Plot each point given in polar coordinates, and find other polar coordinates (r, θ) of the point

Plot each point given in polar coordinates, and find other...

Key Concepts

Non-Unique Representations in Polar Coordinates

A single point in the plane can have multiple polar coordinate representations. For example, changing the sign of the radius while adding ? to the angle gives an equivalent point. Recognizing and identifying these equivalent representations is crucial when working with polar coordinates.

Periodicity of Angles in Polar Coordinates

The angle in polar coordinates is periodic with a period of 2? radians, meaning that adding or subtracting multiples of 2? to the angle yields the same point. This periodicity is a key property that leads to the non-unique representations of points in polar coordinates.

Polar Coordinates

Polar coordinates represent points in a plane using a distance from the origin and an angle from a fixed direction. This system provides an alternative to Cartesian coordinates and is particularly useful in problems with circular or rotational symmetry.

Conversion Between Polar and Cartesian Coordinates

Converting between polar and Cartesian coordinates involves using the relationships x = r cos(?) and y = r sin(?). This conversion is essential for plotting points and understanding the geometric interpretation of the coordinate systems.

Transcript

00:01 All right.

00:05 In problem 39, we are given the polar coordinates 1 and pi over 2.

00:12 And we are to find three equivalent pairs of coordinates using the conditions provided in a, b, and c.

00:23 Okay? they're very easy to do.

00:25 You just need to remember a couple things about coterminal angles and that's.

00:32 About it.

00:33 It's pretty straightforward.

00:34 All right, so let's see where our angle is first.

00:37 One and pi over two.

00:39 I'll do that one.

00:42 So pi over two is right here, 90 degrees.

00:48 And then we would go one ring out from the pole.

00:53 So that would put us right there if we're doing those, that scale.

00:57 And that's point p.

01:02 Use cake.

01:04 All right.

01:05 So our task is to find other coordinates that put us in that same exact place.

01:13 Oops.

01:14 I forgot i still had erase selected.

01:16 Let me see if i can get our y -axis back in there a little bit.

01:27 So again, we are tasked with finding coordinates that will put us right in that same exact spot.

01:36 Actually, p was one, one, not at two, right? okay.

01:42 All right.

01:44 So part a says the r value should be, positive but the angle needs to be between zero and negative two pi all right no problem well if the r value is positive then it's going to be the same as are given our value so there's part of our answer and now all we got to do is figure out what is the equivalent negative angle to pi over two so if you look at the picture that would be this angle right here if that that's positive pi over two, then the equivalent negative angle is negative three pi or two.

02:29 So if we go to negative three pi over two's terminal side and go out one ring from the pole, we end up in the same place where p is.

02:39 So there's that one...

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