Question

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

Accepted Answer

Recall the three types of 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 $$ r = 2 + 3 \sin 2	heta $$, substitute $$ -	heta  to $$get $$ r = 2 + 3 \sin(-2	heta) . To $$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 the equation to get $$ r = 2 + 3 \sin 2(\pi - 	heta) . To $$test symmetry about the pole (origin), replace $$ r $$ with $$ -r $$ and $$ 	heta $$ with $$ 	heta + \pi $$, then check if the equation remains unchanged. Substitute these into the equation to get $$ -r = 2 + 3 \sin 2(	heta + \pi) . $$After determining the symmetries, sketch the graph by plotting points for various values of $$ 	heta $$ between 0 and $$ 2\pi $$, calculating corresponding $$ r $$ values using the equation $$ r = 2 + 3 \sin 2	heta $$, and then plotting these points in polar coordinates.

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

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주요 개념

Polar Coordinates and Equations

Symmetry Tests in Polar Graphs

Trigonometric Functions in Polar Equations