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

$r = 10, \ \theta = 53.13^\circ$

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

$r = 5.66, \ \theta = 315^\circ$

$\theta = \arctan\left(\frac{y}{x}\right) + 180^\circ$

$\theta = \arctan\left(\frac{y}{x}\right) - 90^\circ$ for $y < 0$

Question 1

Given a point in rectangular coordinates $(x, y)$, which formula correctly converts it to polar coordinates $(r, 	heta)$?

Accepted Answer

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

Question 2

A ship is located at the point $(6, 8)$ on a map. What is its distance from the origin and the angle it makes with the positive $x$-axis?

Accepted Answer

$r = 10, \ 	heta = 53.13^\circ$

Question 3

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

Accepted Answer

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

Question 4

A drone is flying at a position given by polar coordinates $(r = 5, 	heta = 120^\circ)$. What are its rectangular coordinates?

Accepted Answer

$x = -2.5, \ y = 4.33$

Question 5

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

Accepted Answer

$135^\circ$

Question 6

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

Accepted Answer

$r = 7, \ 	heta = 270^\circ$

Question 7

Which of the following statements is true about the relationship between polar and rectangular coordinates?

Accepted Answer

Every point in the plane can be represented in both polar and rectangular coordinates.

Question 8

A rescue helicopter is at $(x, y) = (4, -4)$. What is the value of $r$ and $	heta$ in degrees?

Accepted Answer

$r = 5.66, \ 	heta = 315^\circ$

Question 9

Which formula would you use to find the angle $	heta$ for a point $(x, y)$ in the second quadrant?

Accepted Answer

$	heta = \arctan\left(\frac{y}{x}ight) + 180^\circ$

Question 10

A weather balloon is at $(r = 10, 	heta = 225^\circ)$. What are its rectangular coordinates?

Accepted Answer

$x = -7.07, \ y = -7.07$

Question 11

A point is given in polar coordinates as $(r = 8, 	heta = 0^\circ)$. What are its rectangular coordinates?

Accepted Answer

$x = 8, \ y = 0$

Question 12

Which of the following is a correct application of the Pythagorean Theorem in converting rectangular to polar coordinates?

Accepted Answer

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

Question 13

A point $(x, y)$ is located at $(0, 5)$. What is its polar angle $	heta$?

Accepted Answer

$90^\circ$

Question 14

A car is parked at $(x, y) = (-6, 0)$. What are its polar coordinates?

Accepted Answer

$r = 6, \ 	heta = 180^\circ$

Question 15

Which of the following is NOT a correct way to express the angle $	heta$ in polar coordinates?

Accepted Answer

$	heta = \arctan\left(\frac{y}{x}ight) - 90^\circ$ for $y < 0$

Converting Between Polar and Rectangular Coordinates

Study Guide - Practice Questions