Problems 64-67 are based on material learned earlier in the...

Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find two pairs of polar coordinates $(r, \theta)$, one with $r>0$ and the other with $r<0$, for the point with rectangular coordinates $(-8,-15)$. Express $\theta$ in radians.

Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find two pairs of polar coordinates $(r, \theta)$, one with $r>0$ and the other with $r<0$, for the point with rectangular coordinates $(-8,-15)$. Express $\theta$ in radians.

Key Concepts

Conversion Between Rectangular and Polar Coordinates

This concept involves using formulas to relate rectangular (Cartesian) coordinates (x, y) to polar coordinates (r, ?). The fundamental relationships are r = ?(x² + y²) which gives the distance from the point to the origin, and ?, typically found through the relation tan(?) = y/x, which describes the angle the line from the origin to the point makes with the positive x-axis.

Polar Coordinate Representations

A point in polar coordinates can have multiple representations due to the periodic nature of angles and the possibility of using a negative radius. This concept emphasizes that the same point can be described by different pairs of (r, ?), and understanding these equivalencies is important when converting between coordinate systems.

Negative Radius in Polar Coordinates

This key idea explains that polar coordinates allow the radius (r) to be negative, which means the point is located in the direction opposite to that indicated by the angle ?. This concept is useful for representing points in alternative forms and underlines the flexibility of polar coordinates compared to Cartesian coordinates.

Angle Determination in Polar Coordinates

Determining the correct angle ? is crucial when converting from rectangular coordinates, as the arctan function alone may not indicate the proper quadrant. This concept involves adjusting the computed angle based on the signs of x and y to ensure that the angle accurately represents the point's position in the plane.

Transcript

00:01 Here we need to find the rectangular coordinates of the polar coordinate 6 comma 2 pi over 3.

00:07 Let me just write it like this, so it might look a little more common.

00:12 Now, what the point 6 comma 2 pi over 3 means is this point is going to be 6 away from the origin at an angle of 2 pi over 3.

00:21 Now 2 pi over 3 is an angle given in radiance.

00:25 To convert this to degrees, we're going to multiply times 180 over pi.

00:29 I feel like that's kind of easy to memorize because degrees.

00:32 Degrees typically don't have pie in them unless it's some kind of really funky question.

00:37 So when you do convert radiance to degrees, you're going to make that pie cancel away, which is a good reminder that pie is on the bottom there.

00:44 Now, three goes into 180, 60 times.

00:47 So we're left with two times 60, which is 120 degrees.

00:51 Therefore, there's zero, there's 90, here's 180.

00:58 120 degrees would be 30 degrees past 90.

01:02 And then this point would be six away from the origin.

01:07 So there's our point, which is six away.

01:09 Now we need to come up with the rectangular coordinates, which are what you've learned first.

01:14 So from the origin, you've got some distance to the left, which will be our x coordinate, and then we have some distance up, which would be the y coordinate.

01:23 It is very important that we know that the triangle that we created here, if this is 30, because that's how far past 90 it went, that makes this angle 60 degrees, this is a 90, this is the 30.

01:38 So the way we're going to do this is to find the x and y coordinate.

01:42 We're going to use our sign and our cosine functions...

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