Q: Given the polar coordinates $$(-2, \pi)$$, where is the point located relative to the pole?

2 units along the ray opposite the terminal side of $$\theta = \pi$$

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

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

Q: Which of the following is a correct method to find all polar coordinates for a given point $$(r, \theta)$$?

Add or subtract integer multiples of $$2\pi to$$ $$\theta$$, or use $$(-r, \theta + \pi).$$

Q: Convert the rectangular equation $$3x - y = 6 to a $$polar equation expressing $$r in $$terms of $$\theta.$$

$$r = \frac{6}{3\cos \theta - \sin \theta}$$

Q: Which of the following is the correct relationship between $$x$$, $$y$$, $$r$$, and $$\theta in $$polar and rectangular coordinates?

$$x = r \cos \theta$$, $$y = r \sin \theta$$, $$r^2$$ = $$x^2$$ + $$y^2$$, $$\tan \theta = \frac{y}{x}$$

Question 1

Which of the following statements correctly describes the pole and the polar axis in the polar coordinate system?

Accepted Answer

The pole is the endpoint of the polar axis, and the polar axis is a horizontal ray extending to the right.

Question 2

Given the polar coordinates $$(-2, \pi)$$, where is the point located relative to the pole?

Accepted Answer

2 units along the ray opposite the terminal side of $$	heta = \pi$$

Question 3

Which of the following is NOT a valid set of polar coordinates for the same point as $$(4, \frac{\pi}{3})$$?

Accepted Answer

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

Question 4

Convert the polar coordinates $$(3, \pi) to $$rectangular coordinates $$(x, y). $$What is the result?

Accepted Answer

$$(-3, 0)$$

Question 5

A point in rectangular coordinates is $$(1, -\sqrt{3}). $$What are its polar coordinates $$(r, 	heta)$$ with $$r \u003e 0$$ and $$0 \u003c 	heta \u003c 2\pi$$?

Accepted Answer

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

Question 6

Which formula correctly converts a point from polar coordinates $$(r, 	heta) to $$rectangular coordinates $$(x, y)$$?

Accepted Answer

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

Question 7

Convert the rectangular equation $$x^2 + (y$$ + $$1)^2 = 1$ to a polar equation expressing r$ in terms of $$	heta$$.

Accepted Answer

$$r(r + 2 \sin 	heta) = 0$$

Question 8

Which of the following rectangular equations is equivalent to the polar equation $$r = 10 \sin 	heta$$?

Accepted Answer

$$x^2$$ + $$y^2 = 10y$

Question 9

A medical imaging device locates a tumor at $$(r, 	heta) = (5, \frac{\pi}{6}) cm in $$polar coordinates. What are the rectangular coordinates of the tumor?

Accepted Answer

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

Question 10

Convert the polar equation $$	heta = \frac{3\pi}{4} to a $$rectangular equation.

Accepted Answer

$$y = -x$$

Question 11

A radar system detects an object at $$(-8, \frac{2\pi}{3}) in $$polar coordinates. What are the rectangular coordinates of the object?

Accepted Answer

$$(4, -4\sqrt{3})$$

Question 12

Which of the following is the correct conversion from rectangular to polar coordinates for the point $$(-3, 3)$$, with $$0  0$$?

Accepted Answer

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

Question 13

Convert the polar equation $$r = -2 \sec 	heta to a $$rectangular equation.

Accepted Answer

$$x = -2$$

Question 14

A chemical reaction occurs at the point $$(0, 4) in $$rectangular coordinates. What are the polar coordinates $$(r, 	heta)$$ with $$0  0$$?

Accepted Answer

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

Question 15

Which of the following is a correct method to find all polar coordinates for a given point $$(r, 	heta)$$?

Accepted Answer

Add or subtract integer multiples of $$2\pi to$$ $$	heta$$, or use $$(-r, 	heta + \pi).$$

Question 16

Convert the rectangular equation $$3x - y = 6 to a $$polar equation expressing $$r in $$terms of $$	heta.$$

Accepted Answer

$$r = \frac{6}{3\cos 	heta - \sin 	heta}$$

Question 17

A drone flies to the point $$(r, 	heta) = (10, \frac{\pi}{2})$$ meters. What are the rectangular coordinates of the drone?

Accepted Answer

$$(0, 10)$$

Question 18

Which of the following is the correct relationship between $$x$$, $$y$$, $$r$$, and $$	heta in $$polar and rectangular coordinates?

Accepted Answer

$$x = r \cos 	heta$$, $$y = r \sin 	heta$$, $$r^2$$ = $$x^2$$ + $$y^2$$, $$	an 	heta = \frac{y}{x}$$

Polar Coordinates and Conversions in Precalculus

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