Plot the point (5,−π3)(5,-\frac{\pi}{3})(5,−3π), then identify which of the following sets of coordinates is the same point.

( − 5 , 2 π 3 ) (-5,\frac{2\pi}{3}) ( − 5 , 3 2 π )

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

In this problem we're asked to plot the point five negative pi over three and then identify which of the following sets of coordinates is at the same point. So let's start by plotting our point five negative pi over three. Now we're going to locate theta first and here since theta is a negative value that just tells me that I'm going to measure my angle clockwise rather than counterclockwise. So looking at my graph here going clockwise pi over three radians, I know that my point should be right along this line. Now since my r value is five, I want to count out one, two, three, four, five and end up with my point right here at five, negative pi over three. So now let's take a look at all of these different sets of coordinates to determine which is actually located at the same point. Now looking at that first coordinate pair, negative five, negative pi over three. Let's think about this for a second. We have the exact same angle as our original point, but now our r value is negative. So is this going to end up at the same point? Well, let's take a look. Now if I go to my same angle negative pi over three but then I have a negative r value, I'm going to have to count out in the opposite direction. So this will definitely not be located at the same point, so this is not my answer here. Now let's move on to our next coordinate pair, negative five, pi over three. Here I do have a negative r value and I have a different angle as well, so maybe this is the answer. But let's go ahead and locate this point on our graph. Now if I go to pi over three radians and count negative five, I'm going to end up here in quadrant three, which is definitely not where my original point is. So this, again, is not my answer here. This is not located at the same point. Let's look at our next one. Negative five, two pi over three. So again, we have this negative r value and we have a different value of theta. But is this the value of theta that will end up at the same spot? Let's see. While locating that angle two pi over three, I should be along this line, but I have that negative r value. So that tells me I need to count back in the opposite direction five and I will end up right back at the same place as my original point. So this is my solution. Negative five two pi over three is located at the same point as five, negative pi over three. It's a different set of coordinates for the same point and this makes sense because our r value is negative here and looking at the difference between these two theta values, I just added pi to it, which is exactly how we come up with a different point given a negative r value. So here I have my solution, but let's double check our last point here and make sure that it's not also located at the same point. This point is negative five five pi over three. So let's first locate five pi over three. Five pi over three is at the exact same spot, the same angle that my original point is. But here I still have a negative r value. So that tells me I'm going to have to count out in the opposite direction here, which is going to land me somewhere in quadrant two. So this is not going to work either. This is not a set of coordinates for the same point. So here my solution is negative five, two pi over three, a different set of coordinates for my same point five, negative pi over three. Let Let me know if you have any questions. Thanks for watching.

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

( − 5 , 2 π 3 ) (-5,\frac{2\pi}{3}) ( − 5 , 3 2 π )

( − 5 , − π 3 ) (-5,-\frac{\pi}{3}) ( − 5 , − 3 π )

( − 5 , π 3 ) (-5,\frac{\pi}{3}) ( − 5 , 3 π )

( − 5 , 5 π 3 ) (-5,\frac{5\pi}{3}) ( − 5 , 3 5 π )

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

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

$(-5,-\frac{\pi}{3})$

$(-5,\frac{\pi}{3})$

$(-5,\frac{2\pi}{3})$

$(-5,\frac{5\pi}{3})$

Verified step by step guidance

Step 1: Understand the polar coordinate system. A point in polar coordinates is represented as (r, θ), where r is the radial distance from the origin and θ is the angle measured counterclockwise from the positive x-axis.

Step 2: Analyze the given point (5, -π/3). Here, r = 5 indicates the distance from the origin, and θ = -π/3 represents the angle measured clockwise (negative direction) from the positive x-axis.

Step 3: Convert the angle -π/3 into its equivalent positive angle by adding 2π (a full rotation). This gives θ = 2π - π/3 = 5π/3. This step ensures the angle is expressed in the standard range [0, 2π).

Step 4: Reflect the point across the origin to find equivalent coordinates with a negative radius. For a point (r, θ), the equivalent point with negative radius is (-r, θ + π). Here, (-5, θ + π) = (-5, 5π/3 + π). Simplify the angle: 5π/3 + π = 2π + 2π/3 = 2π/3.

Step 5: Compare the transformed coordinates (-5, 2π/3) with the given options. The correct answer is the set of coordinates that matches (-5, 2π/3).

5:32m

Watch next

Master Intro to Polar Coordinates with a bite sized video explanation from Patrick

Start learning

Polar Coordinates practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice