# Polar Coordinates Worksheet

Polar Coordinates Worksheet
• Page 1
1.
Which of the following is correct?
 a. Polar coordinate system is a line b. Polar coordinate system is a Point c. Polar coordinate system is a plane d. Polar coordinate system is a space

#### Solution:

A polar coordinate system is a plane on which every point is associated with two polar coordinates.
[Definition.]

2.
What is the number of coordinates of a point in a polar coordinate system?
 a. Infinite b. 4 c. 3 d. 1

#### Solution:

Every point of a plane in a polar coordinate system has infinite number of polar coordinates.

3.
What are the polar coordinates of the pole?
 a. ($\theta$, 0) b. (0, 0) c. (1, 2) d. (0, $\theta$)

#### Solution:

Pole has polar coordinates (0, θ), where θ is any angle.

4.
Which of the following is correct?
 a. Every point of a polar coordinate system has unique pair of polar coordinates. b. Each pair of polar coordinates determine a unique point. c. Every point of a polar coordinate system has 2 pairs of polar coordinates. d. Each pair of polar coordinates does not determine a unique point.

#### Solution:

For each pair of polar coordinates, (r, θ), there exists a unique point in the plane whose directed distance from the pole is r and whose directed angle is θ.

However, the polar coordinates of a point with directed distance r from the pole, directed angle θ are many like (r, θ), (r, 2π + θ), (r, 4π + θ).

So, each pair of polar coordinates determine a unique point and for a point, the pairs of polar coordinates are not unique.

5.
From the polar coordinate system shown, the coordinates of P are

 a. (7, 60o) b. (5, 60o) c. (4, 60o) d. (6, 100o)

#### Solution:

The directed distance of the point P from the pole is r = 5.
[From the figure.]

The measure of the angle made by the line OP with polar axis in the counter clockwise direction is 60o.

So, the polar coordinates of P are (5, 60o).

6.
The polar coordinates of point P are ($\sqrt{2}$, $\frac{\pi }{4}$). Which of the following are the rectangular coordinates for point P?
 a. (- 1, - 1) b. (1, 1) c. (- 1, 1) d. (1, - 1)

#### Solution:

Polar coordinates of P are (r, θ) = (2, π4).

r = 2, θ = π4

Let (x, y) be the rectangular coordinates of P.

So, x = r cos θ = 2 cos π4 = 1 and y = r sin θ = 2 sin π4 = 1

(x, y) = (1, 1).

7.
Let P be a point in a polar coordinate system. If $r$ is the directed distance from the pole O to P and $\theta$ is the directed angle measured in counter clockwise direction with the initial side on the polar axis and the terminal side on the line OP, then write the polar coordinates of P.
 a. (- $r$, $\theta$) b. ($r$, $\theta$) c. (- $\theta$, $r$) d. ($\theta$, $r$)

#### Solution:

The first polar coordinate of a point in a polar coordinate system is its directed distance from the pole O.
[Use the definition of directed distance.]

Here for P, the directed distance from the pole O is r.

The second polar coordinate of a point in a polar coordinate system is its directed angle whose initial side is on the polar axis and the terminal side is on the line OP.
[Use the definition of the directed angle.]

Here for P, the directed angle is θ.

So, the polar coordinates of P are (r, θ).

8.
The polar coordinates of a point P are ($r$, $\theta$). Which of the following is the polar coordinate pair of P other than ($r$, $\theta$)?
 a. ($r$, $\theta$ + $\frac{3\pi }{2}$) b. ($r$, $\theta$ + 2$\pi$) c. ($r$, $\theta$ + $\frac{\pi }{2}$) d. ($r$, $\theta$ - $\frac{\pi }{2}$)

#### Solution:

If (r, θ) are the polar coordinates of a point, then any other polar coordinates of the point must be of the form (r, θ + 2nπ) or (- r, θ + (2n + 1)π), where n is any integer.

For n = 1, (r, θ + 2nπ) = (r, θ + 2π)

So, among the choices, (r, θ + 2π) are the polar coordinates of P other than (r, θ).

9.
Which of the following is correct?
 a. Both (2, $\frac{\pi }{6}$) and (2, $\frac{7\pi }{6}$) represent the same point. b. Both (2, $\frac{\pi }{6}$) and (2, $\frac{13\pi }{6}$) do not represent the same point. c. Both (2, $\frac{\pi }{6}$) and (2, $\frac{7\pi }{6}$) do not represent the same point. d. Both (2, $\frac{\pi }{6}$) and (2, $\frac{13\pi }{6}$) represent the same point.

#### Solution:

For integral values of n, both (r, θ + 2nπ) and (r, θ) represent the same point in a polar coordinate system.

Then (r, θ + 2π) and (r, θ) represent the same point.
[Substitute n = 1.]

(2, 13π6) = (2, π6 + 2π)
[Write 13π6 as π6 + 2π.]

So, both (2, π6) and (2, 13π6) represent the same point.

10.
Which of the following is correct?
 a. Both (3, $\frac{\pi }{3}$) and (- 3, $\frac{\pi }{3}$) represent the same point. b. Both (3, $\frac{\pi }{3}$) and (- 3, - $\frac{\pi }{3}$) represent the same point. c. Both (3, $\frac{\pi }{3}$) and (- 3, - $\frac{2\pi }{3}$) represent the same point. d. Both (3, $\frac{\pi }{3}$) and (3, - $\frac{\pi }{3}$) represent the same point.

#### Solution:

For integral values of n, both (r, θ), (- r, θ + (2n + 1) π) represent the same point in a polar coordinate system.

So, (- r, θ - π) and (r, θ) represent the same point.
[Substitute n = -1.]

(-3, - 2π3) = (-3, π3 - π)
[Write - 2π3 as π3 - π.]

So, both (3, π3) and (- 3, - 2π3) represent the same point.