Odd Angles Formula Trivia
12-21-2018, 01:10 PM
Post: #2
 Albert Chan Senior Member Posts: 2,634 Joined: Jul 2018
RE: Odd Angles Formula Trivia
(12-20-2018 05:38 PM)Albert Chan Wrote:  Challenge: prove above pattern continues: if sin(4nx ± x) = f(s,c), then cos(4nx ± x) = ± f(c,s)

Let angle A = 4kx + x, s = sin(x), c = cos(x):

sin(A+2x) = sin(A) cos(2x) + cos(A) sin(2x) = f(s,c) (c² - s²) + f(c,s) (2 s c) = g(s,c)
cos(A+2x) = cos(A) cos(2x) - sin(A) sin(2x) = f(c,s) (c² - s²) - f(s,c) (2 s c) = -g(c,s)

sin(A+4x) = sin(A+2x) cos(2x) + cos(A+2x) sin(2x) = g(s,c) (c² - s²) - g(c,s) (2 s c) = h(s,c)
cos(A+4x) = cos(A+2x) cos(2x) - sin(A+2x) sin(2x) = -g(c,s) (c² - s²) - g(s,c) (2 s c) = h(c,s)

Angles A+2x and A+4x is same as 4(k+1)x ± x
Post #1 examples already shows n=1 work, and by induction, if n=k work, n=k+1 also work.

QED
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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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