Question

Identify, graph and state the symmetries for each polar equation.

r=9 cos (5 theta)
r=2 cos theta

Answer 1

Both equation are symmetric about the x-axis.

Explanation:

If (r,θ) can be replaced by (r,-θ),then the graph is symmetric about the x-axis.

If (r,θ) can be replaced by (-r,-θ),then the graph is symmetric about the y-axis.

If (r,θ) can be replaced by (-r,θ),then the graph is symmetric about the origin.

The given equation is

`r=9\cos(5\theta)`

Replace the value of (r,θ) by (r,-θ).

`r=9\cos(5(-\theta))=9\cos(5\theta)=r`

It is symmetric about the x-axis.

Replace the value of (r,θ) by (-r,-θ).

`-r=9\cos(5(-\theta))=9\cos(5\theta)=r\\eq -r`

It is not symmetric about the y-axis.

Replace the value of (r,θ) by (-r,θ).

`-r=9\cos(5(\theta))=r\\eq -r`

It is not symmetric about the origin.

Therefore the first equation is symmetric about the x-axis.

The given equation is

`r=2\cos(\theta)`

Replace the value of (r,θ) by (r,-θ).

`r=2\cos(-\theta)=2\cos(\theta)=r`

It is symmetric about the x-axis.

Replace the value of (r,θ) by (-r,-θ).

`-r=2\cos(-\theta)=2\cos(\theta)=r\\eq -r`

It is not symmetric about the y-axis.

Replace the value of (r,θ) by (-r,θ).

`-r=2\cos(\theta)=r\\eq -r`

It is not symmetric about the origin.

Therefore the second equation is symmetric about the x-axis.