Worksheet: Conversion between Polar and Rectangular Coordinates

In this worksheet, we will practice using the transformation equations relating polar and Cartesian coordinates to convert points from polar to rectangular coordinates.

Q1:

Given that the polar coordinates of point 𝐴 are ( 4 , 1 2 0 ) , find the Cartesian coordinates of 𝐴 .

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

Q2:

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

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

Q3:

Answer the questions using the figure shown.

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

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

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

Q4:

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

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

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

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

Q5:

William and Hannah are learning about polar coordinates. They have been given the point 𝐴 , which is 4 , 1 1 𝜋 6 , in polar coordinates, and the point 𝐵 , 2 3 , 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 3 , 2
  • B 2 , 2 3
  • C 2 , 2 3
  • D 2 3 , 2

William decides to convert point 𝐵 into polar coordinates. He does his calculations and gets an answer of 4 , 𝜋 6 . Is his answer correct?

  • AYes
  • BNo

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

  • AWilliam
  • BHannah

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

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

Q6:

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

Q7:

Given that the polar coordinates of point 𝐴 are ( 3 , 6 0 ) , find the Cartesian coordinates of 𝐴 .

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

Q8:

Given that the polar coordinates of point 𝐴 are ( 7 , 1 2 0 ) , find the Cartesian coordinates of 𝐴 .

  • A 7 2 , 7 3 2
  • B 7 3 2 , 7 2
  • C 7 2 , 7 3 2
  • D 7 2 , 7 3 2

Q9:

Given that the polar coordinates of point 𝐴 are ( 6 , 1 5 0 ) , find the Cartesian coordinates of 𝐴 .

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

Q10:

Given that the polar coordinates of point 𝐴 are ( 1 , 1 5 0 ) , find the Cartesian coordinates of 𝐴 .

  • A 3 2 , 1 2
  • B 1 2 , 3 2
  • C 3 2 , 1 2
  • D 3 2 , 1 2

Q11:

Given that the polar coordinates of point 𝐴 are ( 5 , 3 0 ) , find the Cartesian coordinates of 𝐴 .

  • A 5 3 2 , 5 2
  • B 5 2 , 5 3 2
  • C 5 3 2 , 5 2
  • D 5 3 2 , 5 2

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