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

$$\theta = \arctan \left( \frac{y}{x} \right)$$

$$\theta = \arctan \left( \frac{8}{-6} \right)$$

$$r \sin \theta = r \cos \theta$$

$$\theta = \arctan \left( \frac{-7}{0} \right)$$

Question 1

Which of the following equations correctly converts from rectangular coordinates $$(x, y) to $$polar coordinates $$(r, 	heta)$$?

Accepted Answer

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

Question 2

Given the rectangular equation $$x^2$$ + $$y^2 = 9$, what is the equivalent equation in polar coordinates?

Accepted Answer

$$r^2 = 9$

Question 3

If a point in rectangular coordinates is $$(3, 4)$$, what is its $$r$$ value in polar coordinates?

Accepted Answer

$$r = \sqrt{3^2$ + 4^2$}$$

Question 4

Which formula gives the angle $$	heta in $$polar coordinates from rectangular coordinates $$(x, y)$$?

Accepted Answer

$$	heta = \arctan \left( \frac{y}{x} ight)$$

Question 5

Convert the polar equation $$r = 2 \cos 	heta to $$rectangular form.

Accepted Answer

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

Question 6

A lighthouse is located at the origin. A boat is at $$(r, 	heta) = (5, \frac{\pi}{3}). $$What are the rectangular coordinates of the boat?

Accepted Answer

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

Question 7

Which of the following is the correct rectangular form of the polar equation $$r = 4 \sin 	heta$$?

Accepted Answer

$$x^2$$ + $$y^2 = 4y$

Question 8

If $$x = r \cos 	heta$$ and $$y = r \sin 	heta$$, what is $$r in $$terms of $$x$$ and $$y$$?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$

Question 9

A drone flies from the origin to the point $$(x, y) = (-6, 8). $$What is the angle $$	heta in $$polar coordinates (in radians)?

Accepted Answer

$$	heta = \arctan \left( \frac{8}{-6} ight)$$

Question 10

Which of the following is a correct step in converting $$r = 3 to $$rectangular coordinates?

Accepted Answer

$$x^2$$ + $$y^2 = 9$

Question 11

A radar detects an object at $$(r, 	heta) = (10, \frac{\pi}{4}). $$What is the $$y-$$coordinate in rectangular form?

Accepted Answer

$$10 \sin \frac{\pi}{4}$$

Question 12

Convert the rectangular equation $$y = x to $$polar form.

Accepted Answer

$$r \sin 	heta = r \cos 	heta$$

Question 13

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

Accepted Answer

$$r \cos 	heta = 2$$

Question 14

A weather balloon is tracked at $$(x, y) = (0, -7). $$What is $$	heta in $$polar coordinates?

Accepted Answer

$$	heta = \arctan \left( \frac{-7}{0} ight)$$

Question 15

Which of the following is the correct rectangular form of the polar equation $$r = 5$$?

Accepted Answer

$$x^2$$ + $$y^2 = 25$

Converting Between Polar and Rectangular Forms in Trigonometry

Study Guide - Practice Questions