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

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

Q: Convert the rectangular equation $$y = 2x to $$polar form and solve for $$r.$$

$$r = 0 or$$ $$\tan\theta = 2$$

Q: Which of the following is the correct polar form for the rectangular equation $$x^2 = y$?

$$r^2$$\$$cos^2$$\theta = r\sin\theta$$

Question 1

Which of the following is the correct rectangular coordinate for the polar point $$\left(5, \frac{5\pi}{3}\right) as $$shown in the first image?

Accepted Answer

$$\left(5\cos\frac{5\pi}{3},\ 5\sin\frac{5\pi}{3}\right) = \left(2.5,\ -4.33\right)$$

Question 2

Given the polar coordinate $$\left(4, \frac{3\pi}{4}\right) as $$shown in the second image, which quadrant does this point lie in?

Accepted Answer

Second quadrant

Question 3

For the polar coordinate $$(-3, 120^\circ) as $$shown in the third image, what is the equivalent polar coordinate with a positive radius?

Accepted Answer

$$(300^\circ, 3)$$

Question 4

Which of the following is an equivalent polar coordinate for $$\left(-3, -\frac{3\pi}{4}\right) as $$shown in the fourth image, with a positive radius and angle between $$0$$ and $$2\pi$$?

Accepted Answer

$$\left(\frac{5\pi}{4}, 3\right)$$

Question 5

Convert the rectangular coordinate $$(-3, 3) to $$polar form. Express your angle in radians.

Accepted Answer

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

Question 6

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

Accepted Answer

$$x^2$$ + $$y^2 = 16$

Question 7

Given the polar equation $$r = 5\sec	heta$$, what is the equivalent rectangular equation?

Accepted Answer

$$x = 5$$

Question 8

Convert the rectangular equation $$x^2$$ + $$y^2 = 2x$ to polar form and solve for r$.

Accepted Answer

$$r = 2\cos	heta$$

Question 9

A point is given in polar coordinates as $$\left(4, -\frac{5\pi}{6}\right). $$What is an equivalent coordinate with a positive radius and angle between $$0$$ and $$2\pi$$?

Accepted Answer

$$\left(4, \frac{7\pi}{6}\right)$$

Question 10

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

Accepted Answer

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

Question 11

Convert the rectangular coordinate $$\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) to $$polar form. Express your angle in radians.

Accepted Answer

$$\left(1, -\frac{\pi}{3}\right)$$

Question 12

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

Accepted Answer

$$y = 2$$

Question 13

A point is given as $$(-2, \frac{3\pi}{2}) in $$polar coordinates. What is an equivalent coordinate with a positive radius and angle between $$0$$ and $$2\pi$$?

Accepted Answer

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

Question 14

Convert the rectangular equation $$y = 2x to $$polar form and solve for $$r.$$

Accepted Answer

$$r = 0 or$$ $$	an	heta = 2$$

Question 15

Which of the following is the correct polar form for the rectangular equation $$x^2 = y$?

Accepted Answer

$$r^2$$\$$cos^2$$	heta = r\sin	heta$$

Question 16

Given the polar equation $$r = 2\csc	heta$$, what is the equivalent rectangular equation?

Accepted Answer

$$y = 2$$

Question 17

A point is given as $$(2, -\frac{\pi}{6}) in $$polar coordinates. What is an equivalent coordinate with a negative radius and angle between $$0$$ and $$2\pi$$?

Accepted Answer

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

Polar Coordinates: Representation, Conversion, and Equivalent Forms 6.3

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