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 3, negative pi over 6, and then identify which of the following sets of coordinates is the same point. So let's start by plotting this point, negative 3, negative pi over 6. Now locating my angle theta first, because it's negative, I know that I should be going clockwise pi over 6 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, 3 for that negative 3 r value, I end up right here at negative 3 negative pi over 6. 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 3, 11 pi over 6, I see that I still have that same r value of negative 3, but I have a theta value of 11 pi over 6. So let's go ahead and figure out where this is. Now on my graph, 11 pi over 6 puts me right along this line. But since I have that negative r value, I'm going to end up going 3 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 6, comparing it to my original angle, negative pi over 6, I see that I just added 2 pi to that original angle in order to get my new angle, 11 pi over 6. 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 35 pi over 6. So let's first locate our angle 5 pi over 6. Now this is 5 pi over 6, 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 4 counting out 3 for that negative 3. 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, 311pi over 6. 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 6. So let's see what happens here. If I come to my angle 11 pi over 6 and count out 1, 2, 3, I'm going to end up right here at quadrant 4, 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 rvalue 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, 3pi over 6. 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 6 radians and then count out 1, 2, 3, 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 3, 11 pi over 6 is a set of coordinates for the same point as negative 3, negative pi over 6. 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 )

9. Polar Equations

Polar Coordinate System

Trigonometry

9. Polar Equations

Polar Coordinate System

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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 that the point is given in polar coordinates, where the first value is the radius (r) and the second value is the angle (θ) in radians.

Step 2: Plot the point (-3, -\frac{\pi}{6}). The negative radius means the point is in the opposite direction of the angle. Start by plotting the angle -\frac{\pi}{6}, which is equivalent to rotating clockwise by \frac{\pi}{6} radians from the positive x-axis.

Step 3: Since the radius is -3, move 3 units in the opposite direction of the angle -\frac{\pi}{6}. This effectively places the point in the direction of \frac{5\pi}{6} radians, as moving in the opposite direction of -\frac{\pi}{6} is equivalent to moving in the direction of \frac{5\pi}{6}.

Step 4: Convert the angle \frac{5\pi}{6} to an equivalent angle by adding 2\pi (a full rotation) to find other possible representations. \frac{5\pi}{6} + 2\pi = \frac{5\pi}{6} + \frac{12\pi}{6} = \frac{17\pi}{6}.

Step 5: Compare the equivalent angles to the given options. The angle \frac{11\pi}{6} is equivalent to \frac{5\pi}{6} when considering a full rotation, thus the point (-3, \frac{11\pi}{6}) is the same as (-3, -\frac{\pi}{6}).

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