Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates

Plot the point whose polar coordinates are given. Then find...

Key Concepts

Conversion Between Polar and Cartesian Coordinates

Converting polar coordinates to rectangular coordinates using the formulas x = r cos ? and y = r sin ? aids in visualizing and verifying the equality of different representations. This conversion is essential when plotting points and understanding how varying r and ? affect the point's location in the plane.

Equivalence of Polar Representations

Multiple pairs of polar coordinates can represent the same point. A common method to see this equivalence is by either adjusting the angle by multiples of 2? while keeping the radius fixed or by using a negative radius with an appropriately shifted angle (adding ? to the original angle) to indicate the same direction.

Polar Coordinates

A two-dimensional coordinate system where each point is determined by a distance from a fixed origin (radius r) and an angle (?) from a fixed direction. This system is fundamental in representing locations in a plane when circular or rotational symmetry is present.

Periodicity of the Angular Component

The angle in polar coordinates is periodic with a period of 2?, meaning that adding or subtracting any integer multiple of 2? to the angle results in the same point. This property is crucial as it allows multiple valid representations for the same point using different angles.

Transcript

00:04 So we want to plot the point 2 comma pi over 3, which represents a polar coordinate.

00:11 So let's think of the first component as being r, which is our radius.

00:16 Second component being theta, which is our angle.

00:20 So we want to move out two units to the right from the origin to move to a circle of radius 2.

00:28 So again, moving to the right by two units.

00:32 And then we want to rotate a positive pi for 3, which would be.

00:37 In the first quadrant.

00:41 So again, we're moving out to the right by 2 and then rotating around 5 or 3 units.

00:46 We just plot the point.

00:48 We don't actually write the movement.

00:51 So 2 comma pi root 3.

00:54 Next, let's find two other representations for this coordinate.

01:00 One using a radius that is greater than 0, and one using a radius that is less than 0.

01:07 For the radius that's greater than zero, all we have to do is add 2 pi or subtract 2 pi to our angle.

01:15 In this case, let's add 2 pi.

01:19 In order to add 2 pi, we need a common denominator.

01:24 So our common denominator is 3.

01:25 So we have pyrid 3 plus 6 pyrid 3, or in other words, 2 .7 pyrid 3 as an equivalent location of the same coordinate.

01:43 Next, let's find what happens if our radius is less than zero.

01:48 So if our radius is less than zero, we would need a negative 2 as our radius.

01:54 So let's think about what that looks like on our graph.

01:57 So now instead of moving right by 2 first, we're going to move to the left by 2.

02:02 We want to end in the same location as our original point.

02:08 So let's think about we had a reference angle of pi or 3.

02:11 So in this case, we want to subtract pi or three from pi.

02:17 So we have pi minus pi for three, which was our reference angle.

02:21 Again, finding a common denominator, get 3 pi minus pi or three.

02:29 Or in other words, negative 2 comma, 2 pi with 3.

02:38 But in this case, we want it to be a negative 2 pi or 3 so that we actually are moving in the negative direction.

02:48 Another option would have been to move positive direction, which would have been the pi plus pi with three.

03:02 So another option, negative two comma pi plus pi with three.

03:14 Or again, finding a common denominator, negative two comma four pi or three.

03:28 So different options for for each case, i'm going to use the second one in this case.

03:37 But either would work.

03:39 Next, let's plot 1 comma negative 3 -5 -4...

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