Graphing Polar Functions Examples

• Without using a calculator, graph the polar function r = cos θ for

Start by finding r at the endpoints. When θ = 0,

r = cos(0) = 1.

When  ,

As θ moves from 0 to , the value of r = cos θ decreases from 1 to 0.

When θ = π, r = cos π = -1.

Remembering how we graph polar coordinates with negative r, we find these points:

When θ is in the second quadrant and r is negative, the point (r, θ) is in the fourth quadrant.
As θ moves closer to π, the points (r, θ) move closer to the positive x-axis.

As θ moves from to π, the value r = cos θ goes from 0 to -1.

The points (r, θ) move further from the origin as θ gets closer to π.

Determine whether the following is a correct graph of r = cos θ for .

When   we should have and we do:

When θ = 0 we should have r = cos(0) = 1, and we do:

For θ in between and 0 the value of r should be positive, and it is:

This appears to be a correct graph of r = cos θ.

Determine whether the following is a correct graph of r = sin θ for .

When

we should have

.

But the point

isn't on the graph we were given.