[Math] Simplify a quick sum of sines

trigonometry

Simplify $\sin 2+\sin 4+\sin 6+\cdots+\sin 88$

I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.

Best Answer

If you know about complex numbers, you may want to use that: \begin{array}{lcl}\sum_{n=1}^N\sin 2n & = &\Re\left(\sum_{n=1}^N\left(\sin 2n+i\cos 2n\right)\right) \\ & = & \Re\left(\sum_{n=1}^Ne^{2ni}\right) \\ & = & \Re\left(e^{2i}\frac{1-e^{2Ni}}{1-e^{2i}}\right).\end{array}

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