Odd Angles Formula Trivia
12-15-2019, 01:49 AM (This post was last modified: 01-22-2020 08:46 PM by Albert Chan.)
Post: #7
 Albert Chan Senior Member Posts: 2,142 Joined: Jul 2018
RE: Odd Angles Formula Trivia
The same trick work for sin(x)n, but only odd n can convert to multiple angle sin

Example, for sin(x)³:

(z - 1/z) ﻿= 2i sin(x)
(z - 1/z)³ = z³ - 3z + 3/z - 1/z³ = (z³ - 1/z³) - 3(z - 1/z)

Divide above (both side) by -2i, we get: 2² sin(x)³ = -sin(3x) + 3 sin(x)

For even n, we get back linear combinations of multiple angles cos:

(z - 1/z)6 = z6 - 6z4 + 15z² - 20 + 15/z² - 6/z4 + 1/z6 = (z6 + 1/z6) - 6(z4 + 1/z4) + 15(z² + 1/z²) - 20

Divide above (both side) by -2, we get: 25 sin(x)6 = -cos(6x) + 6 cos(4x) - 15 cos(2x) + 20/2

Update: we can deduce the signs without z's. From right to left, sign pattern is + − + − + − ...
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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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