Plot the point (−3,−π6)(-3,-\frac{\pi}{6})(−3,−6π), then identify which of the following sets of coordinates is the same point.

( − 3 , 11 π 6 ) (-3,\frac{11\pi}{6}) ( − 3 , 6 11 π )

Plot the point ( − 3 , − π 6 ) (-3,-\frac{\pi}{6}) ( − 3 , − 6 π ) , then identify which of the following sets of coordinates is the same point.

In this problem, we're asked to plot the point negative three negative pi over six and then identify which of the following sets of coordinates is the same point. So let's start by plotting this point, negative three negative pi over six. Now locating my angle theta first, because it's negative, I know that I should be going clockwise pi over six radians to end me up along this line here. But since my r value is negative, that tells me that I'm going to count out in the opposite direction instead of out towards my angle. So counting that opposite direction three for that negative three r value, I end up right here at negative three negative pi over six. Now that we have our point plotted, let's determine which of these sets of coordinates is the same point. Now looking at that first coordinate pair negative three eleven pi over six, I see that I still have that same r value of negative three, but I have a theta value of 11 pi over six. So let's go ahead and figure out where this is. Now on my graph, 11 pi over six puts me right along this line, but since I have that negative r value, I'm going to end up going three out in that opposite direction, landing me right back at my original point. So this is a different set of coordinates for that same point. Now this makes sense because looking at my angle theta, 11 pi over six, comparing it to my original angle, negative pi over six, I see that I just added two pi to that original angle in order to get my new angle 11 pi over six. Now let's double check all of our other points here to make sure that they're not also located at the same point. Now my next point is negative three five pi over six. So let's first locate our angle five pi over six. Now this is five pi over six so it looks like I might end up being at the same point. But here I have a negative r value so I'm not going to end up in that quadrant. I'm going to end up here in quadrant four counting out three for that negative three. So this is not at the same point, this is not a set of coordinates for my same original point. Now let's look at our next point here, three eleven pi over six. Here it looks like our r value has changed. It is now positive instead of negative and I have an angle of 11 pi over six. So let's see what happens here. If I come to my angle 11 pi over six and count out one, two, three, I'm going to end up right here at quadrant four, which is not where my original point is. So this is not a set of coordinates that represents my same point. Now thinking about this and thinking about what it should be if I change that r value from what it originally was, I should have taken my original angle theta and added pi to get my new value of theta so that this could be located at the same point. But that's not what happened here. So this is not a set of coordinates at the same point. Now let's look at our final coordinate pair here, three pi over six. Now here I did change my r value to be positive instead of negative, and it looks like my theta value is also now positive. So let's see what happens here. Now if I go to pi over six radians and then count out one, two, three, I'm gonna end up right here, which is still not at the same point and this is not another coordinate pair that represents the same point. So my answer here is that negative three eleven pi over six is a set of coordinates for the same point as negative three negative pi over six. Thanks for watching and let me know if you have questions.

Plot the point ( − 3 , − π 6 ) (-3,-\frac{\pi}{6}) ( − 3 , − 6 π ) , then identify which of the following sets of coordinates is the same point.

( − 3 , 11 π 6 ) (-3,\frac{11\pi}{6}) ( − 3 , 6 11 π )

( − 3 , 5 π 6 ) (-3,\frac{5\pi}{6}) ( − 3 , 6 5 π )

( 3 , 11 π 6 ) \left(3,\frac{11\pi}{6}\right) ( 3 , 6 11 π )

( 3 , π 6 ) (3,\frac{\pi}{6}) ( 3 , 6 π )

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

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Multiple Choice

Plot the point $(-3,-\frac{\pi}{6})$ , then identify which of the following sets of coordinates is the same point.

$(-3,\frac{11\pi}{6})$

$(-3,\frac{5\pi}{6})$

$\left(3,\frac{11\pi}{6}\right)$

$(3,\frac{\pi}{6})$

Verified step by step guidance

Step 1: Understand the problem. The point (-3, -π/6) is given in polar coordinates, where -3 is the radius (r) and -π/6 is the angle (θ). The task is to identify an equivalent point from the given options.

Step 2: Recall that polar coordinates are periodic in the angular component. Specifically, angles can be expressed as θ + 2nπ, where n is any integer. This means that -π/6 is equivalent to -π/6 + 2π = 11π/6 in the standard interval [0, 2π).

Step 3: Note that the radius (r) can be negative in polar coordinates. A negative radius means the point is located in the opposite direction of the angle. For example, (-3, -π/6) is equivalent to (3, π/6) because flipping the radius changes the direction by π radians.

Step 4: Compare the transformed coordinates to the given options. (-3, 11π/6) matches the equivalent point derived from the periodicity of the angle, and (3, π/6) matches the equivalent point derived from flipping the radius.

Step 5: Verify the other options. (-3, 5π/6) and (3, 11π/6) do not match the transformations of (-3, -π/6). Thus, the correct equivalent coordinates are (-3, 11π/6).

5:32m

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