The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Polar Coordinates

The position of a point in the plane can be described by its distance and direction from the origin. In measuring direction we take the x-axis as the starting point. Let X be the point (1, 0) on the x-axis and let P be a point in the plane as in Figure 7.7.1.

07_trigonometric_functions-412.gif

Figure 7.7.1

A pair of polar coordinates of P is given by (r, θ) where r is the distance from the origin to P and θ is the angle XOP.

Each pair of real numbers (r, θ) determines a point P in polar coordinates. To find P we first rotate the line OX through an angle θ, forming a new line OX', and then go out a distance r along the line OX'. If θ is negative then the rotation is in the negative, or clockwise direction. If r is negative the distance is measured along the line OX' in the direction away from X' (see Figure 7.7.2).

07_trigonometric_functions-413.gif

Figure 7.7.2

Example 1: Ploting Points
Example 2: Circle and Straight Line
Example 3: Spiral of Archimedes

An equation in rectangular coordinates can readily be transformed into an equation in polar coordinates with the same graph by using x = r cos θ, y = r sin θ.

Here are the polar equations for various types of straight lines. Examples of their graphs are shown in Figure 7.7.7.

(1)

Line through the origin (not vertical).

07_trigonometric_functions-421.gif

Rectangular equation: y = mx.

Polar equation: r sin θ = mr cos θ ,

or:

tan θ = m.

(2)

Horizontal line (not through origin).

07_trigonometric_functions-422.gif

Rectangular equation: y = b.

Polar equation: r sin θ = b,

or:

r = b csc θ.

(3)

Vertical line (not through origin).

07_trigonometric_functions-423.gif

Rectangular equation: x = a.

Polar equation: r cos θ = a,

or:

r = a sec θ.

(4)

Vertical line through origin.

Rectangular equation: x = 0.

Polar equation: r cos θ = 0,

or:

θ = π/2.

(5)

Other lines.

07_trigonometric_functions-424.gif

Figure 7.7.7

Rectangular equation: y = mx + b.

Polar equation: r sin θ = mr cos θ + b,

or:

07_trigonometric_functions-420.gif

Example 4: Parabola
Example 5: Hyperbola
Example 6: Sine
Example 7: Spiral and Circle


Last Update: 2006-11-15