# Multiply complex numbers to show trigonometric addition formulas

Use the rules for multiplication of two complex numbers written in the form $r(\cos\theta +i\sin\theta)$ to show that $\sin(\theta_1 +\theta_2)=\sin\theta_1\cos\theta_2 +\sin\theta_2\cos\theta_1$ and $\cos(\theta_1 +\theta_2)=\cos\theta_1\cos\theta_2 −\sin\theta_1\sin\theta_2$.

I have no idea how to do this. It's part of a complex number worksheet but I can't find a way to do it. I thought about using Euler's formula but got no where. Thank you in advance

• If you already know that $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix}+e^{-ix}}{2}$ then the rest is just grouping thing. – Stephen Dedalus Dec 16 '13 at 12:52
• @Baranovskiy: I don't think that's what she's supposed to do here. Simply multiplying $e^{i\theta_1}$ with $e^{i\theta_2}$ will do. – Henning Makholm Dec 16 '13 at 12:53
• @HenningMakholm I misunderstood question but after editing I can see that you are right. – Stephen Dedalus Dec 16 '13 at 13:00

$\exp(i\theta_1)\exp(i\theta_2)=\exp i(\theta_1+\theta_2)=\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)=(\cos(\theta_1)+i\sin(\theta_1)\dot{}(\cos(\theta_2)+i\sin(\theta_2)$
bacause you have: $\exp(i\phi)=cos(\phi)+i\sin(\phi)$