1-2 Plot the point whose polar coordinates are given. Then...

1-2 Plot the point whose polar coordinates are given. Then find
two other pairs of polar coordinates of this point, one with $r>0$
and one with $r<0 .$
$$\begin{array}{lll}{\text { (a) }(1,7 \pi / 4)} & {\text { (b) }(-3, \pi / 6)} & {\text { (c) }(1,-1)}\end{array}$$

Key Concepts

Handling Negative Radius

In polar coordinates, a negative radius is accommodated by reversing the direction of the vector indicated by the given angle. This is mathematically remedied by adding ? to the angle, resulting in an equivalent representation with a positive radius, thereby demonstrating the non-uniqueness of the polar coordinates for a point.

Equivalence of Polar Coordinates

Because of the periodic nature of the angle and the possibility of negative radii, multiple pairs of polar coordinates can correspond to the same point in the plane. Adjustments like adding or subtracting multiples of 2? to the given angle or changing the sign of the radius while adding ? to the angle allow one to find different polar forms of the same point.

Conversion between Polar and Cartesian Coordinates

This is the method by which one converts polar coordinates (r, ?) to Cartesian coordinates (x, y) using the formulas x = r cos? and y = r sin?. This conversion is essential for plotting points well-known in a familiar rectangular coordinate system.

Polar Coordinate Representation

This concept describes how a point in the plane can be specified by a distance from the origin (r) and an angle (?) measured from the positive x-axis. The representation is unique only up to an addition of multiples of 2? to the angle, and as a result, a single point can have infinitely many polar coordinate representations.

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