Upload
others
View
368
Download
5
Embed Size (px)
Citation preview
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.