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 . 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 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 the same way that we did with coterminal angles.
So, for some point in polar coordinates , it will be located at the exact same point as . This is not the only way that we can go about finding multiple ordered pairs for the same point because we can also change by making it negative. Now if I make negative, that means that I need to take my angle and add 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 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 and we're asked to find multiple ordered pairs that are all located at this point, we can keep the same. And if we keep the same, we can simply add or subtract multiples of from our angle in order to get multiple ordered pairs. Or if we want to change 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 .
So with this in mind, let's go ahead and find even more coordinates for this point, . Now here, we're actually given specific criteria to find these coordinates. Here, we're told that 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 , I'm going to keep that same positive 4. So is 4. What should be? Well, here we're told that should be between -2π and 0. And remember that if we keep the same or if we keep the same, we can take and add or subtract multiples of . So if I take my angle and I subtract in order to give me -11π/6, this is my new angle 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 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 should be less than or equal to 0. That is to say, it should be negative. So here I'm going to make my value negative 4. Then I want to be between 0 and . Now looking at my original angle, , 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, ). Now we want this to be located at the same point, so let's verify that that's true. Now coming to on my graph here, I know that I end up on this line. But since that 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 . So what exactly happened? Well, remember that whenever we change to be negative, we have to add to our original angle . So let's see what happens if I do that. If I take my original angle and add 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 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 . 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 and add or subtract as many multiples of 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.