Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ

Accepted Answer

Recall the three common symmetry tests for polar equations: symmetry about the polar axis (x-axis), symmetry about the line $$ 	heta = \frac{\pi}{2}  (y-$$axis), and symmetry about the pole (origin). To test symmetry about the polar axis, replace $$ 	heta $$ with $$ -	heta  in $$the equation and check if the equation remains unchanged. For $$ r = 4 \sin 3	heta $$, substitute $$ -	heta  to $$get $$ r = 4 \sin(-3	heta) . To $$test symmetry about the line $$ 	heta = \frac{\pi}{2} $$, replace $$ 	heta $$ with $$ \pi - 	heta $$ and check if the equation remains unchanged. Substitute $$ \pi - 	heta $$ into $$ r = 4 \sin 3	heta  to $$get $$ r = 4 \sin 3(\pi - 	heta) . To $$test symmetry about the pole, replace $$ r $$ with $$ -r $$ and $$ 	heta $$ with $$ 	heta + \pi $$ and check if the equation remains unchanged. Substitute these into the equation to get $$ -r = 4 \sin 3(	heta + \pi) . $$After determining the symmetries, sketch the graph by plotting points for values of $$ 	heta $$ from 0 to $$ 2\pi $$, using the equation $$ r = 4 \sin 3	heta . $$Note that the factor 3 inside the sine function indicates the graph will have multiple petals (specifically, 3 petals if $$ n  is $$odd in $$ r = a \sin n	heta ).$$

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ

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Key Concepts

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Graphing Rose Curves