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How can I expand $sin5x$ easily in terms of sine.... ie knowing how many terms and in what degree. plus their constans.

Thanks. hope i've explained myself ok. or do i need to take 10 mins to expand everything in terms of sin?

Thanks. hope i've explained myself ok. or do i need to take 10 mins to expand everything in terms of sin?

You can use De Moivre's Theorem: $(\cos \theta + i\sin \theta)^{n}=\cos n\theta + i\sin n\theta$

Expand this with the binomial theorem, take the imaginary part and then if necessary subsitute $cos^{2} x = 1-\sin ^2 x$ to get it in terms of $\sin x$ if required.

Unparseable latex formula:

\sin 5x = Im(\cos 5x + i\sin 5x) = Im(\cos x + i\sin x)^{5}\\[br]\sin 5x = Im \cos ^{5} x (1+i\tan x)^{5}

Expand this with the binomial theorem, take the imaginary part and then if necessary subsitute $cos^{2} x = 1-\sin ^2 x$ to get it in terms of $\sin x$ if required.

Goldenratio

How can I expand $sin5x$ easily in terms of sine.... ie knowing how many terms and in what degree. plus their constans.

Thanks. hope i've explained myself ok. or do i need to take 10 mins to expand everything in terms of sin?

Thanks. hope i've explained myself ok. or do i need to take 10 mins to expand everything in terms of sin?

(cosx+isinx)

(cosx+isinx)

from (i) and (ii)

sin5x=5cos

=5(1-2sin

this is how to do it, but I might have got the wrong answer

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