General form for \(\displaystyle{\sin{{\left({k}{x}\right)}}}\) in terms of

Rex Maxwell

Rex Maxwell

Answered question

2022-03-23

General form for sin(kx) in terms of sin(x)  and  cos(x)
Identities for sin(2x) and sin3x, as well as their cosine counterparts are very common, and can be used to synthesize identities for sin4x and above. Given some integer k, is there an equation to find sin(kx) in terms of sinx and cos(x)?
For example, if I wanted to find the identity for sin(1000x), what would it be?
Identies used:
sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
cos(a+b)=cos(a)cos(b)+sin(a)sin(b)

Answer & Explanation

Boehm98wy

Boehm98wy

Beginner2022-03-24Added 18 answers

Use the De Moivre formula to calculate sin(nx) and cos(nx) for large n
Lesly Fernandez

Lesly Fernandez

Beginner2022-03-25Added 16 answers

Hint. One has
sinnθ=Un1(cosθ)sinθ
with
Un(x)=k=0|n2|(1)k ((nk),(k)) (2x)n2k
these polynomials are known as Chebyshev polynomials of the second kind.

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