$$x = r\cos \theta$$, so $$r = 3\cos \theta$$ becomes $$x = 3$$

Create a table of values for $$\theta$$ and $$r$$

Minimum $$r = 0 at$$ $$\theta = \frac{3\pi}{2}$$

Plot points for $$\theta$$ from $$0 to$$ $$2\pi$$ and connect them smoothly

Question 1

Given the polar equation $$r = 2 + 2\sin 	heta$$, what is the general shape of its graph?

Accepted Answer

A cardioid

Question 2

Which of the following is the correct method to convert the polar equation $$r = 3\cos 	heta to $$rectangular (Cartesian) form?

Accepted Answer

$$x = r\cos 	heta$$, so $$r = 3\cos 	heta$$ becomes $$x = 3$$

Question 3

If a polar equation is given as $$r = 2\sin 	heta$$, what is the corresponding rectangular equation?

Accepted Answer

$$y = 2$$

Question 4

A polar graph $$r = 4\cos 	heta$$ represents which of the following in rectangular coordinates?

Accepted Answer

$$x^2$$ + $$y^2 = 4x$

Question 5

Which of the following best describes the graph of $$r = 2\sin 	heta$$?

Accepted Answer

A circle tangent to the y-axis

Question 6

What is the first step in graphing a polar equation such as $$r = 1 + \cos 	heta$$?

Accepted Answer

Create a table of values for $$	heta$$ and $$r$$

Question 7

Given the polar equation $$r = 2\cos 	heta$$, what is the center and radius of the corresponding circle in rectangular coordinates?

Accepted Answer

Center at $$(1, 0)$$, radius $$1$$

Question 8

If $$r = 1 - \sin 	heta$$, what is the minimum value of $$r$$ and at what angle does it occur?

Accepted Answer

Minimum $$r = 0 at$$ $$	heta = \frac{3\pi}{2}$$

Question 9

Which of the following polar equations represents a circle centered at $$(0, a/2)$$ with radius $$a/2$$?

Accepted Answer

$$r = a\sin 	heta$$

Question 10

A student is asked to graph $$r = 1 + \sin 	heta. $$Which of the following is a correct step in the process?

Accepted Answer

Plot points for $$	heta$$ from $$0 to$$ $$2\pi$$ and connect them smoothly

Question 11

Which of the following is the rectangular form of the polar equation $$r = 2$$?

Accepted Answer

$$x^2$$ + $$y^2 = 4$

Question 12

If $$r = 2\cos 	heta$$, what is the value of $$r$$ when $$	heta = \frac{\pi}{3}$$?

Accepted Answer

$$r = 1$$

Question 13

Which of the following best describes the graph of $$r = 1 + \cos 	heta$$?

Accepted Answer

A limaçon with a dimple

Question 14

A real-world application: A lighthouse rotates its beam in a circle described by $$r = 5. $$What is the area covered by the beam?

Accepted Answer

$$25\pi$$

Question 15

Which of the following is the correct rectangular form for $$r = 3\sin 	heta$$?

Accepted Answer

$$x^2$$ + $$y^2 = 3y$

Graphing Trigonometric Functions in Polar Equations

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