Determine the symmetries of the curve.
-$$r = - 3 - 4 \sin \theta$$
A) x-axis only
B) No symmetry
C) y-axis only
D) Origin only

Question

Quizplus · Accepted Answer

The polar curve $r=-3-4\sin	heta$ is symmetric about the $y$-axis. To see why, suppose $(r,	heta)$ is a point on the curve. Then $(-r,	heta+\pi)$ is also a point on the curve since
\begin{align*}
-3 - 4\sin (	heta+\pi) &= -3 - 4\sin	heta\cos(\pi)-4\cos	heta\sin(\pi) \
& = - (-3 -4\sin	heta).
\end{align*}
Thus, if $(r,	heta)$ is on the curve, then $(-r,	heta+\pi)$ is on the curve as well, and these two points must be symmetric about the $y$-axis. Since the curve is not symmetric about the $x$-axis or the origin, the answer is $\boxed{	extbf{(C)}}.$

Determine the Symmetries of the Curve r=−3−4sin⁡θr = - 3 - 4 \sin \thetar=−3−4sinθ

Determine the Symmetries of the Curve $r = - 3 - 4 \sin \theta$