Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Lesson: Conversion between Polar and Rectangular Coordinates

Sample Question Videos

Worksheet • 11 Questions • 1 Video

Q1:

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

  • A ο€» βˆ’ 2 , 2 √ 3 
  • B ο€» 2 √ 3 , βˆ’ 2 
  • C ο€» βˆ’ 2 , βˆ’ 2 √ 3 
  • D ο€» 2 , βˆ’ 2 √ 3 

Q2:

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

  • A ( 5 . 3 9 , 1 . 9 5 )
  • B ( 5 . 3 9 , 2 . 7 6 )
  • C ( 2 9 , 2 . 7 6 )
  • D ( 2 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 ο€Ό 3 , 3 πœ‹ 4  , ο€Ό 3 , 1 1 πœ‹ 4  , ο€Ό 3 , βˆ’ 5 πœ‹ 4 
  • B ο€Ό 3 , 3 πœ‹ 2  , ο€Ό 3 , 1 1 πœ‹ 2  , ο€Ό 3 , βˆ’ 5 πœ‹ 2 
  • C ο€Ό 3 , 3 πœ‹ 4  , ο€Ό 3 , 1 1 πœ‹ 4  , ο€Ό 3 , 5 πœ‹ 4 
  • D ο€Ό 4 , 3 πœ‹ 4  , ο€Ό 4 , 1 1 πœ‹ 4  , ο€Ό 4 , βˆ’ 5 πœ‹ 4 
  • E ο€Ό 4 , 3 πœ‹ 4  , ο€Ό 4 , 1 1 πœ‹ 4  , ο€Ό 4 , 5 πœ‹ 4 

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

  • A ο€Ώ βˆ’ 3 √ 2 , 3 √ 2 
  • B ο€Ώ βˆ’ 3 √ 2 , βˆ’ 3 √ 2 
  • C ο€Ώ 3 √ 2 , βˆ’ 3 √ 2 
  • D ο€Ό 3 2 , βˆ’ 3 2 
  • E ο€Ό βˆ’ 3 2 , 3 2 

Q4:

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

Calculate the distance π‘Ÿ of this point from the origin. Give your answer in exact form.

  • A π‘Ÿ = √ 6 5
  • B π‘Ÿ = 9
  • C π‘Ÿ = 6 5
  • D π‘Ÿ = √ 3 3
  • E π‘Ÿ = 3

Find the angle πœƒ that 𝑂 𝐴 makes with the positive π‘₯ -axis, giving your answer in radians to two decimal places.

  • A πœƒ = 2 . 0 9
  • B πœƒ = βˆ’ 0 . 5 2
  • C πœƒ = βˆ’ 1 . 0 5
  • D πœƒ = 2 . 6 2
  • E πœƒ = 1 . 0 5

Given that point 𝐴 can be expressed in polar form as ( π‘Ÿ , πœƒ ) , which of the following is also a legitimate polar form for point 𝐴 ?

  • A ( π‘Ÿ , πœƒ βˆ’ 2 πœ‹ )
  • B ( βˆ’ π‘Ÿ , πœƒ + 2 πœ‹ )
  • C ( π‘Ÿ , πœƒ βˆ’ πœ‹ )
  • D ( π‘Ÿ , πœƒ + 3 πœ‹ )
  • E ( βˆ’ π‘Ÿ , πœƒ βˆ’ 2 πœ‹ )

Q5:

William and Hannah are learning about polar coordinates. They have been given the point 𝐴 , which is ο€Ό 4 , βˆ’ 1 1 πœ‹ 6  , in polar coordinates, and the point 𝐡 , ο€» 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 ο€» 2 √ 3 , 2 
  • B ο€» 2 , 2 √ 3 
  • C ο€» βˆ’ 2 , 2 √ 3 
  • D ο€» 2 √ 3 , βˆ’ 2 

William decides to convert point 𝐡 into polar coordinates. He does his calculations and gets an answer of ο€» 4 , πœ‹ 6  . Is his answer correct?

  • ANo
  • BYes

William concludes that points 𝐴 and 𝐡 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 ( 3 . 6 1 , 5 6 . 3 ) ∘
  • B ( 3 . 6 1 , 3 3 . 7 ) ∘
  • C ( 1 3 , 3 3 . 7 ) ∘
  • D ( 1 3 , 5 6 . 3 ) ∘

Q7:

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

  • A ο€Ώ 3 2 , 3 √ 3 2 
  • B ο€Ώ 3 √ 3 2 , 3 2 
  • C ο€Ώ 3 2 , βˆ’ 3 √ 3 2 
  • D ο€Ώ βˆ’ 3 2 , βˆ’ 3 √ 3 2 

Q8:

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

  • A ο€Ώ βˆ’ 7 2 , βˆ’ 7 √ 3 2 
  • B ο€Ώ βˆ’ 7 √ 3 2 , βˆ’ 7 2 
  • C ο€Ώ βˆ’ 7 2 , 7 √ 3 2 
  • D ο€Ώ 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 ο€» βˆ’ 3 √ 3 , 3 
  • B ο€» 3 , βˆ’ 3 √ 3 
  • C ο€» βˆ’ 3 √ 3 , βˆ’ 3 
  • D ο€» 3 √ 3 , βˆ’ 3 

Q10:

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

  • A ο€Ώ βˆ’ √ 3 2 , 1 2 
  • B ο€Ώ 1 2 , βˆ’ √ 3 2 
  • C ο€Ώ βˆ’ √ 3 2 , βˆ’ 1 2 
  • D ο€Ώ √ 3 2 , βˆ’ 1 2 

Q11:

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

  • A ο€Ώ 5 √ 3 2 , 5 2 
  • B ο€Ώ 5 2 , 5 √ 3 2 
  • C ο€Ώ 5 √ 3 2 , βˆ’ 5 2 
  • D ο€Ώ βˆ’ 5 √ 3 2 , βˆ’ 5 2 
Preview