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Converting Complex Numbers From Rectangular Form to Trigonometric - Problem 1 2,112 views

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

We're converting from rectangular form to trigonometric form. Here I have the complex number z equals root 3 plus i. Now, if I would have graphed this, root 3 is bigger than 1. Here the b value is 1, and so my number would look something like this.

So what I need to do is find r, it's the modulus of the numbers. So distance from the origin of the number, and theta, which is the argument of the number; the angle it makes with the positive real axis. So let me find r first.

I've got the formula r equals the square root of a², plus b². A is root 3. So it's root 3 squared, and b is 1, 1² is 1. So I'm going to get 3 plus 1, root 4 which is 2.

Next I have to find theta. Theta is the argument of the complex number. I get theta using the fact that cosine of theta equals a over r. Now a is root 3, r is 2, and sine theta equals b over r. B is 1, and r is 2. So what angle has a cosine of root 3 over 2 and a sine of 1/2? That's pi over 6. So my trigonometric form for this complex number is z equals 2 times the quantity, cosine of pi over 6 plus i sine pi over 6, and that's my answer.

Now, notice we didn't really have to worry too much about the requirement, that theta is between 0 and 2 pi. Remember the way the trig functions work, they're periodic with period 2 pi. So this number could just as easily be represented by an argument of pi over 6, plus 2 pi. But that means that they are infinitely many ways to represent this number in trig form.

So your teacher is probably going to do something like this. They'll say, give me an argument between 0 and 2 pi. So that there will be only answer. You always want to give r positive, because r is this square root of the sum of two squares. It's going to end up being positive. Anyway, this is our final answer.