Rational trig identities?

10122021, 04:05 PM
Post: #7




RE: Rational trig identities?
(10102021 08:02 PM)Albert Chan Wrote: Pattern appeared, (t+1)^n, odd powers goes on top, even powers bottom, alternate sign. 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  10102021, 04:42 PM
RE: Rational trig identities?  Albert Chan  10102021, 06:21 PM
RE: Rational trig identities?  Albert Chan  10102021, 08:02 PM
RE: Rational trig identities?  Albert Chan  10122021 04:05 PM
RE: Rational trig identities?  Albert Chan  10102021, 09:25 PM
RE: Rational trig identities?  John Keith  10112021, 01:08 PM
RE: Rational trig identities?  Albert Chan  10122021, 02:09 PM

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