Polar Form in Complex Numbers

Question

I've recently started getting in touch with complex numbers and I found it is quite easier to operate with them (multiplication, division, powers) in polar form rather than algebraically developing the whole thing. Would it be recommended? The main problem I'm having is that as I'm computing the angles with the calculator, it sometimes approximates angles and therefore some results which could be written as a fraction (for example, $5^{1/2}/2$), are given in a decimal (approximated) way. Any tip to work this out with the lesser error possible? Thank you very much.

Answer 1

Yes: don't compute the angles with a calculator. Why would you want to do that? For instance, if you want to compute$$\left(4\left(\cos\left(\frac34\right)+\sin\left(\frac34\right)i\right)\right)\times\left(3\left(\cos\left(\frac23\right)+\sin\left(\frac23\right)i\right)\right),$$then you will just get$$12\left(\cos\left(\frac{17}{12}\right)+\sin\left(\frac{17}{12}\right)i\right).$$No need to compute those sines and cosines.

Answer 2

Well you could write, for instance, $\cos1$ instead of the approximation. Or $re^{i\theta}=r\cos\theta+ir\sin\theta$, using Euler's formula, in general.