Simplifying An Exponential Function Which Includes Complex Numbers

Question

In there j is the complex number. In signals and systems lecture we generally use 'j' instead of 'i'.

In my book solution it's includes one equation like that. But i didn't understand this conversion or simplification.

$$1-e^{-j*0.4*\pi} = 1.1756*e^{j*0.3*\pi}$$

Answer 1

You can use Euler's Formula and the Cartesian to Polar coordinate transformation as follows:

$1-e^{-j*0.4\pi}=1-(\cos(0.4\pi)-j\sin(0.4\pi))$

Which equals

$1-\cos(0.4\pi)+j\sin(0.4\pi)$

Now we can calculate the absolute value of this value and the angle it makes with the real axis and plug these into our exponential ($re^{j\theta}$).

$r=\sqrt{(1-\cos(0.4\pi))^{2}+(\sin(0.4\pi))^{2}}=1.1756$

$\theta=\arctan(\frac{\sin(0.4\pi)}{1-cos(0.4\pi)})=0.3\pi$

Hope this helps.