# Sine and Cosine series and complex numbers

Write the following term $\cos(3 \varphi + \psi)$ in form of $m \cdot(cos \, \varphi)^{n_1} \cdot (sin \, \varphi)^{n_2} \cdot (cos \, \varphi)^{n_3} \cdot (\sin \, \psi)^{n_4}$ with $m \in \mathbb{Z}$ and $n_i \in \mathbb{N}_0$. Try only to use the sine and cosine series:

$cos(\varphi) = \sum\limits_{n=0}^{\infty} (-1)^n \dfrac{(\varphi)^{2n}}{2n!} \quad$ and $\quad sin(\varphi) = \sum\limits_{n=0}^{\infty} (-1)^n \dfrac{(\varphi)^{2n+1}}{(2n+1)!} \quad$ and $\quad e^x = \sum\limits_{n=0}^{\infty} \dfrac{x^n}{n!}$

What i tried was to use is $cos(a+b) = cos(a) \cdot cos(b) - sin(a) \cdot sin(b)$ but the minus get me confused. Also the m coefficient remembers me at the radius of a complex number. Using complex numbers would also fit to the actual topic of my lecture.

## 1 Answer

As you are using complex numbers, $$\cos(3φ+ψ)=Re\Bigl(\exp(i·(3φ+ψ))\Bigr)=Re\Bigl((\exp(iφ))^3·(\exp(iψ))\Bigr) \\ =Re\Bigl((\cosφ+i\sinφ)^3(\cosψ+i\sinψ)\Bigr)$$ Now apply binomial theorems etc.