Question

How do you graph `r=3costheta`?

Answer 1

Multiply both sides by r
Substitute `x^2 + y^2` for `r^2` and x for `rcos(theta)`
Complete the square and write in the standard form of a circle.
Use a compass to draw the circle.

Explanation:

Multiply both sides by r:

`r^2 = 3rcos(theta)`

Substitute `x^2 + y^2` for `r^2` and x for `rcos(theta)`

`x^2 + y^2 = 3x" [1]"`

The standard Cartesian form for a circle is:

`(x-h)^2+(y-k)^2=r^2" [2]"`

Insert a -0 in the numerator of second term of equation [1]:

`x^2 + (y-0)^2 = 3x" [3]"`

Subtract 3x from both sides:

`x^2-3x + (y-0)^2 = 0" [4]"`

We want to complete the square, using the pattern `(x - h)^2 = x^2 - 2hx + h^2`, therefore, we add `h^2` to both sides of equation [4]:

`x^2 - 3x + h^2 + (y-0)^2= h^2" [5]"`

Set the middle term on the right side of the pattern equation to the middle term on the left side of equation [5]:

`-2hx = -3x`

Solve for h:

`h = 3/2`

Substitute `3/2` for h in equation [5]:

`x - 3x + (3/2)^2 + (y-0)^2 = (3/2)^2" [6]"`

We know that the 3 terms on the left side of equation [6] are same as the left side of the pattern with `h = 3/2`:

`(x - 3/2)^2 + (y-0)^2 = (3/2)^2" [7]"`

Equation [7] is the standard Cartesian form for the equation of a circle with its center at `(3/2, 0)` and a radius of `3/2`

To graph equation [7], set your compass to a radius of 3/2, put the center at `(3/2,0)` and draw a circle.