HW 6.6.1: Polar Coordinates

Plot each point and convert the given polar coordinates to Cartesian coordinates.

1. 77,6π⎛ ⎞

⎜ ⎟⎝ ⎠ 2. 3,π

6⎛⎝⎜

⎞⎠⎟ 3. 74,

4π⎛ ⎞

⎜ ⎟⎝ ⎠ 4. 2, 5π

3⎛⎝⎜

⎞⎠⎟

5. ⎟⎠⎞⎜

⎝⎛ −

4, 6 π 6. 7,− 2π

3⎛⎝⎜

⎞⎠⎟ 7. 3,

2π⎛ ⎞

⎜ ⎟⎝ ⎠ 8. 1,−π( )

9. 3,6π⎛ ⎞−⎜ ⎟⎝ ⎠

10. −2,0( ) 11. (3, 2) 12. 0,π4

⎛⎝⎜

⎞⎠⎟

Convert the given Cartesian coordinates to polar coordinates.

13. 5 32,− 52

⎛⎝⎜

⎞⎠⎟

14. − 7 22, 7 22

⎛

⎝⎜⎞

⎠⎟ 15. −5,0( ) 16. 0,−6( )

17. 4,4 3( ) 18. −3 2,3 2( ) 19. (4, 2) 20. 8,0( )

21. ( 4, 6)− 22. −3,2( ) 23. (3, 5)− 24. 7,−1( )

Selected Answers: 1. The Cartesian coordinates are (x, y) = (r cos(θ), r sin(θ))

= 7 cos !!!, 7 sin !!

!= −7 cos !

!,−7 sin !

!= − ! !

!,− !

!

3. The Cartesian coordinates are (x, y) = (r cos(θ), r sin(θ))

= 4 cos !!!, 4 sin !!

!= 4 cos !

!,−4 sin !

!= ! !

!,− ! !

!= 2 2,− 2 2

5. The Cartesian coordinates are (x, y) = (r cos(θ), r sin(θ))

= 6 cos − !!, 6 sin − !

!= 6 cos !

!,−6 sin !

!= ! !

!,− ! !

!= 3 2,− 3 2

7. The Cartesian coordinates are (x, y) = (r cos(θ), r sin(θ)) = 3 cos !!, 3 sin !

!= 0, 3

9. The Cartesian coordinates are (x, y) = (r cos(θ), r sin(θ))

= −3 cos !!,−3 sin !

!= − ! !

!,− !

!

11. The Cartesian coordinates are (x, y) = (r cos(θ), r sin(θ)) = (3 cos(2), 3 sin(2)) ≈ (- 1.2484, 2.7279)

13. 5,− π6

⎛⎝⎜

⎞⎠⎟ 15. 5,π( ) 17. 8,π

3⎛⎝⎜

⎞⎠⎟

19. (4, 2) = (x, y) = (r cos(θ), r sin(θ)). Then tan(θ) = !! = !

! = !

!. Since (x, y) is located in the first

quadrant, where 0 ≤ θ ≤ !!, θ = tan-1 !

! ≈ 0.46365. And r2 = x2 + y2 = 42 + 22 = 20 à r = 20 = 2 5.

21. (-4, 6) = (x, y) = (r cos(θ), r sin(θ)). Then tan(θ) = !! = !

!! = − !

!. Since (x, y) is located in the

second quadrant, where !! ≤ θ ≤ π, and tan(θ) = tan(θ + π), θ = tan-1 − !

! + π ≈ -0.9828 + π ≈ 2.1588.

And r2 = x2 + y2 = (-4)2 + 62 = 52 à r = 2 13.

23. (3, -5) = (x, y) = (r cos(θ), r sin(θ)). Then tan(θ) = !! = !!

!. Since (x, y) is located in the fourth

quadrant, where !!!

≤ θ ≤ 2π, θ = tan-1 (− !!) + 2π ≈ -1.0304 + 2π ≈ 5.2528. And r2 = x2 + y2 = 32 + (-5)2

= 34 à r = 34.