Odd Angles Formula Trivia

10132021, 03:40 AM
(This post was last modified: 10132021 09:05 AM by Albert Chan.)
Post: #9




RE: Odd Angles Formula Trivia
(12202018 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) An elegant proof (induction proof work, but messy, and does not explain why ...) Let y = pi/2  x Let f3(cos(x),sin(x)) = cos(3x) → f3(sin(x),cos(x)) = f3(cos(y),sin(y)) = cos(3y) = cos(3*(pi/2)3x) = −sin(3x) Let f5(cos(x),sin(x)) = cos(5x) → f5(sin(x),cos(x)) = f5(cos(y),sin(y)) = cos(5y) = cos(5*(pi/2)5x) = +sin(5x) We can extend angle to (4n ± 1)*x and get same pattern , because 4n*(pi/2) = 2n*pi  This is inspired by swapping of complex parts function I post here z = x+y*i swap(z) = y+x*i = i*conj(z) swap(z)^3 = (i*conj(z))^3 = i*conj(z^3) = −swap(z^3) swap(z)^5 = (i*conj(z))^5 = i*conj(z^5) = +swap(z^5) Since i^4 = 1, same (4n±1) pattern happened here as well. (cos(x) + i*sin(x))^(4n±1) = cos((4n±1)*x) + i * sin((4n±1)*x) (sin(x) + i*cos(x))^(4n±1) = ±sin((4n±1)*x) + i*±cos((4n±1)*x) This already proved it ! Think of LHS the "f". For effect of swapping parts, read RHS vertically. 

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Messages In This Thread 
Odd Angles Formula Trivia  Albert Chan  12202018, 05:38 PM
RE: Odd Angles Formula Trivia  Albert Chan  12212018, 01:10 PM
RE: Odd Angles Formula Trivia  Albert Chan  08142019, 01:54 PM
RE: Odd Angles Formula Trivia  Albert Chan  08152019, 01:55 AM
RE: Odd Angles Formula Trivia  Albert Chan  08152019, 03:42 PM
RE: Odd Angles Formula Trivia  Albert Chan  12152019, 01:03 AM
RE: Odd Angles Formula Trivia  Albert Chan  12152019, 01:49 AM
RE: Odd Angles Formula Trivia  Albert Chan  01222020, 12:45 AM
RE: Odd Angles Formula Trivia  Albert Chan  10132021 03:40 AM

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