How do you find the rectangular equation for $r=2costheta$ ?

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Answer 1 · 2016-11-07T23:48:56

Please see the explanation for steps leading to the rectangular equation.

Multiply both sides by r:

$r^2 = 2rcos(theta)$

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

$x^2 + y^2 = 2x$

Add $h^2 - 2x$ to both sides:

$x^2 - 2x + h^2 + y^2 = h^2$

Use the right side of the pattern $(x - h)^2 = x^2 - 2hx + h^2$ equal to the first 3 terms to find the value of h:

$x^2 - 2hx + h^2 = x^2 - 2x + h^2$

$-2hx = -2x$

$h = 1$

Substitute the left side of the pattern with h = 1 and the right side becomes $1^2$

$(x - 1)^2 + y^2 = 1^2$

It is a circle with center of $(1, 0)$ and radius of 1:

Here is a graph

enter image source here

How do you find the rectangular equation for r=2costhetar=2costheta?