Q: For the polar function $$r = 2\sin(\theta)$$, what is the maximum value of $$r$$ and at what angle does it occur?

Maximum $$r = 2 at$$ $$\theta = \frac{\pi}{2}$$

Question 1

Which of the following is the correct conversion from polar coordinates $$\left(2, \frac{\pi}{3}\right) to $$rectangular coordinates $$(x, y)$$?

Accepted Answer

$$(1, \sqrt{3})$$

Question 2

Given the polar equation $$r = 2 + 2\cos(	heta)$$, what type of graph does this equation produce?

Accepted Answer

Limacon with an inner loop

Question 3

If a point in polar coordinates is $$(-3, \frac{\pi}{4})$$, where is the point located?

Accepted Answer

3 units from the pole in the direction opposite $$\frac{\pi}{4}$$

Question 4

Convert the rectangular coordinates $$(-4, 4) to $$polar coordinates $$(r, 	heta)$$, with $$r \u003e 0.$$

Accepted Answer

$$(4\sqrt{2}, \frac{3\pi}{4})$$

Question 5

Which of the following is the correct rectangular form of the complex number $$z = 2	ext{cis}\left(\frac{\pi}{3}ight)$$?

Accepted Answer

$$1 + \sqrt{3}i$$

Question 6

The graph of $$r = 5\sin(2	heta) is $$known as which type of polar graph?

Accepted Answer

Rose curve with 4 petals

Question 7

For the polar function $$r = 2\sin(	heta)$$, what is the maximum value of $$r$$ and at what angle does it occur?

Accepted Answer

Maximum $$r = 2 at$$ $$	heta = \frac{\pi}{2}$$

Question 8

Which of the following equations represents a lemniscate?

Accepted Answer

$$r^2 = 4$$\sin(2	heta)$$

Question 9

If $$r = 3 + 2\sin(	heta)$$, what is the minimum value of $$r$$?

Accepted Answer

$$r_{min} = 1$$

Question 10

Which of the following is true about the rate of change of $$r$$ for the polar function $$r = 2 + 2\cos(	heta) on $$the interval $$0 \u003c 	heta \u003c \frac{\pi}{2}$$?

Accepted Answer

The graph is moving toward the origin

Question 11

Which of the following rectangular coordinates corresponds to the polar coordinates $$(5, \pi)$$?

Accepted Answer

$$(-5, 0)$$

Question 12

What is the general equation for a rose curve with 7 petals?

Accepted Answer

$$r = a\sin(7	heta)$$

Question 13

Which of the following is the correct polar form for the rectangular coordinates $$(-3, -3)$$ with $$r \u003e 0$$?

Accepted Answer

$$(3\sqrt{2}, \frac{5\pi}{4})$$

Question 14

Which of the following statements is true about the graph of $$r = 1 - \sin(	heta) as$$ $$	heta$$ increases on $$0 \u003c 	heta \u003c \frac{\pi}{2}$$?

Accepted Answer

The points on the graph are getting closer to the origin

Question 15

Which of the following is the correct rectangular form for the complex number $$z = 3	ext{cis}\left(\frac{\pi}{2}ight)$$?

Accepted Answer

$$3i$$

Question 16

What is the symmetry of the graph $$r = 4\sin(3	heta)$$?

Accepted Answer

Symmetric about the pole

Question 17

Which of the following is the correct polar form for the rectangular coordinates $$(0, -8)$$ with $$r \u003e 0$$?

Accepted Answer

$$(8, \frac{3\pi}{2})$$

Question 18

If $$r = 2\cos(	heta)$$, what is the shape of the graph?

Accepted Answer

Circle

Question 19

Which of the following is the correct conversion from rectangular coordinates $$(6, 8) to $$polar coordinates $$(r, 	heta)$$ with $$r \u003e 0$$?

Accepted Answer

$$(10, \arctan(\frac{4}{3}))$$

Study Notes: Polar Coordinates, Polar Graphs, and Complex Plane in Precalculus

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