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### Transcript of Chapter 9 Polar Coordinates; Vectors shubbard/HPCanswers/PR_9e_ISM... Chapter 9 Polar Coordinates;...

• 865

Chapter 9

Polar Coordinates; Vectors Section 9.1

1.

The point lies in quadrant IV.

2. 2

6 9

2   =   

3. b

a

4. 4

π−

5. pole, polar axis

6. True

7. False

8. cosr θ ; sinr θ

9. A

10. B

11. C

12. C

13. B

14. D

15. A

16. D

17.

18.

19.

20.

21.

22.

23.

• Chapter 9: Polar Coordinates; Vectors

866

24.

25.

26.

27.

28.

29.

30.

31.

a. 4

0, 2 0 5, 3

r θ π > − π ≤ < −   

b. 5

0, 0 2 5, 3

r θ π < ≤ < π −   

c. 8

0, 2 4 5, 3

r θ π > π ≤ < π    

32.

a. 5

0, 2 0 4, 4

r θ π > − π ≤ < −   

b. 7

0, 0 2 4, 4

r θ π < ≤ < π −   

c. 11

0, 2 4 4, 4

r θ π > π ≤ < π    

33.

a. ( )0, 2 0 2, 2r θ> − π ≤ < − π b. ( )0, 0 2 2,r θ< ≤ < π − π c. ( )0, 2 4 2, 2r θ> π ≤ < π π

34.

a. ( )0, 2 0 3,r θ> − π ≤ < − π b. ( )0, 0 2 3, 0r θ< ≤ < π − c. ( )0, 2 4 3, 3r θ> π ≤ < π π

• Section 9.1: Polar Coordinates

867

35.

a. 3

0, 2 0 1, 2

r θ π > − π ≤ < −   

b. 3

0, 0 2 1, 2

r θ π < ≤ < π −   

c. 5

0, 2 4 1, 2

r θ π > π ≤ < π    

36.

a. ( )0, 2 0 2,r θ> − π ≤ < − π

b. ( )0, 0 2 2, 0r θ< ≤ < π −

c. ( )0, 2 4 2, 3r θ> π ≤ < π π

37.

a. 5

0, 2 0 3, 4

r θ π > − π ≤ < −   

b. 7

0, 0 2 3, 4

r θ π < ≤ < π −   

c. 11

( 0, 2 4 3, 4

r θ π > π ≤ < π    

38.

a. 5

0, 2 0 2, 3

r θ π > − π ≤ < −   

b. 4

0, 0 2 2, 3

r θ π < ≤ < π −   

c. 7

0, 2 4 2, 3

r θ π > π ≤ < π    

39. cos 3cos 3 0 0 2

x r θ π= = = ⋅ =

sin 3sin 3 1 3 2

y r θ π= = = ⋅ =

Rectangular coordinates of the point 3, 2

π     

are

( )0, 3 .

40. 3

cos 4cos 4 0 0 2

x r θ π= = = ⋅ =

3 sin 4sin 4 ( 1) 4

2 y r θ π= = = ⋅ − = −

Rectangular coordinates of the point 3

4, 2

π     

are

( )0, 4− .

41. cos 2cos 0 2 1 2x r θ= = − = − ⋅ = − sin – 2sin 0 – 2 0 0y r θ= = = ⋅ =

Rectangular coordinates of the point ( )– 2, 0 are ( )2, 0− .

42. cos 3cos 3( 1) 3x r θ= = − π = − − = sin 3sin 3 0 0y r θ= = − π = − ⋅ =

Rectangular coordinates of the point ( )3,− π are ( )3, 0 .

43. 3

cos 6cos150º 6 3 3 2

x r θ  

= = = − = −    

1 sin 6sin150º 6 3

2 y r θ= = = ⋅ =

Rectangular coordinates of the point ( )6, 150º are ( )3 3, 3− .

44. 1 5

cos 5cos300º 5 2 2

x r θ= = = ⋅ =

3 5 3 sin 5sin 300º 5

2 2 y r θ

  = = = − = −  

 

• Chapter 9: Polar Coordinates; Vectors

868

Rectangular coordinates of the point ( )5, 300º

are 5 5 3

, 2 2

  −  

  .

45. 3 2

cos 2cos 2 2 4 2

x r θ  π= = − = − − =    

3 2 sin 2sin 2 2

4 2 y r θ π= = − = − ⋅ = −

Rectangular coordinates of the point 3

2, 4

π −   

are ( )2, 2− .

46. 2 1

cos 2cos 2 1 3 2

x r θ π  = = − = − − =   

2 3 sin 2sin 2 3

3 2 y r θ π= = − = − ⋅ = −

Rectangular coordinates of the point 2

2, 3

π −   

are ( )1, 3− .

47. 1 1

cos 1cos 1 3 2 2

x r θ π = = − − = − ⋅ = −   

3 3 sin 1sin 1

3 2 2 y r θ

 π = = − − = − − =       

Rectangular coordinates of the point 1, 3

π − −   

are 1 3

, 2 2

  −    

.

48. 3 2 3 2

cos 3cos 3 4 2 2

x r θ  π = = − − = − − =       

3 2 3 2 sin 3sin 3

4 2 2 y r θ

 π = = − − = − − =       

Rectangular coordinates of the point 3

3, 4

π − −   

are 3 2 3 2

, 2 2

      

.

49. ( )cos 2cos( 180º ) 2 1 2x r θ= = − − = − − = sin – 2sin( 180º ) – 2 0 0y r θ= = − = ⋅ =

Rectangular coordinates of the point

( )– 2, 180º− are ( )2, 0 .

50. cos 3cos( 90º ) 3 0 0x r θ= = − − = − ⋅ = sin – 3sin( 90º ) 3( 1) 3y r θ= = − = − − =

Rectangular coordinates of the point ( )– 3, 90º− are ( )0, 3 .

51. cos 7.5cos110º 7.5( 0.3420) 2.57x r θ= = ≈ − ≈ − sin 7.5sin110º 7.5(0.9397) 7.05y r θ= = ≈ ≈

Rectangular coordinates of the point ( )7.5, 110º are about ( )2.57, 7.05− .

52. cos 3.1cos182º 3.1( 0.9994) 3.10x r θ= = − ≈ − − ≈ sin 3.1sin182º 3.1( 0.0349) 0.11y r θ= = − ≈ − − ≈

Rectangular coordinates of the point ( )3.1, 182º− are about ( )3.10, 0.11 .

53. ( )cos 6.3cos 3.8 6.3( 0.7910) 4.98x r θ= = ≈ − ≈ − ( )sin 6.3sin 3.8 6.3( 0.6119) 3.85y r θ= = ≈ − ≈ −

Rectangular coordinates of the point ( )6.3, 3.8 are about ( )4.98, 3.85− − .

54. ( )cos 8.1cos 5.2 8.1(0.4685) 3.79x r θ= = ≈ ≈ ( )sin 8.1sin 5.2 8.1( 0.8835) 7.16y r θ= = ≈ − ≈ −

Rectangular coordinates of the point ( )8.1, 5.2 are about ( )3.79, 7.16− .

55. 2 2 2 2

1 1 1

3 0 9 3

0 tan tan tan 0 0

3

r x y

y

x θ − − −

= + = + = =    = = = =       

Polar coordinates of the point (3, 0) are (3, 0) .

56. 2 2 2 20 2 4 2r x y= + = + = =

1 1 2tan tan 0

y

x θ − −   = =   

   

Since 2

0 is undefined,

2 θ π=

Polar coordinates of the point (0, 2) are 2, 2

π     

.

• Section 9.1: Polar Coordinates

869

57. 2 2 2 2

1 1 1

( 1) 0 1 1

0 tan tan tan 0 0

1

r x y

y

x θ − − −

= + = − + = =    = = = =   −   

The point lies on the negative x-axis, so θ = π . Polar coordinates of the point ( 1, 0)− are ( )1, π .

58. 2 2 2 2

1 1

0 ( 2) 4 2

2 tan tan

0

r x y

y

x θ − −

= + = + − = = −   = =   

   

Since 2

0

− is undefined,

2 θ π= .

The point lies on the negative y-axis, so 2

θ π= − .

Polar coordinates of the point (0, 2)− are 2, 2

π −   

.

59. The point (1, 1)− lies in quadrant IV. 2 2 2 2

1 1 1

1 ( 1) 2

1 tan tan tan ( 1)

1 4

r x y

y

x θ − − −

= + = + − = − π   = = = − = −   

   

Polar coordinates of the point (1, 1)− are

2, 4

π −   

.

60. The point ( 3, 3)− lies in quadrant II. 2 2 2 2

1 1 1

( 3) 3 3 2

3 tan tan tan ( 1)

3 4

r x y

y

x θ − − −

= + = − + = π   = = = − = −   −   

Polar coordinates of the point ( )3, 3− are 3

3 2, 4

π     

.

61. The point ( )3, 1 lies in quadrant I. ( )22 2 2

1 1

3 1 4 2

1 tan tan

63

r x y

y