Geometry and Measurement

2.1 Coordinate Geometry


Although the most common way to represent a point in a plane is in terms of its distances from two perpendicular axes, there are several other ways. One such way is in terms of the distance of the point from the origin and the angle between the positive x -axis and the ray emanating from the origin going through the point.

In the figure, the regular rectangular coordinates of P are (x,y) and the polar coordinates are . If r > 0, it is measured along the ray of the terminal side of . If r < 0, it is measured in the direction of the opposite ray.

Since  and , there is an easy relationship between rectangular and polar coordinates:

x2 + y2 = r2, using the Pythagorean theorem.

Unlike the case involving rectangular coordinates, each point in the plane can be named by an infinite number of polar coordinates.


1. (2,30°), (2,390°), (2,–330°), (–2,210°), (–2,–150°) all name the same point.

In general, a point in the plane represented by  can also be represented by  or , where n is an integer.

2. Express point P, whose rectangular coordinates are in terms of polar coordinates.

Therefore,  = 60°and (6,60°) are the polar coordinates of P .

3. Describe the graphs of (A) r = 2 and (B).

(A) r2 = x2 + y2

r = 2

Therefore, x2 + y2 = 4, which is the equation of a circle whose center is at the origin and whose radius is 2.

(B) Since  and y = 1. The domain of  is all real  that are not multiples of .

Thus,  is the equation of a horizontal line with holes in it at multiples of  one unit above the x -axis.

Both r = 2 and  are examples of polar functions. Such functions can be graphed on a graphing calculator by using POLAR mode. Enter 2 into r1 and graph using ZOOM 6, the standard window.

Although the graph is accurate for the scale given, the shape looks like that of an ellipse, not a circle. This is because the standard screen has a scale on the x -axis that is larger than that on the y-axis, resulting in a distorted graph. The graph below is obtained by using the ZOOM ZSquare command.


1. A point has polar coordinate (2,60°). The same point can be represented by

     (A)  (–2,240°)

     (B)  (2,240°)

     (C)  (–2,60°)

     (D)  (2,–60°)

     (E)  (2,–240°)

2. The polar coordinates of a point P are (2,200°). The rectangular coordinates of P are

     (A)  (–1.88,–0.68)

     (B)  (–0.68,–1.88)

     (C)  (–0.34,–0.94)

     (D)  (–0.94,–0.34)

     (E)  (–0.47,–0.17)

3. Describe the graph of .

     (A)  a parabola

     (B)  an ellipse

     (C)  a circle

     (D)  a vertical line

     (E)  the x -axis