Polar Coordinates: Master the Art of Graphing and Equations

Question 1

Find the equation of the polar curve that represents a circle with radius 2 centered at the origin.

Accepted Answer

r = 2

Answer

r = 2cos(theta)

Answer

r = 2sin(theta)

Answer

r^2 = 2

Question 2

Find the area enclosed by the polar curve r = 2sin(2theta).

Accepted Answer

4pi

Answer

pi

Answer

2pi

Answer

8pi

Question 3

Which of the following is the polar equation of the line y = x?

Accepted Answer

r = cos(theta) + sin(theta)

Answer

r = sin(theta) + cos(theta)

Answer

r = tan(theta) + cot(theta)

Answer

r = cot(theta) + tan(theta)

Question 4

Find the points of intersection between the polar curves r = 2 and r = 1 + cos(theta).

Accepted Answer

(1, pi/3), (1, 5pi/3)

Answer

(2, pi/3), (2, 5pi/3)

Answer

(1, pi/2), (1, 3pi/2)

Answer

(2, pi/2), (2, 3pi/2)

Question 5

Which of the following represents the polar coordinate of the rectangular coordinate (3,4)?

Accepted Answer

(3, π/4)

Answer

(4, π/3)

Answer

(5, π/6)

Answer

(6, π/4)

Question 6

Convert the polar coordinates (2, 5π/6) to rectangular coordinates.

Accepted Answer

(√3, 1)

Answer

(-√3, 1)

Answer

(-1, √3)

Answer

(1, √3)

Question 7

Which of the following is the graph of the polar equation r = 2 - cos(θ)?

Accepted Answer

A cardioid

Answer

A circle

Answer

A lemniscate

Answer

A rose curve

Question 8

Find the polar coordinates of the point ( (-2, 2sqrt{3}) ).

Accepted Answer

(2, 60°)

Answer

(2, -60°)

Answer

(2, 120°)

Answer

(2, -120°)

Question 9

Graph the polar equation ( r = 2 + cos(	heta) ).

Accepted Answer

A cardioid with a cusp at ( (2, 0) ).

Answer

A circle with the center at ( (1, 0) ).

Answer

A limaçon with an inner loop.

Answer

A rose curve with 4 petals.

Question 10

What is the length of the curve ( r = 	heta ) from ( 	heta = 0 ) to ( 	heta = pi/2 )?

Accepted Answer

π

Answer

π/2

Answer

2π

Answer

4π

Question 11

Which of the following is a polar equation of a circle?

Accepted Answer

r = a

Answer

r = a cos(	heta)

Answer

r = a sin(	heta)

Answer

r = a cos(	heta) + b sin(	heta)

Question 12

Find the polar equation of the line ( y = x + 1 ).

Accepted Answer

r = sec(	heta) + 1

Answer

r = csc(	heta) - 1

Answer

r = 	an(	heta) - 1

Answer

r = cot(	heta) + 1

Question 13

Which of the following is the polar form of the complex number 3 + 4i?

Accepted Answer

(5, 36.9°)

Answer

(3, 4)

Answer

(5, -36.9°)

Answer

(4, 3)

Question 14

Find the rectangular coordinates of the point (5, π/3).

Accepted Answer

(2.5, 4.33)

Answer

(5, 0)

Answer

(-2.5, 4.33)

Answer

(-5, 0)

Question 15

Which of the following is the equation of a horizontal line passing through the point (2, π/2)?

Accepted Answer

θ = π/2

Answer

r = 2

Answer

r = 0

Answer

θ = 0

Question 16

Which of the following is the equation of a spiral that starts at the origin and spirals outward?

Accepted Answer

r = e^θ

Answer

r = θ

Answer

r = 1/θ

Answer

r = θ^2

Question 17

Which of the following is the polar equation of the circle centered at the origin with radius 5?

Accepted Answer

r = 5

Answer

θ = 5

Answer

r = 5cos(θ)

Answer

r = 5sin(θ)

Question 18

Which of the following is the polar equation for the limaçon with inner loop?

Accepted Answer

r = 2 - cos(θ)

Answer

r = 2 + cos(θ)

Answer

r = 2 + 2cos(θ)

Answer

r = 2 - 2cos(θ)

Question 19

Convert the complex number -1 + i to polar form (r, θ).

Accepted Answer

(1, -π/4)

Answer

(1, π/4)

Answer

(√2, π/4)

Answer

(√2, -π/4)

Question 20

Find the rectangular form of the polar coordinate (3, 5π/6).

Accepted Answer

(-3√3/2, 3/2)

Answer

(-3/2, -3√3/2)

Answer

(3√3/2, 3/2)

Answer

(3/2, -3√3/2)

Question 21

Which polar equation represents a circle with radius 5 centered at the origin?

Accepted Answer

r = 5

Answer

θ = 5

Answer

r = 5sinθ

Answer

r = 5cosθ

Question 22

Which inequality represents the shaded region in the polar coordinate plane bounded by the circle r ≤ 3?

Accepted Answer

r ≤ 3

Answer

r ≥ 3

Answer

0 ≤ θ ≤ π/2

Answer

π/2 ≤ θ ≤ π

Question 23

Identify the graph of the polar equation r = 2 - sin(2θ).

Accepted Answer

A cardioid

Answer

A circle with radius 1

Answer

A rose with 4 petals

Answer

A rose with 8 petals

Question 24

Which polar equation represents a circle with radius 4?

Accepted Answer

r = 4

Answer

r = 2

Answer

r = 4cos(θ)

Answer

r = 4sin(θ)

Question 25

Identify the polar equation of the limaçon with outer loop.

Accepted Answer

r = 2 + cos(θ)

Answer

r = 2 - cos(θ)

Answer

r = 1 + 2cos(θ)

Answer

r = 1 - 2cos(θ)

Question 26

Which of the following represents the polar form of the complex number 2 + 3i?

Accepted Answer

(√13, π/3)

Answer

(2, π/3)

Answer

(2, π/6)

Answer

(√13, π/4)

Question 27

Graph the polar equation r = 2cos(θ) and identify the shape it represents.

Accepted Answer

A cardioid with a cusp at the origin

Answer

A circle with radius 2 centered at the origin

Answer

A limaçon with a loop at the origin

Answer

A rose with 4 petals

Question 28

Write the polar equation of the circle with radius 3 centered at (2, π/4).

Accepted Answer

r = 3cos(θ - π/4)

Answer

r = 3

Answer

r = 3sin(θ - π/4)

Answer

r = 3(cos(θ) + sin(θ))

Question 29

Find the area of the sector of the circle r = 5 bounded by the rays θ = π/4 and θ = 3π/4.

Accepted Answer

25π/8

Answer

5π/4

Answer

10π/4

Answer

50π/8

Question 30

Find all points of intersection between the polar curves r = 2 and r = 1 + cos(θ).

Accepted Answer

(1, 0), (1, π/2), (2, π/3), (2, 5π/3)

Answer

(1, π/3), (1, 5π/3)

Answer

(2, 0), (2, π)

Answer

(1, π/4), (1, 3π/4)

Question 31

Express the complex number 3 + 4i in polar form.

Accepted Answer

(5, arctan(4/3))

Answer

(3, 4)

Answer

(4, arctan(3/4))

Answer

(7, π/4)

Question 32

Determine the polar equation of the circle with radius 3 centered at the origin.

Accepted Answer

r = 3

Answer

r = 3sinθ

Answer

r = 3cosθ

Answer

r = 3tanθ

Question 33

Identify the type of polar curve represented by the equation r = 2 + cosθ.

Accepted Answer

Cardioid

Answer

Circle

Answer

Line

Answer

Rose curve

Question 34

Which of the following is the graph of the polar equation $r = 2 - 2\sin(\theta)$?

Accepted Answer

A cardioid

Answer

A circle

Answer

A rose curve

Answer

A spiral

Question 35

Which point is represented by the polar coordinates (3, π/3)?

Accepted Answer

(3, √3)

Answer

(-3, √3)

Answer

(-3, -√3)

Answer

(3, -√3)

Question 36

Which graph represents the polar equation r = 3cos(2θ)?

Accepted Answer

A limaçon with inner loop radius 3

Answer

A circle with radius 3

Answer

A cardioid with radius 3

Answer

A rose curve with 6 petals

Question 37

Determine the polar equation for the circle with a radius of 5 centered at the origin.

Accepted Answer

r = 5

Answer

x + y = 5

Answer

x^2 + y^2 = 5

Answer

r = 5 sin θ

Question 38

Convert the rectangular coordinates (-2, 3) to polar coordinates.

Accepted Answer

r ≈ 3.61, θ ≈ 123.7°

Answer

r = 5, θ = 180°

Answer

r = 3, θ = 90°

Answer

r ≈ 4.24, θ ≈ 153.4°

Question 39

Sketch the polar graph of r = 1 + cos θ.

Accepted Answer

A cardioid with a dimple at the pole

Answer

A spiral that approaches the origin as θ increases

Answer

A circle with radius 1 centered at the origin

Question 40

What are the parametric equations for the polar curve r = 2 sin θ?

Accepted Answer

x = 2 cos θ, y = 2 sin θ

Answer

x = 2 sin θ, y = 2 cos θ

Answer

x = 2 cos θ, y = 2 sec θ

Answer

x = 2 cos θ, y = 2 tan θ

Question 41

What are the polar coordinates of the point with rectangular coordinates (1, √3)?

Accepted Answer

(2, 60°)

Answer

(√2, 45°)

Answer

(√3, 30°)

Question 42

Which polar equation represents a circle centered at the origin with a radius of 4?

Accepted Answer

r = 4

Answer

r = 4 cos θ

Answer

r = 4 sin θ

Answer

θ = 4

Question 43

What is the rectangular form of the polar equation r = 2 cos θ?

Accepted Answer

x² + y² = 2x

Answer

x² - y² = 2

Answer

x² + y² = 4

Answer

y² = 2x

Question 44

Which graph represents the polar equation r = 1 + sin θ?

Accepted Answer

Cardioid

Answer

Rose curve

Answer

Lemniscate

Answer

Circle

Question 45

Which rectangular coordinates correspond to the polar point (2, 150°)?

Accepted Answer

(−√3, 1)

Answer

(−√3, −1)

Answer

(√3, 1)

Answer

(√3, −1)

Question 46

The polar equation r = 2 represents:

Accepted Answer

A circle centered at the origin with radius 2

Answer

A line passing through the origin

Answer

A spiral

Answer

A cardioid

Question 47

Which of the following polar equations represents a cardioid: a heart-shaped curve?

Accepted Answer

r = 1 + cos(θ)

Answer

r = θ

Answer

r = 2cos(θ)

Answer

r² = 9

Question 48

Which of the following polar equations represents a rose curve with 3 petals?

Accepted Answer

r = 3cos(3θ)

Answer

r = 3sin(θ)

Answer

r = 3cos(θ)

Question 49

Which of the following represents the polar coordinates of the point with rectangular coordinates (3, 4)?

Accepted Answer

(5, 53.13°)

Answer

(7, 53.13°)

Answer

(5, 36.87°)

Answer

(7, 36.87°)

Question 50

What is the rectangular form of the polar equation r = 2cos(θ)?

Accepted Answer

(x-1)² + y² = 1

Answer

(x-2)² + y² = 4

Answer

x² + (y-2)² = 4

Answer

x² + (y-1)² = 1

Question 51

A point in the polar coordinate system is represented as (4, 120°). What is the angle of the point in radians?

Accepted Answer

2π/3

Answer

4π/3

Answer

5π/3

Answer

π/3

Question 52

Which of the following polar equations represents a circle centered at the origin with a radius of 3?

Accepted Answer

r = 3

Answer

r = 3cos(θ)

Answer

r = 3sin(θ)

Answer

θ = 3

Question 53

What is the maximum value of r for the polar equation r = 4sin(2θ)?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 54

The polar coordinates (2, π/6) and (-2, 7π/6) represent the same point in the plane. This illustrates which property of polar coordinates?

Accepted Answer

Non-uniqueness

Answer

Periodicity

Answer

Asymptotes

Answer

Symmetry

Question 55

Which of the following describes the graph of the polar equation r = 2 + 2cos(θ)?

Accepted Answer

Cardioid

Answer

Lemniscate

Answer

Limaçon with inner loop

Answer

Rose curve

Question 56

What is the area enclosed by one loop of the polar curve r = 2sin(3θ)?

Accepted Answer

π/3

Answer

π/2

Answer

π/6

Answer

π

Question 57

Which of the following is a key advantage of using polar coordinates over rectangular coordinates in certain situations?

Accepted Answer

They can simplify equations involving circular or rotational symmetry.

Answer

They make it easier to represent points in three dimensions.

Answer

They are always more accurate than rectangular coordinates.