How do you find the rectangular equation for $r=-6sintheta$ ?

Question

7853 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-11-12T17:09:30

Please see the explanation for steps leading to the equation of a circle:

$(x - 0)^2 + (y - -3)^2 = 3^2$

Multiply both sides of the equation by r:

$r^2 = -6rsin(theta)$

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

$x^2 + y^2 = - 6y$

Add $6y + k^2$ to both sides:

$x^2 + y^2 + 6y + k^2 = k^2$

Use the right side of the pattern $(y - k)^2 = y^2 - 2ky + k^2$ , to complete the square for the y terms:

$y^2 - 2ky + k^2 = y^2 + 6y + k^2$

$-2ky = 6y$

$k = -3$ and $k^2 = 9 = 3^2$

Replace the y terms with the left side of the pattern but with $k = -3$ :

$x^2 + (y - -3)^2 = k^2$

To put this into the standard form for a circle, substute $3^2$ for $k^2$ (not 9) )and insert a $-0$ in the x term:

$(x - 0)^2 + (y - -3)^2 = 3^2$

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

How do you find the rectangular equation for r=-6sinthetar=-6sintheta?