$$x = r \cdot \cos \theta$$, $$y = r \cdot \sin \theta$$

$$(r, \theta + 2\pi n)$$, where $$n is $$any integer

$$r = \sqrt{x^2$ + y^2$}$$, $$\theta = \arctan \frac{y}{x}$$

Measure angle $$\theta$$ counterclockwise from the polar axis, then move $$r$$ units from the pole

$$\theta = \arctan \frac{y}{x}$$

Question 1

Which of the following correctly describes the relationship between rectangular coordinates $$(x, y)$$ and polar coordinates $$(r, 	heta)$$?

Accepted Answer

$$x = r \cdot \cos 	heta$$, $$y = r \cdot \sin 	heta$$

Question 2

A point is given in polar coordinates as $$(4, \frac{\pi}{2}). $$What are its rectangular coordinates?

Accepted Answer

$$(0, 4)$$

Question 3

Which of the following is a valid alternate representation for the polar point $$(5, \frac{3\pi}{4})$$?

Accepted Answer

$$(-5, -\frac{\pi}{4})$$

Question 4

If a point in polar coordinates is $$(r, 	heta)$$, which of the following also represents the same point?

Accepted Answer

$$(r, 	heta + 2\pi n)$$, where $$n is $$any integer

Question 5

Given the polar point $$(-3, \frac{\pi}{6})$$, which of the following is an equivalent representation?

Accepted Answer

$$(3, \frac{7\pi}{6})$$

Question 6

A ship is located at $$(6, \frac{\pi}{3}) in $$polar coordinates. What is its $$x-$$coordinate in rectangular form?

Accepted Answer

$$6 \cdot \cos \frac{\pi}{3} = 3$$

Question 7

Which of the following points is NOT equivalent to $$(2, \frac{5\pi}{6})$$?

Accepted Answer

$$(2, -\frac{7\pi}{6})$$

Question 8

A drone is flying at $$(8, \frac{\pi}{2}) in $$polar coordinates. What is its $$y-$$coordinate in rectangular form?

Accepted Answer

$$8 \cdot \sin \frac{\pi}{2} = 8$$

Question 9

Which of the following is the correct formula to convert rectangular coordinates $$(x, y) to $$polar coordinates $$(r, 	heta)$$?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$, $$	heta = \arctan \frac{y}{x}$$

Question 10

A lighthouse is located at $$(-7, \frac{2\pi}{3}) in $$polar coordinates. Which of the following is an equivalent representation?

Accepted Answer

$$(7, \frac{5\pi}{3})$$

Question 11

If a point is given as $$(r, 	heta) in $$polar coordinates, which of the following is NOT a valid representation for the same point?

Accepted Answer

$$(r, 	heta + \pi)$$

Question 12

A point is given as $$(5, \frac{\pi}{2}) in $$polar coordinates. What is its $$x-$$coordinate in rectangular form?

Accepted Answer

$$5 \cdot \cos \frac{\pi}{2} = 0$$

Question 13

Which of the following sets of polar coordinates represent the same point as $$(3, \frac{\pi}{4})$$?

Accepted Answer

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

Question 14

A point is given as $$(-2, \frac{\pi}{3}) in $$polar coordinates. What is its equivalent positive $$r$$ representation?

Accepted Answer

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

Question 15

Which of the following is the correct way to plot the point $$(r, 	heta) in $$polar coordinates?

Accepted Answer

Measure angle $$	heta$$ counterclockwise from the polar axis, then move $$r$$ units from the pole

Question 16

A point is given as $$(4, \frac{5\pi}{6}) in $$polar coordinates. Which of the following is NOT an equivalent representation?

Accepted Answer

$$(4, -\frac{7\pi}{6})$$

Question 17

A point is given as $$(2, \frac{\pi}{3}) in $$polar coordinates. What is its $$y-$$coordinate in rectangular form?

Accepted Answer

$$2 \cdot \sin \frac{\pi}{3} = \sqrt{3}$$

Question 18

Which of the following is the correct formula for the angle $$	heta in $$polar coordinates given $$(x, y)$$?

Accepted Answer

$$	heta = \arctan \frac{y}{x}$$

Polar Coordinate System: Concepts, Plotting, and Equivalent Coordinates

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