Rational trig identities?
10-12-2021, 04:05 PM
Post: #7
 Albert Chan Senior Member Posts: 2,142 Joined: Jul 2018
RE: Rational trig identities?
(10-10-2021 08:02 PM)Albert Chan Wrote:  Pattern appeared, (t+1)^n, odd powers goes on top, even powers bottom, alternate sign.

tan(n*x) = (n*t - binom(n,3)*t^3 + ...) / (1 - binom(n,2)*t^2 + ...)

What has this property ? Complex number !
odd powers of i goes imaginery, even powers of i goes real, i*i = -1

Assuming |θ| < pi/2, we have atan(tan(θ)) = θ

z = 1 + tan(θ)*i = r * cis(θ)
Z = z^n = r^n * cis(nθ)
tan(nθ) = im(Z) / re(Z)

If θ = atan(1/n), tan(θ) = 1/n
To keep Z parts integer, we scale up Z: (1+i/n)^n → (n+i)^n

CAS> (i+2)^2 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 3+4i → a(2) = 4/3
CAS> (i+3)^3 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 18+26i → a(3) = 26/18 = 13/9
CAS> (i+4)^4 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 161+240i → a(4) = 240/161
CAS> (i+5)^5 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1900+2876i → a(5) = 2876/1900 = 719/475
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 Messages In This Thread Rational trig identities? - John Keith - 10-10-2021, 04:42 PM RE: Rational trig identities? - Albert Chan - 10-10-2021, 06:21 PM RE: Rational trig identities? - Albert Chan - 10-10-2021, 08:02 PM RE: Rational trig identities? - Albert Chan - 10-12-2021 04:05 PM RE: Rational trig identities? - Albert Chan - 10-10-2021, 09:25 PM RE: Rational trig identities? - John Keith - 10-11-2021, 01:08 PM RE: Rational trig identities? - Albert Chan - 10-12-2021, 02:09 PM

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