Polar Coordinate System - Video Tutorials & Practice Problems

Intro to Polar Coordinates

Hey, everyone at this point, we're used to plotting points using a rectangular or Cartesian coordinate system where we're given ordered pairs X comma Y. But here we're going to take a look at a different coordinate system called the polar coordinate system where instead of X and Y, we are now going to plot points in terms of R and theta. Now we've been working with circles and angles a lot throughout this course and polar ordinates are no different. So here I'm going to walk you through exactly what polar coordinates are using a lot of what we've already learned. And then we'll practice plotting some points on our polar coordinate system together. So let's go ahead and get started. Now, when working in rectangular coordinates, we know that we have our X and our Y axis and our origin located at the 0.00. Then when plotting an ordered pair, I would go X units over and Y units up or down. Like here we have the 0.3 comma two. So how does this work in polar coordinates or remember that I said that we're going to work with polar coordinates in terms of R and theta now R is going to be the distance from the pole, which is what we think of as being our origin. But here in polar coordinates, this center point is referred to as the pole located at R equals zero. Then as we get further and further away from that pole R increases. So here R would be one, here R would be two and so on R is the radius of each of these circles. Then we have theta and theta is going to be the angle from the polar axis, which is what we think of as being our positive X axis. Now we're going to measure theta counter clockwise from this polar axis, the same way that we would on the unit circle. So if I go pi over two radiance away from that polar axis, we reach what we think of as being our positive Y axis. But here in polar coordinates, this is just the line theta equals pi over two. So what about ordered pairs? Well, let's take a look at this point here we see that this point is 12345 units away from that center pole. So I have an R value of five, then this is pi over six radiance away from that polar axis. So I have a theta value of pi over six and ordered pairs. And polar coordinates are always going to be written in this order R theta. So here that we see in rectangular coordinates. So we would go over and up or down based on our X and Y values. And now in polar coordinates, we're instead going to go around and out based on our values for theta and R. Now, here we just identified an ordered pair. But more often you'll be given an ordered pair and asked to plot it on your polar coordinate system. So let's go ahead and get some practice with that together. Now, the first point that we're asked to plot here is four comma pi over three. So I have an R value of four and theta is equal to pi over three. Now, when plotting points in polar coordinates, even though R comes first in that ordered pair, we actually want to locate theta first. So here, since theta is pi over three, I want to come over to my polar coordinate system and locate that angle measured from the polar axis pi over three radiance. Now, once we have located theta, we can then count our units away from the pole here. Since RS four, I'm going to count 12, 34 units away from that pole in order to plot this first point here located at four comma pi over three. Now here are values for R and theta were both positive, but that won't always be the case. So let's go ahead and take a look at another point here. Now here the second point that we're asked to plot is five comma negative pi over three. So here we have an R value of five and theta is negative pi over three. Now again, we want to locate theta first, but here theta is negative. So how are we going to do with that? Well, remember that when working with our unit circle, whenever we were faced with a negative angle, we would simply measure that angle clockwise instead of counterclockwise. And we're going to do the exact same thing here. So here, since I have a theta value of a negative pi over three, I'm going to measure this angle clockwise from this polar axis. So clockwise from this polar axis, pi over three radiance, I end up right along this line. Now, I can just use that R value to plot this point counting five units away from that center point. My pull in order to plot the second point located at five negative pi over three. Now, whenever you have a negative value for theta, you can also think of this as being a reflection over the polar axis where we measure that angle from. Now here, our R value was still positive. But again, this won't always be the case. So let's take a look at one final point here. Here we asked to plot the point negative three pi over six. So I have an R value of negative three and theta here is pi over six. Now, remember we want to locate that angle theta first. So here my angle pi over six measuring that counter clockwise from that polar axis. I am along this line here. Now, since this R value is negative, what exactly are we going to do here? Well, whenever we're faced with a negative R value, we're simply going to count from our pole in the opposite direction. So typically, if this was a positive three, I would go ahead and start counting out this way towards my angle. But now I'm going to count in the opposite direction because this R value is negative. So I'm going to count 123 units away from the pole in that opposite direction for this negative R value in order to plot that final point at negative three pi over six. Now, whenever faced with a negative R value, we can also think of this as being a reflection over the pole. Now, as we saw here, it's going to be really important to pay attention to the signs of both your R and theta values. So let's keep this in mind as we continue to practice. Thanks for watching and I'll see you in the next one.

Intro to Polar Coordinates Example 1

Hey, everyone, let's plot these points in polar coordinates together. Now we first looked at polar coordinates using angle measures given in radiance. But you might also see angles given in degrees like we see here with point A at 560 degrees. Now, here I have an R value of five and A theta value of 60 degrees. Remember when plotting polar coordinates. So we want to locate our angle first. So coming over here to my polar coordinate system, I'm going to first locate my angle 60 to degrees or pi over three radiance. Then I'm going to count out on that line using that R value in this case five. So 12345 out to this point point A. Now looking at point B, this is located at 390 degrees. So I have an R value three A theta value of 90 degrees. So remember locating that angle first, which in this case is right here at 90 degrees or pi over two radiance. And then counting out using R value 123, here is my point B. Now let's move on to point C. Now point C is zero negative five pi over three radiance. So I have an R value of zero and a theta value of negative five pi over three. So remember when we're dealing with a negative value of theta that just tells us that we need to measure our ankle clockwise rather than counterclockwise. So here I'm going to locate my angle. First, of course, go around clockwise to five pi over three radians which will land me right back along this line. But here our R value is zero. So I'm actually gonna stay right there at the pole or what we think of as our origin on an XY coordinate system. And this is my point C. Now moving on to our final point here at 27 pi over three. So I have an R value of two and A theta value of seven pi over three. Now here, if I measure around seven pi over three radiant, I actually do a complete rotation and then go another pi over three radiant to land me right back at this same line. That point A is located on. Now here using my radius value counting out to two. I end up right here at point D. So here we've plotted these four points in polar coordinates. Let's continue practicing together. Thanks for watching and let me know if you have questions.

Plot the point on the polar coordinate system.

$(5,210°)$

• A.
• B.
• C.
• D.

Plot the point on the polar coordinate system.

$(-3,-90°)$

• A.
• B.
• C.
• D.

Plot the point on the polar coordinate system.

$(6,-\frac{11\pi}{6})$

• A.
• B.
• C.
• D.

Plot the point on the polar coordinate system.

$(-2,\frac{2\pi}{3})$

• A.
• B.
• C.
• D.

Determining Different Coordinates for the Same Point

Hey, everyone in working with the unit circle, we found that multiple different angle measures could all be located at the exact same position like say pi over six and 13 pi over six. Now, these were referred to as co terminal angles and we found them by adding or subtracting multiples of two pi. Now, the same exact idea actually applies to points on our polar coordinate system. See one single point can be represented by multiple different or repairs. And this is actually something that you'll be asked explicitly to find multiple different order pairs that haul map back to the same point. Now, this might sound like it's going to be complicated, but all we're going to do here is continue using our knowledge of co terminal angles along with what we know about polar coordinates in order to find an infinite number of ordered pairs all located right at the same point. Now here, I'm gonna show you exactly how to find these different ordered pairs. So let's go ahead and get started. Now taking a look at our graph here, I see that I have the point for pi over six. Now looking at this point? What if I wanted to represent it with a different ordered pair? How could I go about that? Well, from my knowledge of co terminal angles, I know that pi over six and 13 pi over six are both located at the same position. So what if I instead represent at this point with the ordered pair for 13 pi over six? Now if I look this angle, 13 pi over six, I know that I'm gonna end up right along that line. Then if I count four units out, I do end up right back at the same point. So these are two different ordered pairs, both located at the exact same point. And I could continue to find even more ordered pairs by going more rotations around just adding multiples of two pi the same way that we did with co terminal angles. So for some, for some point in polar coordinates, R theta, it will be located at the exact same point as R theta plus or minus two pi N. Now this is not the only way that we can go about finding multiple ordered pairs for the same point because we can also change R by making it negative. Now, if I make R negative, that means that I need to take my ankle theta and add pi to it in order to ensure that it's located at the same position as my original point, then I can continue to add or subtract multiples of two pi as we did before. Now seeing this all written out can look a little bit complicated, but it's actually rather simple. So let's think about this. If we're given a point R theta and we're asked to find a multiple ordered pairs that are all located at this point, we can keep R the same. And if we keep R the same, we can simply add or subtract multiples of two pi from our angle theta in order to get multiple ordered pairs. Or if we want to change the by making it negative, all we have to do is add pi to our angle. Then again, we can continue adding or subtracting multiples of two pi. So with this in mind, let's go ahead and find even more coordinates for this 0.4 pi over six. Now, here we're actually given specific criteria to find these coordinates. So let's take a look here here. We're told that R should be greater than, or equal to zero. That is to say it should be positive as it was in our original point. So our original 0.4 pi over six, I'm gonna keep R that same positive four. So R is four. What should the be? Well, here, we're told that the should be in between negative two pi and zero. And remember that if we keep the, the same or if we keep R the same, we can take theta and add or subtract multiples of two pi. So if I take my angle pi over six, I subtract two pi in order to give me negative 11 pi over six, this is my new angle theta within the specified interval. So here I have my new ordered pair or negative 11 pi over six. Now this should be located at the exact same point as my original point. So let's verify that that's true. Coming back up to my graph year measuring to my angle negative 11 pi over six, clockwise from that polar axis, I end up right along this line. Then if I count four units out since my R value is still four, I do end up right back at that same point. So let's find one final set of coordinates here. Now, here our criteria tells us that R should be less than or equal to zero. That is to say it should be negative. So here I'm going to make my R value negative four. Then I want the to be in between zero and two pi now looking at my original angle pi over six, I know that this is already within my spec interval. So what if I just kept it the same? So here I have my point negative four pi over six. Now we want this to be located at the same point. So let's verify that that's true. Now, coming to ne to pi over six on my graph here, I know that I end up on this line. But since that R value is negative, I would actually be counting out in the opposite direction from the pole. So I would count out in this direction and end up right about here in quadrant three. Now this is definitely not at the same point as my original ordered pair for pi over six. So what exactly happened? Well, remember that whenever we change R to be negative, we have to add pi to our original angle theta. So let's see what happens if I do that. If I take my original angle pi over six and add pi to it in order to give me seven pi over six, that gives me the ordered pair negative 47 pi over six. Now locating seven pi over six radiance on my graph here I end up along this line. But since this R value is negative, I'm going to count out in the opposite direction from the pole in this direction four units. And now I do end right back up at that same point. So we need to be really careful and remember to add our factor of pi if we're changing the sign of R. Now, here I have a bunch of different coordinates that are all located at the exact same point. Now, whenever you're asked to find ordered pairs, you may not always be given specific criteria to find these ordered pairs and that's totally fine. You can continue to find them in this way. Or you can just take your angle theta and add or subtract as many multiples of two pie as you want to get more angles. Now that we know how to find different ordered pairs that all map back to the same point. Let's continue practicing together. Thanks for watching and I'll see you in the next one.

Plot the point $(3,\frac{\pi}{2})$ & find another set of coordinates, $(r,θ)$, for this point, where:

(A) $r≥0,2π≤θ≤4π$,

(B) $r≥0,-2π≤θ≤0$,

(C) $r≤0,0≤θ≤2π$.

• A. $(3,\frac{5\pi}{2}),(-3,-\frac{3\pi}{2}),(-3,\frac{3\pi}{2})$
• B. $(3,\frac{5\pi}{2}),(3,-\frac{3\pi}{2}),(-3,\frac{3\pi}{2})$
• C. $(-3,\frac{5\pi}{2}),(-3,-\frac{3\pi}{2}),(-3,\frac{\pi}{2})$
• D. $(3,\frac{5\pi}{2}),(3,-\frac{3\pi}{2}),(-3,\frac{\pi}{2})$

Determining Different Coordinates for the Same Point Example 2

Hey, everyone in this problem, we're asked to plot the point negative 25 pi over three and then find three sets of coordinates for this point. Now we're not given any criteria here. So we really have an endless amount of possibilities in finding these other sets of coordinates. But let's start by of course, plotting this point at negative 25 pi over three. Remember when plotting in polar coordinates, we want to locate our angle theta first which in this case is five pi over three. So coming to my graph here, I know that my point should be along this line. But since my R value is negative, that tells me that I should actually be counting out in the opposite direction. So this way instead of out towards my angle, so counting two because it is negative two right here gives me my point at negative 25 pi over three. Now, from here, we can proceed in finding these different sets of coordinates for this point. Now my original point is negative 25 pi over three. So how can we change this to come up with some different sets of coordinates? Now, remember that whenever we are faced with doing this, we can always just add a bunch of multiples of two pi to our angle and it will still be located at the same point, just different numbers of rotations around. So if I want to keep my R value the same negative two and just take my angle five pi over three and add two pi to it giving me 11 pi over three. This will be located at the exact same point negative 2, 11 pi over three. So this gives my first set of coordinates still located at that same point. So what else should I do? Well, let's still keep that R value the same. What else could I do to my angle here? So that it's still located at that same point? Well, I could just add another multiple of two pi. So if I take that 11 pi over three and I add another multiple of two pi, this will give me 17 pi over three and it will still be located at the same angle just having gone another rotation around. So another set of coordinates here could be negative 2, 17 pi over three and that's still located at the exact same point. Now let's mix it up here and actually change our R value. Remember we can change R to make it negative but here R is already negative. So if I add a negative in front of it negative negative two, this will actually be positive two. So here my R value has turned to positive. So what do I need to do to my theta value to make sure that this is still located at the same point? Remember that whenever we make our R value negative or we add a negative to our R value in this case, making it positive, that tells me that I need to take my original angle. In this case, five pi over three and add pi to it. Now, in adding pi to five pi over three, this gives me a pi over three and this gives me another set of coordinates located at that same 0.28 pi over three. And now I have three different sets of coordinates all located at the same point. Now, if you tried this on your own, you could have gotten some different answers in me and that's totally OK because there are infinite number of different coordinates that we could come up with. So just be sure to double check by plotting your point, they should all be located at the exact same spot. Thanks for watching and I'll see you in the next one.

Plot the point $(5,-\frac{\pi}{3})$, then identify which of the following sets of coordinates is the same point.

• A. $(-5,-\frac{\pi}{3})$
• B. $(-5,\frac{\pi}{3})$
• C. $(-5,\frac{2\pi}{3})$
• D. $(-5,\frac{5\pi}{3})$

Plot the point $(-3,-\frac{\pi}{6})$, then identify which of the following sets of coordinates is the same point.

• A. $(-3,\frac{11\pi}{6})$
• B. $(-3,\frac{5\pi}{6})$
• C. $\left(3,\frac{11\pi}{6}\right)$
• D. $(3,\frac{\pi}{6})$