When is \frac{1}{n}\sin(nt)=\frac{1}{n+2}\sin((n+2)t) ?

Abiha Bellamy

Abiha Bellamy

Answered question

2022-02-25

When is
1nsin(nt)=1n+2sin((n+2)t) ?

Answer & Explanation

faraidz3i

faraidz3i

Beginner2022-02-26Added 10 answers

Square your identity:
(n+2)2sin2nt=n2sin2(n+2)t
Convert from sines to cosines:
(n+2)2+n2cos2(n+2)t=n2+(n+2)2cos2nt
Now let x=cost to get
(n+2)2+n2Tn+2(x)2=n2+(n+2)2Tn(x)2
where the Tm are the Chebyshev polynomials. We get a polynomial equation of degree 2(n+2) in x which you can solve to find all solutions.

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