# Polar coordinates

## Polar coordinates

### Polar co-ordinates

#### All complex numbers can be shown in polar form

Consider a complex number

$$z=a+bi$$

We can write this as:

$$z=r\cos(\theta ) + ir\sin(\theta )$$

#### Polar forms are not unique

Because the functions loop:

$$ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))$$

$$ae^{i\theta }=a(\cos(\theta +n\tau )+i\sin(\theta +n\tau ))$$

$$ae^{i\theta }= ae^{i\theta +n\tau}$$

$$ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))$$

$$ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))$$

$$ae^{i\theta }=-a(\cos(\theta )-i\sin(\theta ))$$

$$ae^{i\theta }=-a(\cos(\theta +\dfrac{\pi }{2})+i\sin(\theta +\dfrac{\pi }{2}))$$

#### Real and imaginary parts of a complex number in polar form

We can extract the real and imaginary parts of this number.

$$Re(z):=r\cos (\theta )$$

$$Im(z):=r\sin (\theta )$$

Alternatively:

$$Re(z)=r\dfrac{e^{i\theta }+e^{-i\theta }}{2}$$

$$Im(z)=r\dfrac{e^{i\theta }-e^{-i\theta }}{2i}$$

### Moving between polar and cartesian coordinates

All polar numbers can be shown as Cartesian

$$ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))$$

$$ae^{i\theta }=a\cos(\theta )+ia\sin(\theta )$$

$$z=a+bi$$

$$e^{i\theta }=$$

$$e^x=\sum^{\infty }_{i=0} \dfrac{x^i}{i!}$$

### Arithmetic of polar coordinates

$$z_3=z_1+z_2$$

$$z_3=a_1e^{i\theta_1}+a_2e^{i\theta_2}$$

$$z_3=a_1[\cos(\theta_1)+i\sin(\theta_1)]+a_2[\cos(\theta_2)+i\sin(\theta_2)]$$

$$z_3=[a_1\cos(\theta_1)+a_2\cos(\theta_2)]+i[a_1\sin(\theta_1)+a_2 \sin(\theta_2)]$$

Multiplication

$$z_3=z_1.z_2$$

$$z_3=a_1e^{i\theta_1}a_2e^{i\theta_2}$$

$$z_3=a_1a_2e^{i(\theta_1+\theta_2)}$$

$$a_3=a_1a_2$$

$$\theta_3=\theta_1+\theta_2$$