Polar Coordinates

Plot the following points, ...

Question

Polar Coordinates

Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.

a. (2, π/2)

Accepted Answer

Hello. In this video, we are going to be plotting the given point with the polar coordinate for pi. And then we want to find all polar coordinates that represent that point. So the given point or the given polar coordinate is going to be 4, i. In order to plot this, let's first compare this to a general polar coordinate. Now, the general polar coordinate is of the form R, theta. Here R represents the radius of the polar coordinate, and theta represents the location of its angle. So here, this polar coordinate is going to have a radius of 4. And the angle that this radius is located at is going to be the angle pi. Now with respect to degrees, pi is equivalent to 180 degrees. So in order to plot this point, there's two ways that we can go about this. We can either first measure out the radius and then rotate that line to the respective angle, or we can first find the angle and then extend out that radius. So in this case here, let's go ahead and first plot our radius. Now, we have a radius of 4. Starting at the origin, we are going to measure out 4 along the starting angle, which is 0 degrees. This is going to be a radius of 4. Then we want to go ahead and rotate 180 degrees from the starting angle. So if we were to rotate 180 degrees, the location of our point is going to be located here. This has a radius of 4, so a length of 4 from the origin to the point, and it is located at the angle theta with respect to the polar axis. So this is going to be the location of the given polar coordinate, but now what we want to do is we want to go ahead and find the equivalent representations of this polar coordinate. So, the first equivalent representation is going to be for any full rotation. Now, for any integer rotation of this point, a full 360 degrees, the general representation of an equivalent angle is going to be theta is equal to, well, theta is equal to theta plus 2 pik. This is going to be the representation of any equivalent angle about 360 degrees or -360 degrees or pretty much any integer multiple of rotations, so that means that our point for full rotations is going to be R, theta plus 2 pik. And so if we plug in our known information, that is going to be 4, i + 2 pik. And here, if we were to factor out api from the right-hand side of the coordinate, we can rewrite the coordinate as 41 + 2k multiplied by pi. This is going to be an equivalent angle for full rotations of the given point. But we also need to think about half rotations. Now for half rotations these points can be represented by a negative radius. On a graph, a negative radius means that you move in the direction opposite of the given angle. So that negative radius is going to be represented at negr. But if we're traveling in a negative direction, then our angle has to take us back to the original point. So we'll need to add half of a rotation to our given angle theta. So the general form of this reflection is going to be theta plus pi plus 2 pik. And if we were to further simplify the right hand side and plug in the known information, then the equivalent representation for half rotations. Is going to be 4.2K. This is going to be an equivalent representation of our point. 4 half rotations. And these are going to be our solutions to the problem. And the overall solution to the problem is going to be option C. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Polar CoordinatesPlot the following points, given in polar coordinates. Then find all the polar coordinates of each point.a. (2, π/2)

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Polar Coordinates

Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.

a. (2, π/2)