Example 6: Sine

r = a sin θ

is the circle one of whose diameters is the line from the origin to a point a above the origin.

This can be seen from Figure 7.7.9, if we remember that a diameter and a point on the circle form a right triangle.

As θ increases, the point (a sin θ, θ) goes around this circle once for every π radians.

Figure 7.7.9

Figure 7.7.10

An equation r = f(θ) in polar coordinates has the same graph as the pair of parametric equations

x= f(θ) cos θ, y = f(θ) sin θ in rectangular coordinates. This can be seen from Figure 7.7.10.