Trigonometry: DeMoivre, Complex Numbers, Powers

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Using DeMoivre's Theorem to Raise a Complex Number to a Power
So let's take a look at this idea of multiplying complex numbers when they're written in polar form and see if we can take that idea to the extreme, because always in mathematics when you have an idea, just try to push it as far as you possibly can.
Suppose I have a complex number written in polar or trig form, . Now remember how multiplication of two numbers works. You multiply the moduli together and that's the new modulus, and then to get the angle--to get the argument--you just add those two angles together and that gives you the new angle. So what if I actually multiply a number by itself? Well, let's do that. So if I were to multiply a number by itself, to take and square it, that's just , and we know the recipe. I take and multiply it by , so that would give me . Cosine--and what's the angle? I take and add it to , so that's 2 plus . So if you square a complex number written in this form, what do you do? You square the modulus, and you just double the angle. That's all there is to it.
Well, let's take that idea to the ultimate extreme and let's take a look at what happens if I take and raise it to the power, where is some positive integer, like it's 2, 3, 4, 5, 6, 7, 8, and so on. What would the answer be? Well, you can see what the answer is. If I keep repeating this process, what would happen to the moduli? I'll have the modulus raised to the power, times cosine of what? Well, if I keep adding all the 's together, I'll have . So that's actually a formula for raising complex numbers to very, very high powers. It's very simple. You just take the modulus and raise it to the power , and then for the angle, you just multiply the angle by . This is known as DeMoivre's formula or DeMoivre's Theorem. Isn't that a great name?
Now let's actually use this to actually do some really--now it's a piece of cake. If I say to you, what is ? Well, the method that you would have to use before we discovered this recipe would be to take and multiply it by itself three times, do a foiling, and then do major expansion, and simplify, and it would take you forever. Now what I see is if I can write this number in polar form, I can immediately report the answer. So let's see if we can covert that number to polar form. Well, I think we can do that with not too much trouble.
Let's see what we have. Well, let's see what that point looks like. Well, over this way and then 1 unit up this way, and there's the point . Well, we can use the Pythagorean Theorem to find the length. is 3; is 1; so , . And I recognize this triangle. It's half of an equilateral triangle, and so in fact what I see is this angle here must be 30° or . So if it's , I see immediately how to write this number here in polar form. What do I do? I take , which is in this case 2, times cosine of--and what's the angle? Well, it's . Plus . And that is that complex number there. And now I take it and raise it to the third power.
So I first converted the number into this polar/trig form. Now, once that number here is in the polar/trig form, if I find the angle and find the length of this distance, well then I just use DeMoivre's Law to say what this equals. What do I do? I take this and I raise it to the third power, so it's . And what do I do with the angles? Well, I just take the angle and multiply it by 3. So this is going to be . . Plus , and what does that equal? Well, what is ? Well, , so that drops out. And what is ? That's just 1, so I just see . So when I multiply everything through by 8, I see zero times 8, which is zero, and I just see . So this number, turns out to be just . And you can check that if you want to by multiplying this out three times, but it's really painful. Look how much easier it was just to convert this into the polar/trig form and then use DeMoivre's Law to just say, okay, I take this number and cube it, and then take this angle and multiply it by 3. That's all there is to it.
Let's try one more together. Suppose I want to take and raise it to the 4^th power. Well, you can multiply out and multiply by itself four times. That's going to be a real pain. Or we can see if we can use this new law by first graphing. This is going to be 2 and this is . So this angle you can see is 45° or . And so if this is 2 and this is 2, what's this length? Well, I can use the Pythagorean Theorem to figure that out. I just take the square root of these things squared. So that would be 4 + 4, which is . So I take and that's the length. I have the angle and I have the length, so now it's sort of a done deal. I can write that number as what? Well, , that's the modulus, times cosine of the angle, which is , 45°, plus . And don't forget, I'm raising all of that to the 4^th power.
Well, now what do I do? Well, now it's a piece of cake, because remember how to proceed? What I do is I take this term and raise it to the 4^th power, so I take and I raise it to the 4^th power. And then what do I do with the angles? I just multiply this angle by 4, so is just . See how easy this is? So multiplication of complex numbers when they're written in this form is a piece of cake. , that's times times times . That's just 8 x 8, which is 64. And what's ? Well, let's take a look and see what is. is -1, so that's -1. Plus , and what's ? The sine of is zero. So plus zero . So what I see is just -64. So if you take this complex number and multiply it by itself four times, the answer you get turns out to be -64.
Look how easy it is to actually perform this kind of power raising, just writing it first in this trig/polar form and then using DeMoivre's Law. Really neat. Up next we'll take a look at variations on this theme when we go backwards. Suppose I want to take roots. How would I take the 4^th root? It would be a similar process. Instead of raising this to a power, I'll take a root of this. And instead of actually multiplying here, I'll do something a little bit different. We'll see that next.