### Author Topic: Solving roots of complex numbers  (Read 424 times)

#### Maria-Clara Eberlein

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##### Solving roots of complex numbers
« on: September 20, 2020, 07:02:56 PM »
Suppose we want to solve for z=a+ib in an equation of the form z^n=w. After we find z in exponential representation, I am unsure of how to convert to z=a+ib form if theta is not one of the "special angles" we know the sin and cos of? Should we write z=rcos(theta)+i(rsin(theta)) without evaulating sin(theta) and cos(theta)?

#### RunboZhang

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##### Re: Solving roots of complex numbers
« Reply #1 on: September 20, 2020, 07:18:56 PM »
I think if the angle is unfamiliar, we can leave it as z=r[cos theta + i sin theta], otherwise we need to compute the value of sin and cos. Also, I think it has no difference with z=a+bi, it is just in the polar form.

#### Victor Ivrii

The worst thing you can do is to use calculator to evaluate the value of, say, $\sin (4\pi/9)$ and $\cos (4\pi/9)$ numerically. But it may be useful to mention that
$\cos (4\pi/9)+i\sin (4\pi/9)$ belongs to the first quadrant and pretty close to $i$. Just draw a little picture.