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CCSS.Math:

let's say that I have the complex number Z and in rectangular form we can write it as negative 3 plus negative 3 plus 2i so first let's think about where this is on the complex plane so this is our imaginary axis imaginary axis and that is our real axis real axis and let's see the real part is negative 3 so we could go 1 2 3 to the left of the origin let me space that out a little bit more evenly so 1 2 3 to the left of the origin so negative 3 and then we have 2 I so we're going to go up 2 so up at 2 in the imaginary direction so 1 2 just like that and so Z is real part negative 3 imaginary part 2 Z would be right over here in the complex plane so that is Z right over there now what I want to think about it 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 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 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 that 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 relation ship between our 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 that 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 into 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 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 this line intersects the unit circle well this is forming an angle of theta with the 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 well how would what would the horizontal and vertical coordinates of this point be it we obviously know they're negative three and they're two but what would they be in terms of cosine theta and sine of theta well look this point right over here is a radius of one away from the origin so this distance right over here that distance right over here is one but now we are are away from the origin where R times as far so for R times as far in that direction that 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 all of we're going to scale everything by R so the horizontal the horizontal coordinate of this point right over here instead of being cosine of theta it's 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 our where our times is far so this point right over here is going to be our sine theta and we already know that that's equal to 2r sine theta so given that can we now figure out what R and theta are so let's first let's first think about let's first think about figuring out what theta is so to do that let's think about let's think about some of our our trig functions so one 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 is equal to sine of theta over cosine of theta over cosine of theta but we could also multiply the numerator denominator here by r that won't change the value so that's the same thing as R sine theta over R cosine of theta and we know R sine theta is going to be equal to two and we know that our cosine of theta is negative three so this whole thing is going to be negative two thirds another way of thinking about it is the tangent of this state of tangent of theta is going to be the same thing as the slope of this line right over here and what's 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 all right so you can say positive through 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 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 two thirds of this thing right over here so let me copy and paste that and we could 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 2/3 which it gets us negative 0.5 8800 on and on and on so if we round to the nearest hundredth of a Radian it'd be negative 0.5 9 so this is this is approximately negative 0.5 9 now is this the right angle is this the theta is this the theta that we are looking for well this theta negative 0.5 9 that's going to get us over here that's this angle right over here that's what the inverse tangent that's what the inverse tangent gave us and it makes sense because this 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 halfway around the circle so if we think in terms of radians it's going to be this angle plus it's going to be this angle plus pi radians so the theta that we're looking for isn't isn't zero it 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 let me say approximately just because I have a rounded right over here so let's let's get the calculator out so let's take our previous response and that's just the previous answer plus PI plus PI because we're going to go in the opposite direction so that gives us two point a 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 that this angle right over here is two point five five radians well this angle if we go straight up that's PI over two radians and pi is 3.14 so that's going to be one point five seven right one point something like that and this is PI rating so that's three point one four five one five nine keep going on and on and on so two point five five 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 could set up a right triangle a right triangle we know that this distance this distance right over here is two because this distance right over here is two and we know that this distance 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 is equal to two squared plus three 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 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 our theta so now we can say that Z is equal to instead of negative three we can write it as R cosine theta so we can rewrite it as square root of 13 square root of 13 times the cosine times the cosine of theta which is we're saying theta is roughly two point five five which we round to the nearest hundredths plus plus two we already know is our sine of theta so R is square root of 13 square root of 13 and then sine sine of theta two point five five and then all of this we could say or it all of this is going to be multiplied times I and if we like to simplify it a little bit or to kind of really make make clear what the R is we could factor it out so we could say that Z is equal to square root of 13 square root of 13 times cosine of 2 point five five and once again two point five 5s in approximation so maybe I should say approximately equal to cosine of two point five five cosine of two point five five plus plus I'll throw the I in front 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 thirteen so this this essentially makes the polar it makes this it makes it clear 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