Q: Which of the following correctly expresses the relationship between Cartesian coordinates $$(x, y)$$ and polar coordinates $$(r, \theta)$$?

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

Q: Which of the following is the correct formula for converting from Cartesian to polar coordinates?

$$r = \sqrt{x^2$ + y^2$}, \quad \theta = \tan$$^{-1}$$\left(\frac{y}{x}\right)$$

Q: Which of the following statements is TRUE about the symmetry of the polar curve $$r = \sin \theta$$?

It is symmetric about the vertical line $$\theta = \frac{\pi}{2}.$$

Q: If $$r \u003c 0 in $$polar coordinates, what does this mean geometrically?

The point lies on the line through the origin at angle $$\theta$$, but on the opposite side of the origin from $$r \u003e 0.$$

Q: Which of the following is the correct expression for $$\frac{dy}{dx} in $$terms of $$r$$ and $$\theta$$ for a polar curve $$r = f(\theta)$$?

$$\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$

Question 1

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

Accepted Answer

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

Question 2

A point in polar coordinates is given as $$\left(2, \frac{\pi}{3}\right). $$What are its Cartesian coordinates?

Accepted Answer

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

Question 3

Which of the following statements about polar coordinates is TRUE?

Accepted Answer

A single point in the plane can have infinitely many polar coordinate representations.

Question 4

Which polar equation represents a circle of radius $$2$$ centered at the origin?

Accepted Answer

$$r = 2$$

Question 5

Convert the Cartesian point $$(-1, 1) to $$polar coordinates with $$r \u003e 0$$ and $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

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

Question 6

Which of the following polar equations represents a cardioid?

Accepted Answer

$$r = 1 + \sin 	heta$$

Question 7

If a polar curve is unchanged when $$	heta is $$replaced by $$-	heta$$, what type of symmetry does the curve have?

Accepted Answer

Symmetry about the polar axis (the $$x-$$axis)

Question 8

Which of the following is a correct representation for the point $$\left(1, \frac{\pi}{4}\right) in $$polar coordinates?

Accepted Answer

$$\left(1, \frac{\pi}{4} + 2\pi n\right)$$ for any integer $$n$$

Question 9

Given the polar equation $$r = 1 + \sin 	heta$$, what is the maximum value of $$r$$?

Accepted Answer

$$2$$

Question 10

For the polar curve $$r = 1 + \sin 	heta$$, as $$	heta$$ increases from $$0 to$$ $$\frac{\pi}{2}$$, what happens to $$r$$?

Accepted Answer

$$r$$ increases from $$1 to$$ $$2$$

Question 11

Which of the following is the correct formula for converting from Cartesian to polar coordinates?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}, \quad 	heta = 	an$$^{-1}$$\left(\frac{y}{x}ight)$$

Question 12

A polar curve is given by $$r = 2 \cos 	heta. $$What is the Cartesian equation of this curve?

Accepted Answer

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

Question 13

Which of the following statements is TRUE about the symmetry of the polar curve $$r = \sin 	heta$$?

Accepted Answer

It is symmetric about the vertical line $$	heta = \frac{\pi}{2}.$$

Question 14

If $$r \u003c 0 in $$polar coordinates, what does this mean geometrically?

Accepted Answer

The point lies on the line through the origin at angle $$	heta$$, but on the opposite side of the origin from $$r \u003e 0.$$

Question 15

Which of the following is the correct expression for $$\frac{dy}{dx} in $$terms of $$r$$ and $$	heta$$ for a polar curve $$r = f(	heta)$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{\frac{dr}{d	heta} \sin 	heta + r \cos 	heta}{\frac{dr}{d	heta} \cos 	heta - r \sin 	heta}$$

Question 16

A medical imaging device scans a circular region of tissue with radius $$3 cm. $$What is the polar equation for the boundary of the scanned region?

Accepted Answer

$$r = 3$$

Question 17

Which of the following points lies in the same quadrant as $$	heta = \frac{5\pi}{6}$$ with $$r \u003e 0$$?

Accepted Answer

$$\left(2, \frac{5\pi}{6}\right)$$

Question 18

A chemical reaction occurs in a circular petri dish of radius $$4 cm. $$Which polar equation describes the edge of the dish?

Accepted Answer

$$r = 4$$

Question 19

For the polar curve $$r = 1 + \sin 	heta$$, at which value of $$	heta$$ does $$r$$ attain its minimum value?

Accepted Answer

$$\frac{3\pi}{2}$$

Section 1.3 - Polar Coordinates

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