Q: Which of the following best describes the effect of a negative radius in polar coordinates?

The point is reflected across the origin from the angle $$\theta.$$

Q: If a point in polar coordinates is $$(r, \theta)$$, which of the following is always true for the Cartesian coordinates $$(x, y)$$?

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

Question 1

Which of the following is the correct Cartesian coordinate for the polar point $$(-1, \frac{3\pi}{4})$$?

Accepted Answer

$$(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$$

Question 2

Given the polar coordinate $$(-4, -\frac{5\pi}{6})$$, what is the equivalent polar coordinate with a positive radius?

Accepted Answer

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

Question 3

Which of the following best describes the effect of a negative radius in polar coordinates?

Accepted Answer

The point is reflected across the origin from the angle $$	heta.$$

Question 4

Convert the polar coordinate $$(-2, -\frac{2\pi}{3}) to $$Cartesian coordinates.

Accepted Answer

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

Question 5

A point is given in polar coordinates as $$(-1, -\frac{\pi}{3}). $$What is the equivalent polar coordinate with a positive radius?

Accepted Answer

$$(1, \frac{2\pi}{3})$$

Question 6

Which of the following polar coordinates represents the same point as $$(2, \frac{\pi}{4})$$?

Accepted Answer

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

Question 7

If a point in polar coordinates is $$(r, 	heta)$$, which of the following is always true for the Cartesian coordinates $$(x, y)$$?

Accepted Answer

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

Question 8

A drone is flying at a distance of 5 units from a central tower at an angle of $$\frac{2\pi}{3}$$ from the positive $$x-$$axis. What are its Cartesian coordinates?

Accepted Answer

$$(-2.5, 4.33)$$

Question 9

Which of the following polar coordinates is NOT equivalent to $$(3, \frac{\pi}{2})$$?

Accepted Answer

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

Question 10

A lighthouse is located at the origin. A boat is at polar coordinates $$(6, \frac{7\pi}{6}). $$What is the $$y-$$coordinate of the boat in Cartesian coordinates?

Accepted Answer

$$-3\sqrt{3}$$

Question 11

Which of the following statements is true about the polar coordinate $$(r, 	heta)$$ and its equivalent $$(r, 	heta + 2\pi)$$?

Accepted Answer

They represent the same point.

Question 12

A point is given as $$(-3, \frac{\pi}{6}) in $$polar coordinates. What is the equivalent Cartesian coordinate?

Accepted Answer

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

Question 13

Which of the following is the correct conversion from Cartesian to polar coordinates for the point $$(-1, 1)$$?

Accepted Answer

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

Question 14

A point is at $$(r, 	heta) = (2, \frac{\pi}{3}). If $$the radius is doubled and the angle is increased by $$\frac{\pi}{3}$$, what are the new Cartesian coordinates?

Accepted Answer

$$(-2, 3.464)$$

Question 15

Which of the following is the correct polar equation for a circle of radius 4 centered at the origin?

Accepted Answer

$$r = 4$$

Guided Practice: Plotting Points in Polar Coordinates

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