How do you graph #r=3costheta#?

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.