Test the equation for symmetry with respect to the given axis, line, or pole.
-$r = 6 + 2 \sin \theta ;$ the line $\theta = \frac { \pi } { 2 }$
A) May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$
B) Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$

Question

Quizplus · Accepted Answer

To test for symmetry with respect to the line $\theta = \frac{\pi}{2}$, we substitute $\pi - \theta$ for $\theta$ in the equation and simplify. If the resulting equation is equivalent to the original equation, the graph is symmetric with respect to the line $\theta = \frac{\pi}{2}$. Substituting $\pi - \theta$ into the given equation $r = 6 + 2\sin\theta$ gives $r = 6 + 2\sin(\pi - \theta)$. Since $\sin(\pi - \theta) = \sin\theta$, the equation remains unchanged, proving symmetry with respect to the line $\theta = \frac{\pi}{2}$.

Test the Equation for Symmetry with Respect to the Given r=6+2sin⁡θ;r = 6 + 2 \sin \theta ;r=6+2sinθ;

Test the Equation for Symmetry with Respect to the Given $r = 6 + 2 \sin \theta ;$