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.