MAT334--2020F > Chapter 1

Solving roots of complex numbers

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Maria-Clara Eberlein:
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:
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.