Replace the polar equation with an equivalent Cartesian equation.
-$r = 40 \sin \theta$
A) $x ^ { 2 } + ( y - 20 ) ^ { 2 } = 400$
B) $\mathrm { y } = 40$
C) $( x - 20 ) ^ { 2 } + y ^ { 2 } = 400$
D) $x ^ { 2 } + ( y - 40 ) ^ { 2 } = 400$

Question

Quizplus · Accepted Answer

The polar equation $r = 40 \sin \theta$ can be converted to Cartesian coordinates using the relationship $r^2 = x^2 + y^2$ and $y = r \sin \theta$. Substituting $r = 40 \sin \theta$ into $y = r \sin \theta$ gives $y = 40 \sin^2 \theta$. Using the identity $\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}$ and the fact that $x = r \cos \theta$, we can manipulate the equation to fit the form of a circle. The correct conversion, without going through the detailed algebraic manipulation here, results in the equation $x^2 + (y - 20)^2 = 400$, which is a circle centered at (0, 20) with a radius of 20.

Replace the Polar Equation with an Equivalent Cartesian Equation r=40sin⁡θr = 40 \sin \thetar=40sinθ

Replace the Polar Equation with an Equivalent Cartesian Equation $r = 40 \sin \theta$