Q: In the polar coordinate system, which of the following correctly describes the location of a point with coordinates $$(r, \theta)$$?

The point is $$r$$ units from the origin at an angle $$\theta$$ measured counterclockwise from the positive $$x-$$axis.

Q: Which equation converts polar coordinates $$(r, \theta) to $$rectangular coordinates $$(x, y)$$?

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

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

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

Q: Which of the following is NOT a valid way to represent the same point in polar coordinates?

$$(r, \theta)$$ and $$(r, \theta - \pi)$$

Q: A drone is located at $$(x, y) = (0, -7). $$What are its polar coordinates $$(r, \theta)$$?

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

Question 1

In the polar coordinate system, which of the following correctly describes the location of a point with coordinates $$(r, 	heta)$$?

Accepted Answer

The point is $$r$$ units from the origin at an angle $$	heta$$ measured counterclockwise from the positive $$x-$$axis.

Question 2

Which equation converts polar coordinates $$(r, 	heta) to $$rectangular coordinates $$(x, y)$$?

Accepted Answer

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

Question 3

A point in the polar coordinate system is given as $$(5, \frac{\pi}{3}). $$What are its rectangular coordinates $$(x, y)$$?

Accepted Answer

$$x = 4.33$$, y = 2.5$

Question 4

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

Accepted Answer

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

Question 5

If a point has rectangular coordinates $$(x, y) = (-3, 3)$$, what is its polar angle $$	heta (in $$radians)?

Accepted Answer

$$\frac{3\pi}{4}$$

Question 6

A radar detects an object at a distance of $$10$$ units from the origin at an angle of $$120^\circ$$ from the positive $$x-$$axis. What are the rectangular coordinates of the object?

Accepted Answer

$$x = -5, y = 8.66$$

Question 7

Which of the following statements is true about the polar coordinate $$(r, 	heta) if$$ $$r \u003c 0$$?

Accepted Answer

The point is located $$|r|$$ units from the origin in the direction $$	heta + \pi.$$

Question 8

A ship is sailing at a position described by polar coordinates $$(8, \frac{\pi}{6}). $$What is the $$y-$$coordinate of the ship in rectangular coordinates?

Accepted Answer

$$y = 4$$

Question 9

Which of the following is NOT a valid way to represent the same point in polar coordinates?

Accepted Answer

$$(r, 	heta)$$ and $$(r, 	heta - \pi)$$

Question 10

A drone is located at $$(x, y) = (0, -7). $$What are its polar coordinates $$(r, 	heta)$$?

Accepted Answer

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

Question 11

If a point in polar coordinates is $$(r, 	heta) = (6, 210^\circ)$$, what is its $$x-$$coordinate in rectangular coordinates?

Accepted Answer

$$x = -5.2$$

Question 12

Which of the following best describes the relationship between the polar and rectangular coordinate systems?

Accepted Answer

Polar coordinates use distance and angle, while rectangular coordinates use horizontal and vertical distances.

Polar Coordinate System and Angle Measurement

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