Find two pairs of polar coordinates, one pair with r>0 and the other pair with r<0, for the poin

Find two pairs of polar coordinates, one pair with r>0 and...

Key Concepts

Cartesian to Polar Coordinate Conversion

This concept involves translating a point from the Cartesian coordinate system, which uses (x, y) values, into the polar coordinate system, which uses a radius r and an angle ?. The conversion relies on the formulas r = ?(x² + y²) and ? = arctan(y/x), with attention paid to the quadrant in which the point lies. This is fundamental in understanding how different coordinate systems can represent the same point in the plane.

Polar Coordinates with Negative Radius

In polar coordinates, a point can be described by a negative radius provided the angle is adjusted appropriately. A negative r signifies that the point is located in the direction opposite to the angle ?. This alternate representation is important because a single point in the plane can have infinitely many polar coordinate descriptions, and choosing a representation with negative r can sometimes simplify problems involving symmetry or transformations.

Angle Normalization in Polar Coordinates

Angle normalization is the process of adjusting the angle ? so that it falls within a specific interval, typically between 0° and 360° (or 0 and 2? radians). This ensures consistency in representing angles and makes it easier to compare different polar coordinates. When converting from Cartesian coordinates or switching between representations (such as from r > 0 to r < 0), normalizing the angle helps maintain a standard and unambiguous description of the point.

Transcript

00:03 In this problem, we have to give all possible polar coordinates for the point minus 3, 0.

00:11 So here, x is equal to minus 3 and y is equal to 0.

00:17 We know that x is equal to r cos theta and y is equal to r sine theta.

00:23 So, x is equal to r cost theta is equal to minus 3 and y is equal to r sine theta is equal to 0.

00:32 Now we have r -square is equal to r -square plus r -square -square -theta plus r -square -square -theta r -cos -theta is equal to minus -3.

00:45 So r -square -qsquare -theta is minus -3 square plus r -sine -theta is 0.

00:54 So r -square -ssquare -that is r -cquare is equal to minus 3 -square plus 0 -square, which is equal to 9 plus 0 which is equal to 9.

01:10 R square is equal to 9 gives us r is equal to plus or minus 3.

01:20 Firstly, we will take the positive coordinate, that is equal to positive 3.

01:32 We have r cos theta is equal to minus 3 and r sine theta is equal to 0.

01:39 That is 3 cost theta is equal to minus 3 and 3 sine theta is equal to 0.

02:04 3 cost theta is equal to minus 3 gives us cos theta is equal to minus 1.

02:16 And 3 sine theta is equal to 0 gives us sine theta is equal to 0.

02:26 So, cost theta is equal to minus 1 and sine theta is equal to 0.

02:33 These equations are satisfied by setting theta is equal to pi...

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