Question

Test for symmetry with respect to a. the polar axis. b. the line θ = π/2. c. the pole. r = 4 + 3 cos θ

Accepted Answer

Recall the tests for symmetry in polar coordinates:

- Symmetry about the polar axis (the horizontal axis) means that if $$(r, 	heta) is on $$the graph, then $$(r, -	heta) is $$also on the graph.
- Symmetry about the line $$	heta = \frac{\pi}{2}$$ means that if $$(r, 	heta) is on $$the graph, then $$(r, \pi - 	heta) is $$also on the graph.
- Symmetry about the pole (origin) means that if $$(r, 	heta) is on $$the graph, then $$(-r, 	heta) or $$equivalently $$(r, 	heta + \pi) is $$also on the graph. To test symmetry about the polar axis, replace $$	heta$$ with $$-	heta in $$the equation and check if the equation remains unchanged. For the given equation:

$$r = 4 + 3 \cos 	heta$$

Replace $$	heta$$ with $$-	heta$$:

$$r = 4 + 3 \cos(-	heta)$$ Use the even property of cosine, which states $$\cos(-	heta) = \cos 	heta$$, so the equation becomes:

$$r = 4 + 3 \cos 	heta$$

Since this is the same as the original equation, the graph is symmetric about the polar axis. To test symmetry about the line $$	heta = \frac{\pi}{2}$$, replace $$	heta$$ with $$\pi - 	heta in $$the equation and check if the equation remains unchanged:

$$r = 4 + 3 \cos(\pi - 	heta)$$ Use the identity $$\cos(\pi - 	heta) = -\cos 	heta$$, so the equation becomes:

$$r = 4 - 3 \cos 	heta$$

Since this is not the same as the original equation, the graph is not symmetric about the line $$	heta = \frac{\pi}{2}.$$

Test for symmetry with respect to a. the polar axis. b. the line θ = π/2. c. the pole. r = 4 + 3 cos θ

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Concetti chiave

Symmetry with Respect to the Polar Axis

Symmetry with Respect to the Line θ = π/2

Symmetry with Respect to the Pole