trivia
08-14-2019, 01:54 PM
Post: #3
 Albert Chan Senior Member Posts: 2,142 Joined: Jul 2018
RE: Odd Angles Formula Trivia
We can show above sign flipping behavior, using Chebyshev Polynomial

By definition, Tn(cos(x)) = cos(n*x)

Tn(cos(pi/2-x)) = cos(n*(pi/2-x)) = cos((n-1)*pi/2 + (pi/2-n*x))
Tn(sin(x)) = sin(n*x + (n-1)*pi/2)

if n=4k+1, (n-1)*pi/2 ≡ 0 (mod 2*pi), then Tn(sin(x)) = sin(n*x+0) = sin(n*x)
if n=4k−1, (n-1)*pi/2 ≡ pi (mod 2*pi), then Tn(sin(x)) = sin(n*x+pi) = - sin(n*x)

QED

Extend above for even n's:

if n=4k+0, n*pi/2 ≡ 0 (mod 2*pi), then Tn(sin(x)) = cos(0-n*x) = cos(n*x)
if n=4k+2, n*pi/2 ≡ pi (mod 2*pi), then Tn(sin(x)) = cos(pi-n*x) = - cos(n*x)
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 Messages In This Thread Odd Angles Formula Trivia - Albert Chan - 12-20-2018, 05:38 PM RE: Odd Angles Formula Trivia - Albert Chan - 12-21-2018, 01:10 PM RE: Odd Angles Formula Trivia - Albert Chan - 08-14-2019 01:54 PM RE: Odd Angles Formula Trivia - Albert Chan - 08-15-2019, 01:55 AM RE: Odd Angles Formula Trivia - Albert Chan - 08-15-2019, 03:42 PM RE: Odd Angles Formula Trivia - Albert Chan - 12-15-2019, 01:03 AM RE: Odd Angles Formula Trivia - Albert Chan - 12-15-2019, 01:49 AM RE: Odd Angles Formula Trivia - Albert Chan - 01-22-2020, 12:45 AM RE: Odd Angles Formula Trivia - Albert Chan - 10-13-2021, 03:40 AM

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