Q: Which of the following correctly describes the location of the point $$(-r, \theta) in $$polar coordinates, where $$r \u003e 0$$?

It lies on the terminal side of angle $$\theta + \pi at a $$distance $$r$$ from the pole.

Q: Convert the rectangular coordinates $$(x, y) = (3, 4) to $$polar coordinates $$(r, \theta).$$

$$(r, \theta) = (5, \arctan \frac{4}{3})$$

Q: Which formula is used to convert a point from polar coordinates $$(r, \theta) to $$rectangular coordinates $$(x, y)$$?

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

Q: If a point $$P$$ lies in Quadrant III with rectangular coordinates $$(x, y)$$, which formula gives the polar angle $$\theta$$?

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

Q: Which of the following is the correct relationship for converting from rectangular to polar coordinates?

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

Q: Which of the following is the polar form of the equation $$y = x$$?

$$r \sin \theta = r \cos \theta$$

Q: A point $$P$$ has rectangular coordinates $$(-2, 2). In $$which quadrant does $$P$$ lie and what is the formula for its polar angle $$\theta$$?

Quadrant II, $$\theta = \arctan \frac{2}{-2} + \pi$$

Question 1

Which of the following correctly describes the location of the point $$(-r, 	heta) in $$polar coordinates, where $$r \u003e 0$$?

Accepted Answer

It lies on the terminal side of angle $$	heta + \pi at a $$distance $$r$$ from the pole.

Question 2

Given the polar coordinates $$(0, 	heta)$$, where $$	heta is $$any angle, what is the rectangular coordinate of the point?

Accepted Answer

$$(0, 0)$$

Question 3

A point has polar coordinates $$(r, 	heta). $$Which of the following is NOT an equivalent representation of the same point?

Accepted Answer

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

Question 4

Convert the rectangular coordinates $$(x, y) = (3, 4) to $$polar coordinates $$(r, 	heta).$$

Accepted Answer

$$(r, 	heta) = (5, \arctan \frac{4}{3})$$

Question 5

A ship is located at $$(r, 	heta) = (10, \frac{\pi}{3}) in $$polar coordinates. What are its rectangular coordinates $$(x, y)$$?

Accepted Answer

$$(x, y) = (10 \cos \frac{\pi}{3}, 10 \sin \frac{\pi}{3})$$

Question 6

Which formula is used to convert a point from polar coordinates $$(r, 	heta) to $$rectangular coordinates $$(x, y)$$?

Accepted Answer

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

Question 7

Given the rectangular coordinates $$(-5, 0)$$, what are the polar coordinates?

Accepted Answer

$$(5, \pi)$$

Question 8

A point $$P$$ has rectangular coordinates $$(0, -7). $$What are its polar coordinates?

Accepted Answer

$$(7, \frac{3\pi}{2})$$

Question 9

If a point $$P$$ lies in Quadrant III with rectangular coordinates $$(x, y)$$, which formula gives the polar angle $$	heta$$?

Accepted Answer

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

Question 10

Which of the following is the correct relationship for converting from rectangular to polar coordinates?

Accepted Answer

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

Question 11

Convert the equation $$x^2$$ + $$y^2 = 9$ from rectangular to polar form.

Accepted Answer

$$r^2 = 9$

Question 12

Which of the following is the polar form of the equation $$y = x$$?

Accepted Answer

$$r \sin 	heta = r \cos 	heta$$

Question 13

A drone flies to the point $$(r, 	heta) = (8, \frac{5\pi}{6}). $$What is its $$y-$$coordinate in rectangular form?

Accepted Answer

$$y = 8 \sin \frac{5\pi}{6}$$

Question 14

Which of the following is a valid polar equation for a circle centered at the pole with radius $$a$$?

Accepted Answer

$$r = a$$

Question 15

If a point has polar coordinates $$(-6, \frac{\pi}{4})$$, what is its equivalent polar coordinate with positive $$r$$?

Accepted Answer

$$(6, \frac{5\pi}{4})$$

Question 16

A lighthouse is located at $$(x, y) = (0, 12). $$What are its polar coordinates?

Accepted Answer

$$(12, \frac{\pi}{2})$$

Question 17

Which of the following is the correct conversion from the polar equation $$r = 4 \cos 	heta to $$rectangular form?

Accepted Answer

$$x = 4$$

Question 18

A point $$P$$ has rectangular coordinates $$(-2, 2). In $$which quadrant does $$P$$ lie and what is the formula for its polar angle $$	heta$$?

Accepted Answer

Quadrant II, $$	heta = \arctan \frac{2}{-2} + \pi$$

Question 19

Which of the following is the correct rectangular form of the polar equation $$r = 2 \sin 	heta$$?

Accepted Answer

$$y = 2$$

Polar Coordinates and Polar Equations: Study Notes

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