Worksheet: Polar Coordinates

In this worksheet, we will practice defining and plotting points given in polar coordinates and converting between the Cartesian and polar coordinates of a point.

Q1:

Consider the points plotted on the graph.

Write down the polar coordinates of 𝐶, giving the angle 𝜃 in the range 𝜋<𝜃𝜋.

  • A 1 , 𝜋 4
  • B 1 , 𝜋 4
  • C 𝜋 4 , 1
  • D 1 , 7 𝜋 4
  • E 1 , 𝜋 4

Q2:

The polar coordinates of Point 𝐸 are (2,80). Which of the points 𝐹(2,380), 𝐺(2,440), 𝐻(2,80), or 𝐼(4,160) is coincident with Point 𝐸?

  • APoint 𝐺
  • BPoint 𝐹
  • CPoint 𝐻
  • DPoint 𝐼

Q3:

The polar coordinates of Point 𝐴 are 3,𝜋4. Which of the points 𝐵6,𝜋4, 𝐶3,5𝜋4, 𝐷3,9𝜋4, or 𝐸32,𝜋8 is coincident with Point 𝐴?

  • APoint 𝐶
  • BPoint 𝐸
  • CPoint 𝐵
  • DPoint 𝐷

Q4:

Which of the ordered pairs (4,30), (4,330), (4,390), and (4,390) does not describe the position of Point 𝐵 in the diagram?

  • A ( 4 , 3 3 0 )
  • B ( 4 , 3 9 0 )
  • C ( 4 , 3 0 )
  • D ( 4 , 3 9 0 )

Q5:

Which of the ordered pairs 5,𝜋7, 5,15𝜋7, 5,8𝜋7, and 5,13𝜋7 does not describe the position of Point 𝑃 in the diagram?

  • A 5 , 𝜋 7
  • B 5 , 8 𝜋 7
  • C 5 , 1 3 𝜋 7
  • D 5 , 1 5 𝜋 7

Q6:

Which of the following ordered pairs does NOT describe the position of Point 𝐴 in the diagram?

  • A 2 , 9 𝜋 4
  • B 2 , 1 3 𝜋 8
  • C 2 , 5 𝜋 4
  • D 2 , 3 𝜋 4

Q7:

Convert (2,5) to polar coordinates. Give the angle in radians and round to three significant figures throughout.

  • A ( 2 9 , 2 . 7 6 )
  • B ( 5 . 3 9 , 1 . 9 5 )
  • C ( 5 . 3 9 , 2 . 7 6 )
  • D ( 2 9 , 1 . 9 5 )

Q8:

Consider the point 𝐴 with rectangular coordinates (4,7).

Calculate the distance 𝑟 of this point from the origin. Give your answer in exact form.

  • A 𝑟 = 6 5
  • B 𝑟 = 3 3
  • C 𝑟 = 6 5
  • D 𝑟 = 9
  • E 𝑟 = 3

Find the angle 𝜃 that 𝑂𝐴 makes with the positive 𝑥-axis, giving your answer in radians to two decimal places.

  • A 𝜃 = 2 . 6 2
  • B 𝜃 = 0 . 5 2
  • C 𝜃 = 1 . 0 5
  • D 𝜃 = 2 . 0 9
  • E 𝜃 = 1 . 0 5

Given that point 𝐴 can be expressed in polar form as (𝑟,𝜃), which of the following is also a legitimate polar form for point 𝐴?

  • A ( 𝑟 , 𝜃 2 𝜋 )
  • B ( 𝑟 , 𝜃 + 3 𝜋 )
  • C ( 𝑟 , 𝜃 + 2 𝜋 )
  • D ( 𝑟 , 𝜃 2 𝜋 )
  • E ( 𝑟 , 𝜃 𝜋 )

Q9:

Answer the questions using the figure shown.

Which of the following are three possible pairs of polar coordinates for the marked point?

  • A 3 , 3 𝜋 2 , 3 , 1 1 𝜋 2 , 3 , 5 𝜋 2
  • B 4 , 3 𝜋 4 , 4 , 1 1 𝜋 4 , 4 , 5 𝜋 4
  • C 3 , 3 𝜋 4 , 3 , 1 1 𝜋 4 , 3 , 5 𝜋 4
  • D 4 , 3 𝜋 4 , 4 , 1 1 𝜋 4 , 4 , 5 𝜋 4
  • E 3 , 3 𝜋 4 , 3 , 1 1 𝜋 4 , 3 , 5 𝜋 4

What are the Cartesian coordinates of this point? Give these exactly.

  • A 3 2 , 3 2
  • B 3 2 , 3 2
  • C 3 2 , 3 2
  • D 3 2 , 3 2
  • E 3 2 , 3 2

Q10:

William and Hannah are learning about polar coordinates. They have been given the point 𝐴, which is 4,11𝜋6, in polar coordinates, and the point 𝐵, 23,2, in rectangular coordinates. They would like to compare these two points.

Hannah decides to convert point 𝐴 into rectangular coordinates. Determine the answer that Hannah will get.

  • A 2 , 2 3
  • B 2 , 2 3
  • C 2 3 , 2
  • D 2 3 , 2

William decides to convert point 𝐵 into polar coordinates. He does his calculations and gets an answer of 4,𝜋6. Is his answer correct?

  • ANo
  • BYes

William concludes that points 𝐴 and 𝐵 are different points, whereas Hannah concludes they are the same point. Who is correct?

  • AHannah
  • BWilliam

Which of the following statements represents a good learning point for William and Hannah as a result of this exercise?

  • APolar coordinates are not unique representations for a point; there are many ways to express a point in polar coordinates.
  • BRectangular coordinates are not unique representations for a point; there are many ways to express a point in rectangular coordinates.

Q11:

Given that the polar coordinates of point 𝐴 are (4,120), find the Cartesian coordinates of 𝐴.

  • A 2 , 2 3
  • B 2 3 , 2
  • C 2 , 2 3
  • D 2 , 2 3

Q12:

Convert (2,3) to polar coordinates. Give the angle in degrees and round to three significant figures throughout.

  • A ( 1 3 , 5 6 . 3 )
  • B ( 3 . 6 1 , 5 6 . 3 )
  • C ( 3 . 6 1 , 3 3 . 7 )
  • D ( 1 3 , 3 3 . 7 )

Q13:

Find the distance between the polar coordinates (2,𝜋) and 3,3𝜋4. Give your answer accurate to three significant figures.

Q14:

Consider the polar coordinates 3,𝜋6 and 4,𝜋3, as shown in the figure.

Find the angle between the two lines.

  • A 𝜋 3
  • B 𝜋 6
  • C 𝜋 2

Use the law of cosines to find the length of 𝐴𝐵. Give your answer accurate to three significant figures.

Q15:

Identify which of the points plotted on the graph has polar coordinates 1,𝜋4.

  • A 𝐴
  • B 𝐶
  • C 𝐸
  • D 𝐷
  • E 𝐵

Q16:

Consider the shown points.

Which one has the polar coordinates 3,15𝜋4?

  • AC
  • BE
  • CA
  • DD
  • EB

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