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 5, negative pi over 3 and then identify which of the following sets of coordinates is at the same point. So let's start by plotting our point, 5 negative pi over 3. 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 my angle clockwise rather than counterclockwise. So looking at my graph here going clockwise, pi over 3 radians, I know that my point should be right along this line. Now since my r value is 5, I wanna count out 1, 2, 3, 4, 5, and end up with my point right here at 5 negative pi over 3. 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 5 negative pi over 3. 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 3, 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 5, pi over 3. 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 3 radians and count negative 5, I'm going to end up here in quadrant 3, 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 3. So, again, we have this negative r value, and we have 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 2 pi over 3, 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, 5, and I will end up right back at the same place as my original point. So this is my solution. Negative 5 2 pi over 3 is located at the same point as 5 negative pi over 3. 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 5, 5, pi over 3. So let's first locate 5 pi over 3. 5 pi over 3 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 2. 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 5, 2 pi over 3, a different set of coordinates for my same point 5, negative pi over 3. 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 )

15. Polar Equations

Polar Coordinate System

Precalculus

15. Polar Equations

Polar Coordinate System

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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 that the given point (5, -π/3) is in polar coordinates, where 5 is the radius (r) and -π/3 is the angle (θ) in radians.

Step 2: Convert the polar coordinates to Cartesian coordinates using the formulas x = r * cos(θ) and y = r * sin(θ). This will help visualize the point on the Cartesian plane.

Step 3: Recognize that polar coordinates can have multiple representations. A point (r, θ) can also be represented as (-r, θ + π) because adding π to the angle and negating the radius will point in the same direction.

Step 4: Apply the concept from Step 3 to the given point (5, -π/3). By converting it to (-5, -π/3 + π), we get the equivalent point (-5, 2π/3).

Step 5: Compare the converted point (-5, 2π/3) with the given options to identify that it matches the option (-5, 2π/3), confirming it is the same point.

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