Convert the polar equation to rectangular form. $r = 2 \sin \theta$
A) $x ^ { 2 } + y ^ { 2 } - 2 y = 0$
B) $x ^ { 2 } + y ^ { 2 } + 2 y = 0$
C) $x ^ { 2 } + y ^ { 2 } - 2 \mathrm { x } = 0$
D) $x ^ { 2 } + y ^ { 2 } + 2 \mathrm { x } = 0$
E) $x ^ { 2 } + y ^ { 2 } - 2 = 0$

Question

Convert the polar equation to rectangular form.  $r = 2 \sin \theta$ 
A) $x ^ { 2 } + y ^ { 2 } - 2 y = 0$
B) $x ^ { 2 } + y ^ { 2 } + 2 y = 0$
C) $x ^ { 2 } + y ^ { 2 } - 2 \mathrm { x } = 0$
D) $x ^ { 2 } + y ^ { 2 } + 2 \mathrm { x } = 0$
E) $x ^ { 2 } + y ^ { 2 } - 2 = 0$

Quizplus · Accepted Answer

The polar equation $r = 2 \sin \theta$ can be converted to rectangular form using the identities $r^2 = x^2 + y^2$ and $y = r \sin \theta$. Substituting $r = 2 \sin \theta$ into $y = r \sin \theta$ gives $y = 2 \sin^2 \theta$. Since $r = \sqrt{x^2 + y^2}$, substituting $r$ into the original equation gives $\sqrt{x^2 + y^2} = 2 \sin \theta$, or $x^2 + y^2 = 4 \sin^2 \theta$. Using the identity $y = r \sin \theta$, we get $x^2 + y^2 = 4(y/r)^2$. Simplifying, we find $x^2 + y^2 = 4y^2/r^2$, which simplifies to $x^2 + y^2 = 2y$, or $x^2 + y^2 - 2y = 0$, matching option A.

Convert the Polar Equation to Rectangular Form r=2sin⁡θr = 2 \sin \thetar=2sinθ

Convert the Polar Equation to Rectangular Form $r = 2 \sin \theta$