Rational trig identities?
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10-10-2021, 08:02 PM
Post: #3
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RE: Rational trig identities?
General case, by expanding tan(n*x), in terms of t = tan(x)
tan(n*x) = (t + tan((n-1)*x)) / (1 - t*tan((n-1)*x)) tan(2*x) = (2t) / (1 - t^2) tan(3*x) = (3t - t^3) / (1 - 3t^2) tan(4*x) = (4t - 4t^3) / (1 - 6t^2 + t^4) tan(5*x) = (5t - 10t^3 + t^5) / (1 - 10t^2 + 5t^4) ... 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 + ...) Better with cotangent c = 1/t: (with x = atan(1/n), we have c = n) tan(n*x) = (n*c^(n-1) - binom(n,3)*c^(n-3) + ...) / (c^n - binom(n,2)*c^(n-2) + ...) Or, just let CAS do the work CAS> a(n) := texpand(tan(n*x)) (tan(x)=1/n) CAS> apply(a, range(1,9)) \(\displaystyle \left[ 1,\frac{4}{3},\frac{13}{9}, \frac{240}{161}, \frac{719}{475}, \frac{42372}{27755}, \frac{92567}{60319}, \frac{14970816}{9722113} \right]\) |
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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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