Polar Coordinates and Complex Numbers
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 the rectangular form of a complex number to trigonometric form. Here I have the example z equals -10. Here you have to remember, -10 although it's a real number, is also a complex number. All real numbers are complex. Now let me plot this number.
-10 would be right on the real axis over here. Now, the two things we have to find when we're converting to trig form are r, the modulus of the number. That's the distance of the number from 0. That's actually pretty easy. You can see that it's 10, so r equals 10.
Theta, the argument, and we specifically wan an argument between 0 and 2 pi. That actually is also pretty easy just from the picture, we know that this angle is pi. So we don't have to use the conversion formulas here. We actually have our answers already. R equals 10, theta equals pi. So z is 10 times the cosine of pi plus i sine pi. That's our final answer.
Don't be afraid to draw a picture. You might be able to get the answer very quickly without using the conversion formulas, if you draw a really accurate picture.