Questions: convert the polar equation to rectangular form and sketch the graph. r = 6 cos θ

Solution Steps

The given polar equation is: $$r = 6 \cos \theta$$

Using the relationships $x = r \cos \theta$ and $r = \sqrt{x^2 + y^2}$, we can convert this to rectangular form.

First, multiply both sides by $r$: $$r^2 = 6r \cos \theta$$

Since $r \cos \theta = x$: $$r^2 = 6x$$

And since $r^2 = x^2 + y^2$: $$x^2 + y^2 = 6x$$

Rearrange to get the standard form of a circle: $$x^2 + y^2 - 6x = 0$$

Complete the square for the $x$ terms: $$(x^2 - 6x + 9) + y^2 = 9$$ $$(x - 3)^2 + y^2 = 3^2$$

This is the equation of a circle with center at $(3, 0)$ and radius 3.

Final Answer