Rational trig identities?

10122021, 02:09 PM
Post: #6




RE: Rational trig identities?
Let θ = atan(1/n).
We do not know tan(nθ), but we do know tan(θ) = 1/n Use sum of tangent formula, tan(A+B) = (tan(A) + tan(B)) / (1  tan(A)*tan(B)) tan(2θ) = 2*tan(θ) / (1  tan(θ)^2) = 2n/(n^21) tan(3θ) = tan(2θ+θ) = (tan(2θ) + tan(θ)) / (1  tan(2θ)*tan(θ)) = (3n^21)/(n^33*n) ... When angle reached nθ, we have the solution. Of course, there are more efficient ways to build tan(nθ) (see post #4)  More food for thought, what is the sequence converging to ? n → inf is same as x = 1/n → 0 atan(x) = (x  x^3/3 + x^5/5  ...) = x*(1  x^2/3 + x^4/5  ...) limit(atan(x)/x, x=0) = 1 For finite n: tan(nθ) < tan(1) = 1.5574077246549 ... 

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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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