what is the value of z/w?

Question

Let $|w|=3$ come out of the origin and and let $|z|=2$ and also come out of the origin. Both lines are in the second quadrant and form an angle of $\frac{\pi}{3}$ degrees. What is the value of $\frac z w$?

Answer 1

$\frac{z}{w}=\frac{2}{3}e^{\frac{\pi i}{3}}$ The magnitude is obvious, while the angle is simply the difference in angle between them.

Answer 2

I completely agree with Herb Steinberg's answer, but would like to focus on why the OP had trouble with this question. Math students are taught how to attack problems like this one in a textbook on complex analysis. Math students normally defer studying complex analysis until they have studied both calculus and real analysis.

To the OP, and others in a similar situation: it is not a good idea to try to "re-invent the wheel", meaning don't try to solve this (or similar problems) on your own. Instead, get a complex analysis textbook.

Further, it is not a good idea to tackle subjects out of order. Get grounded in calculus and real analysis before tackling complex analysis.