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.

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).

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).
	Rectangular equation: y = mx.
	Polar equation: r sin θ = mr cos θ ,
or:	tan θ = m.
(2)	Horizontal line (not through origin).
	Rectangular equation: y = b.
	Polar equation: r sin θ = b,
or:	r = b csc θ.
(3)	Vertical line (not through origin).
	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.	Figure 7.7.7
	Rectangular equation: y = mx + b.
	Polar equation: r sin θ = mr cos θ + b,
or:

Example 4: Parabola

Example 5: Hyperbola

Example 6: Sine

Example 7: Spiral and Circle