How do you convert the polar equation $r = 3 sin θ$ $r=3sintheta$ into rectangular form?

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Answer 1 · 2016-10-05T15:27:20

$x^{2} + {(y - \frac{3}{2})}^{2} = \frac{9}{4}$ $x^2+(y-3/2)^2=9/4$

$x = r cos (θ)$ $x=rcos(theta)$
$y = r sin (θ)$ $y=rsin(theta)$
$r^{2} = x^{2} + y^{2}$ $r^2=x^2+y^2$

Multiply the equation by $r$ $r$

$r \cdot r = 3 r sin (θ)$ $r*r=3rsin(theta)$

Simplify

$r^{2} = 3 r sin (θ)$ $r^2=3rsin(theta)$

Make appropriate substitutions

$x^{2} + y^{2} = 3 y$ $x^2+y^2=3y$

Gather all of the terms to the same side

$x^{2} + y^{2} - 3 y = 0$ $x^2+y^2-3y=0$

Complete square using the coefficient of $y$ $y$ variable

${(- \frac{3}{2})}^{2} = \frac{9}{4}$ $(-3/2)^2=9/4$

Add $\frac{9}{4}$ $9/4$ to both sides the equation to keep it balanced. The constant $\frac{9}{4}$ $9/4$ allows you make a perfect square trinomial.

$x^{2} + y^{2} - 3 y + \frac{9}{4} = \frac{9}{4}$ $x^2+y^2-3y+9/4=9/4$

Rewrite

$x^{2} + {(y - \frac{3}{2})}^{2} = \frac{9}{4}$ $x^2+(y-3/2)^2=9/4$

Check out a tutorial on converting an equation from polar to rectangular

Check out a tutorial on completing the square graphically

Check out a tutorial on completing the square analytically

How do you convert the polar equation r=3sinthetar=3sinθr=3sintheta into rectangular form?