Polar Coordinate System - Online Tutor, Practice Problems & Exam Prep

In the polar coordinate system, points are represented as ordered pairs (r, θ), where r is the distance from the pole and θ is the angle from the polar axis. To plot points, locate θ first, then count r units from the pole. Negative r values indicate counting in the opposite direction. Multiple ordered pairs can represent the same point by adding or subtracting multiples of 2π to θ or changing r to negative and adding π to θ. Understanding these concepts is crucial for mastering polar coordinates and their applications in various mathematical contexts.

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, 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. We've been working with circles and angles a lot throughout this course, and polar coordinates 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 point (0,0). Then, when plotting an ordered pair, I would go x units over and y units up or down, like here we have the point (3, 2). So how does this work in polar coordinates? Well, remember that I said 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=0. As we get further away from that pole, r increases. Here r would be 1, here r would be 2, 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 we think of as being our positive x-axis. Now, we're going to measure theta counterclockwise from this polar axis the same way that we would on the unit circle. So if I go π2 radians 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 θ=π2.

Now, let's take a look at this point here. We see that this point is 1, 2, 3, 4, 5 units away from that center pole. So I have an r value of 5. Then this is π6 radians away from that polar axis, so I have a theta value of π6. And ordered pairs in polar coordinates are always going to be written in this order, r, theta. 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 (4, π3). Here I have an r value of 4, and theta is equal to π3. 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 π3, I want to come over to my polar coordinate system and locate that angle measured from the polar axis, π3 radians. Now once we have located theta, we can then count r units away from the pole. Here, since r is 4, I'm going to count 1, 2, 3, 4 units away from that pole to plot this first point here located at (4, π3).

Now here our 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 (5, -π3). Here we have an r value of 5, and theta is -π3. Again, we want to locate theta first, but here theta is negative. So how are we going to deal 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 -π3, I'm going to measure this angle clockwise from this polar axis. So clockwise from this polar axis, π3 radians, I end up right along this line. Now I can just use that r value to plot this point, counting 5 units away from that center point, my pole, to plot the second point located at (5, -π3).

Now whenever you have a negative value for theta, you can also think of this as being a reflection over the polar axis from which we measure that angle. 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're asked to plot the point (-3, π6). I have an r value of -3, and theta here is π6. Now remember, we want to locate that angle theta first, so here my angle π6 measuring that counterclockwise 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. Typically, if this was a positive 3, 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 1, 2, 3 units away from the pole in that opposite direction for this negative r value in order to plot that final point at (-3, π6). Now, whenever faced with a negative r value, we can also think of this as being a reflection over the pole.

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've first looked at polar coordinates using angle measures given in radians, but you might also see angles given in degrees like we see here with point A at 5, 60 degrees. Now here I have an r value of 5 and a θ value of 60 degrees. Remember, when plotting polar coordinates, 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 degrees or π3 radians. Then I'm going to count out on that line using that r value, in this case, 5, so 1, 2, 3, 4, 5, out to this point, point A.

Now looking at point B, this is located at 390 degrees. So I have an r value of 3, a θ value of 90 degrees. So, remember locating that angle first, which in this case is right here at 90 degrees or π2 radians, and then counting out using that r value 1, 2, 3, here is my point B.

Now let's move on to point C. Now point C is 0, -5π3 radians. So I have an r value of 0 and a θ value of -5π3. So, remember when we're dealing with a negative value of θ, that just tells us that we need to measure our angle clockwise rather than counterclockwise. So here I'm going to locate my angle first, of course, going around clockwise to 5π3 radians, which will land me right back along this line. But here our r value is 0, so I'm actually going to 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 27π3. So I have an r value of 2 and a θ value of 7π3. Now here, if I measure around 7π3 radians, I actually do a complete rotation and then go another π3 radians to land me right back at this same line that point A is located on. Now here, using my radius value, counting out to 2, I end up right here at point D.

So here we've plotted these 4 points in polar coordinates. Let's continue practicing together. Thanks for watching, and let me know if you have any 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, π/6 and 13π/6. These were referred to as coterminal angles, and we found them by adding or subtracting multiples of 2π. Now, the same exact idea actually applies to points on our polar coordinate system. See, one single point can be represented by multiple different ordered pairs. This is actually something that you'll be asked explicitly to find: multiple different ordered pairs that all map back to the same point. This might sound like it's going to be complicated, but all we're going to do here is continue using our knowledge of coterminal 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 going to 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 \( 4\frac{\pi}{6} \). 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 coterminal angles, I know that \( \frac{\pi}{6} \) and 13π/6 are both located at the same position. So what if I instead represented this point with the ordered pair (4, 13π/6)? Now if I locate this angle 13π/6 I know that I'm going to end up right along that line. Then if I count 4 units out, I do end up right back at the same point. So these are 2 different ordered pairs both located at the exact same point. And I can continue to find even more ordered pairs by going more rotations around, just adding multiples of \( 2\pi \) the same way that we did with coterminal angles.

So, for some point in polar coordinates \( (r, \theta) \), it will be located at the exact same point as \( (r, \theta \pm 2\pi n) \). 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 angle \( \theta \) 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 \( 2\pi \) 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 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 \( 2\pi \) from our angle \( \theta \) in order to get multiple ordered pairs. Or if we want to change \( \theta \) by making it negative, all we have to do is add π to our angle. Then again, we can continue adding or subtracting multiples of \( 2\pi \).

So with this in mind, let's go ahead and find even more coordinates for this point, \( 4\frac{\pi}{6} \). Now here, we're actually given specific criteria to find these coordinates. Here, we're told that \( r \) should be greater than or equal to 0. That is to say, it should be positive, as it was in our original point. So our original point \( 4\frac{\pi}{6} \), I'm going to keep \( r \) that same positive 4. So \( r \) is 4. What should \( \theta \) be? Well, here we're told that \( \theta \) should be between -2π and 0. And remember that if we keep \( \theta \) the same or if we keep \( r \) the same, we can take \( \theta \) and add or subtract multiples of \( 2\pi \). So if I take my angle \( \frac{\pi}{6} \) and I subtract \( 2\pi \) in order to give me -11π/6, this is my new angle \( \theta \) within the specified interval. So here I have my new ordered pair (4, -11π/6). 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 here, measuring to my angle -11π/6 clockwise from that polar axis, I end up right along this line. Then if I count 4 units out since my \( r \) value is still 4, 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 0. That is to say, it should be negative. So here I'm going to make my \( r \) value negative 4. Then I want \( \theta \) to be between 0 and \( 2\pi \). Now looking at my original angle, \( \frac{\pi}{6} \), I know that this is already within my specified interval. So what if I just kept it the same? So here I have my point (-4, \( \frac{\pi}{6} \)). Now we want this to be located at the same point, so let's verify that that's true. Now coming to \( \frac{\pi}{6} \) 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 3. Now this is definitely not at the same point as my original ordered pair \( 4\frac{\pi}{6} \). 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 \( \frac{\pi}{6} \) and add \( \pi \) to it in order to give me 7π/6, that gives me the ordered pair -4, 7π/6. Now locating 7π/6 radians 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, 4 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 π 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 \( 2\pi \) 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 (-2, \( \frac{5\pi}{3} \)), and then find three sets of coordinates for this point. We're not given any criteria, so we have an endless amount of possibilities in finding these other sets of coordinates. Let's start by plotting this point, (-2, \( \frac{5\pi}{3} \)). Remember when plotting in polar coordinates, we want to locate our angle \(\theta\) first, which in this case is \( \frac{5\pi}{3} \). 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 be counting out in the opposite direction. So counting 2, because it is negative 2 right here, gives me my point at \(-\frac{25\pi}{3}\). Now from here, we can proceed in finding different sets of coordinates for this point. My original point is \(-\frac{25\pi}{3}\). To come up with some different sets of coordinates, remember that we can add multiples of \(2\pi\) to our angle, and it will still be located at the same point, just with different numbers of rotations around. If I want to keep my \(r\) value the same, negative 2, and just take my angle \( \frac{5\pi}{3} \) and add \(2\pi\) to it, giving me \( \frac{11\pi}{3} \), this will be located at the exact same point: (-2, \( \frac{11\pi}{3} \)). This gives me my first set of coordinates still located at the same point.

Let's keep the \(r\) value the same. What else could I do to my angle here so that it's still located at the same point? Well, I could just add another multiple of \(2\pi\). So if I take \( \frac{11\pi}{3} \) and add another multiple of \(2\pi\), this will give me \( \frac{17\pi}{3} \), and it will still be located at the same angle, just having gone another rotation around. Another set of coordinates here could be (-2, \( \frac{17\pi}{3} \)), and it's still located at the exact same point.

Now let's mix it up 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 2, this will make it positive 2. Here my \(r\) value has turned positive. So what do I need to do to my \(\theta\) value to make sure that this is still located at the same point? 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 \( \frac{5\pi}{3} \), and add \(\pi\) to it. Adding \(\pi\) to \( \frac{5\pi}{3} \) gives me \( \frac{8\pi}{3} \), providing another set of coordinates located at the same point, (2, \( \frac{8\pi}{3} \)). Now I have three different sets of coordinates all located at the same point.

If you tried this on your own, you could have gotten different answers than me. That's totally okay because there are an infinite number of different coordinates that we could come up with. Be sure to double-check by plotting your points; 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. $(3,\frac{\pi}{6})$

Here’s what students ask on this topic:

What are polar coordinates and how do they differ from Cartesian coordinates?

Polar coordinates represent points using a distance from a reference point (the pole) and an angle from a reference direction (the polar axis). They are written as (r, θ), where r is the radius and θ is the angle. In contrast, Cartesian coordinates use perpendicular axes (x and y) to define points as (x, y). The key difference is that polar coordinates use a radial distance and angle, while Cartesian coordinates use horizontal and vertical distances.

How do you plot a point in the polar coordinate system?

To plot a point in the polar coordinate system, follow these steps: First, locate the angle θ on the polar grid, measured counterclockwise from the polar axis. Next, from the pole (origin), move r units along the direction of θ. If r is positive, move outward; if r is negative, move in the opposite direction. For example, to plot (4, π/3), find π/3 on the grid and move 4 units outward from the pole.

How can multiple ordered pairs represent the same point in polar coordinates?

In polar coordinates, a single point can have multiple representations. This is achieved by adding or subtracting multiples of 2π to the angle θ, or by changing the sign of r and adding π to θ. For example, the point (4, π/6) can also be represented as (4, 13π/6) or (-4, 7π/6). These transformations leverage coterminal angles and reflections to map back to the same point.

What is the significance of negative r values in polar coordinates?

Negative r values in polar coordinates indicate that the point is located in the direction opposite to the angle θ. To plot a point with a negative r, first locate the angle θ, then move |r| units in the direction opposite to θ. For example, to plot (-3, π/6), find π/6 on the grid and move 3 units in the opposite direction. This effectively reflects the point over the pole.

How do you convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), use the following formulas: r = √(x² + y²) and θ = arctan(y/x). The radius r is the distance from the origin to the point, and θ is the angle formed with the positive x-axis. Ensure to adjust θ based on the quadrant of the point. For example, the Cartesian point (3, 4) converts to polar coordinates as (5, arctan(4/3)).