Converting Complex Numbers From Trigonometric Form to Rectangular - Problem 1

We're converting complex numbers from trigonometric form to polar form. Here is an example; Z equals 2 root 3 cosine 5 pi over 6 plus i sine 5 pi over 6.

In the trigonometric form first of all, remember that this is the modulus of the complex number which means it's distance from the origin to root 3. This is the argument of the complex number, 5 pi over 6. It's the angle that it makes with the positive real axis.

So if I were to graph this complex number, 5 pi over 6 would bring us over here. I have to go 2 pi over 3 away from the origin. So that would be my number z. Anyway, let's convert this to rectangular form. I'll distribute the 2 root 3. I get 2 root 3 cosine 5 pi over 6, plus i times 2 root 3 sine 5 pi over 6.

Cosine of 5 pi over 6 is negative root 3 over 2. So this is 2 root 3, times negative root 3 over 2. Sine of 5 pi over 6 is 1/2, so this is 2 root 3 times 1/2. So we get some cancellation; the 2's cancel. Root 3 times root 3 is 3, so I get -3 plus i times, and then the 2s cancel here, and I get i times root 3. That's my final answer. That's the rectangular form of z.