Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ

Accepted Answer

Recall that to test for symmetry in polar equations, we check three types: symmetry about the polar axis (x-axis), symmetry about the line $$ 	heta = \frac{\pi}{2}  (y-$$axis), and symmetry about the pole (origin). For symmetry about the polar axis, replace $$ 	heta $$ with $$ -	heta  in $$the equation and see if the equation remains unchanged. That is, check if $$ r = 1 - 3 \sin(-	heta) $$ simplifies to the original equation. For symmetry about the line $$ 	heta = \frac{\pi}{2} $$, replace $$ 	heta $$ with $$ \pi - 	heta $$ and check if the equation remains unchanged. So, check if $$ r = 1 - 3 \sin(\pi - 	heta) $$ simplifies to the original equation. For symmetry about the pole, replace $$ r $$ with $$ -r $$ and $$ 	heta $$ with $$ 	heta + \pi $$, then check if the equation remains unchanged. That is, check if $$ -r = 1 - 3 \sin(	heta + \pi) $$ can be rearranged to the original form. After determining the symmetries, sketch the graph by plotting points for various values of $$ 	heta $$ between 0 and $$ 2\pi $$, using the equation $$ r = 1 - 3 \sin 	heta $$, and then reflect the graph according to the symmetries found.

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ

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주요 개념

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Graphing Polar Equations Involving Sine