# Phasor notation of a complex number with apparently non-constant radius

I'm asked to find the phasor of this signal:

$$x_1[n] = 3\sin\left(\frac \pi7 n\right) + 4j\cos\left(\frac \pi7 n\right),\quad (0\le n \le 20) \,.$$

The phasor is defined as:

$$G z_{0}^{n}= Ae^{j\varphi}\cdot r^{n}\cdot e^{j\theta*n} = Ar^{n}[\cos(\theta n+\varphi)+ j\sin(\theta n+\varphi)]$$

But I really don't know how to make the conversion! I mean, one of them has a radius of 3, the other is 4. How can I do this?

## 1 Answer

Hint: a phasor is defined by its magnitude and phase. The magnitude can be obtained by computing

$$\sqrt{\left(\mathcal{Re}\left\{x_1[n]\right\}\right)^2 + \left(\mathcal{Im}\left\{x_1[n]\right\}\right)^2}$$

And the phase can be obtained with

$$\tan^{-1}{\frac{\mathcal{Im}\left\{x_1[n]\right\}}{\mathcal{Re}\left\{x_1[n]\right\}}}$$