Odd Angles Formula Trivia
12-15-2019, 01:03 AM (This post was last modified: 12-15-2019 01:50 AM by Albert Chan.)
Post: #6
 Albert Chan Senior Member Posts: 2,649 Joined: Jul 2018
RE: Odd Angles Formula Trivia
A trick to convert cos(x)n as linear combinations of multiple angle cos

Example: cos(x)³ to linear combinations of cos(3x) and cos(x)

Let complex number z = exp(i x)

(z + 1/z) ﻿= 2 cos(x)
(z + 1/z)³ = z³ + 3z + 3/z + 1/z³ = (z³ + 1/z³) + 3(z + 1/z)

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

We can skip above steps, and write formula directly !
Example: 6th row of pascal triangle = 1 6 15 20 15 6 1

25 cos(x)6 = cos(6x) + 6 cos(4x) + 15 cos(2x) + 20/2
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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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