Sal rewrites the complex number -3+2i (which is given in rectangular form) in its polar form. Created by Sal Khan.
Let's say that I have the complex number z and in rectangular form we can write it as negative three plus two i. So first let's think about where this is on the complex plane. So, this is our imaginary axis and that is our real axis. And let's see, the real part is negative three, so we could go one, two, three to the left of the origin. Let me space that out a little bit more evenly. So one, two, three to the left of the origin. So negative three. And then we have two i's. We're going to go up two. So up two in the imaginary direction. So one, two just like that. So z is real part negative three, imaginary part two. z would be right over here in the complex plane. So that is z right over there. Now what I want to think about are other ways to essentially specify the location of z. And in particular, instead of giving the real and imaginary parts, essentially the coordinates here, let's think about giving a direction and a distance to get to z. So for example we could give the distance from the origin to z, so let's call this distance r, but that distance by itself isn't enough to say where z is. You have to say in what direction do you have to go a distance of r to get to z. and so to specify the direction, we will have this angle, theta, in radians, this angle between the positive real axis and this line right over here, this line or the segment that connects the origin and z. So if someone gave you this angle and this distance, then you could get to z. Now what I want you to do right now is pause this video and see if you can find a relationship between r theta and negative three and two. In fact, given this complex number in rectangular form, can you figure out what r and theta are? Well let's think through this a little bit, and to help us let's remind ourselves of the unit circle definition of trig functions because we are going to use some trig functions to relate r, theta, and two, and negative three. So I'm going to construct a unit circle here. So construct a unit circle. So this right here's a unit circle, a circle of radius one. So construct a unit circle. That's a unit circle. So by definition what are the horizontal and vertical coordinates of this point right over here where this line intersects the unit circle? Well, this is forming an angle of theta with a positive real axis and so the horizontal coordinate over here by definition is going to be cosine of theta. Cosine of theta. That's the unit circle definition of cosine of theta, and the vertical coordinate is going to be sine of theta. Sine of theta. And so what would the horizontal and vertical coordinates of this point be? We obviously know they're negative three and two, but what would they be in terms of cosine theta and sine theta? Well look, this point right over here is a radius of one away from the origin. So this distance right over here is one, but now we are r away from the origin. We're r times as far. So if we're r times as far in that direction, then we're going to be r times as far in the vertical direction and r times as far in the horizontal direction. We're going to scale everything by r. So the horizontal coordinate of this point right here instead of being cosine of theta is going to be r times cosine of theta. So this point right over here, which we know is negative three, is going to be equal to r cosine of theta, and by the same logic, this point over here, the vertical coordinate, we're going to scale up sine theta by r, we're r times as far. So this point right over here is going to be r sine theta and we already know that that's equal to two. r sine theta. So given that, can we now figure out what r and theta are? So let's first think about figuring out what theta is. So to do that, let's think about some of our trig functions. So one trig function that involves sine theta and cosine theta is tangent theta. So for example we could say tangent theta, tangent of our angle, tangent of theta, is equal to sine of theta over cosine of theta. We could also multiply the numerator and denominator here by r. That won't change the value. So that's the same thing as r sine theta over r cosine theta and we know r sine theta is going to be equal to two and we know that r cosine theta is negative three. So this whole thing is going to be negative 2/3. Another way of thinking about it is the tangent of theta is going to be the same thing as the slope of this line right over here. And what is the slope of that line? Well if you start at z and you want to go to the origin you're going to go positive three in the x direction and then you're going to go negative two, or I should say positive three in the horizontal direction, and then you go negative two in the vertical direction. So the slope is your change in vertical over change in horizontal. It's negative two over three. But now we can use this to solve for theta. To solve for theta, we just take the inverse tangent of both sides and we get theta is equal to inverse tangent of negative 2/3, of this thing right over here. So we copy and paste that and we can get our calculator out to figure out what this actually is. So let's turn it on. Let me make sure that I am in radian mode. I am. and so I can take the inverse tangent of negative two divided by three, which gets us negative point five eight eight zero zero, on and on and on. So if we round to the nearest hundredth of a radian, it would be negative point five nine. So this is approximately negative zero point five nine. Now is this the right angle? Is this the theta that we are looking for? Well this theta, negative zero point five nine, that's going to get us over here. That's this angle right over here. That's what the inverse tangent gave us, and it makes sense because this ray is a continuation of this ray right over here. Together they would form a line. It has the same slope. But that's not the theta that we are looking for. The theta that we are looking for is going in the opposite direction. It's half way around the circle. So if we think in terms of radians, it's going to be this angle plus pi radians. So the theta that we're looking for isn't going to be what our calculator gives us for this. It's going to be this value plus pi. So theta, the theta that we care about, is going to be negative zero point five nine, roughly, plus pi. I'm going to say approximately just because I have rounded right over here. So let's get the calculator out. So let's take our previous response, and that's just the previous answer, plus pi because we're going to go in the opposite direction. So that gives us roughly two point five five radians. So this is approximately two point five five radians is our claim for what theta is. Now does that make sense, that this angle is two point five five radians? Well, this angle if we go straight up, that's pi over two radians, and pi is three point one four, so that's going to be one point five seven, something like that, and this is pi radians, so that's three point one four one five nine, keep going on and on and on. So two point five nine is indeed in the right quadrant. So that is right. This is going to be an angle of two point five five radians. Now we just have to figure out the length, what r is. And that we can just use the Pythagorean theorem for. We can set up a right triangle. We know that this distance right over here is two because this distance right over here is two, and we know that this distance is, the coordinate here is negative three, but the distance here is just three. So we know from the Pythagorean theorem that r squared is equal to two squared plus three squared or that r is equal to the square root of four plus nine, which is 13. And so there we have it. We have negative three, so we know r is equal to the square root of 13 and theta is two point five five radians. So using that information, we can now rewrite z in terms of r and and our theta. So now we can say that z is equal to, instead of negative 3, we can write it as r cosine theta, so we can write it as the square root of thirteen times the cosine of theta, which is, we're saying theta is roughly two point five five if we round to the nearest hundredth, plus two, we already know is r sine of theta is the square root of thirteen, and then sine of theta, two point five five, and then all of this is going to be multiplied times i, and if we like to simplify it a little bit, or kind of really make clear what the r is here, we could factor it out. So we could say that z is equal to the square root of 13 times cosine of two point five five, and once again two point five five is an approximation, so maybe I should say approximately equal to, cosine of two point five five plus i times sine of two point five five. And so when I rewrote z in this format right over here, it makes it much clearer what the direction I have to go in is, the direction is two point five five radians counterclockwise from the positive real axis. It also says how far I need to go, I need to go square root of 13. This essentially makes the polar, it makes it clearer how we get there in kind of a more, I guess you could say, polar mindset, and that's why this form of the complex number, writing it this way is called rectangular form, while writing it this way is called polar form.