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# Converting Complex Numbers From Trigonometric Form to Rectangular - Problem 3 1,898 views

###### 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 complex numbers from trigonometric form to rectangular form. Here is an example. Z equals 8 times the quantity; cosine 3 pi over 2 plus i sine 3 pi over 2. Just for practice, let's plot this number.

Now the argument of z is 3 pi over 2, that's the angle that it makes with the positive real axis. 3 pi over 2 would put us here. 8 is the modulus or absolute value of z. That's its distance from 0, so this is the origin 0, and I would go down eight units. This is 3 pi over 2.

You can kind of see once you've plotted it, exactly what the rectangular form is going to be. It's going to be -8i. You can also get that by doing the computation. Distribute the 8 through, 8 cosine 3 pi over 2 plus i times 8 sine 3 pi over 2.

Now cosine of 3 pi over 2 is 0. So this is 8 times 0 plus i times, and sine of 3 pi over 2 is -1, so 8 times 1. This just simplifies again to -8i. So z equals -8i, that's our rectangular form.