Converting Complex Numbers From Trigonometric Form to Rectangular - Problem 3

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.