Solutions to Exercises, Section 5 - UW-Madison park/Fall2014/precalculus/5.2sol.pdfInstructor’s...

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Transcript of Solutions to Exercises, Section 5 - UW-Madison park/Fall2014/precalculus/5.2sol.pdfInstructor’s...

  • Instructors Solutions Manual, Section 5.2 Exercise 1

    Solutions to Exercises, Section 5.2

    In Exercises 18, convert each angle to radians.

    1. 15

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 15, obtaining

    15 = 15180

    radians = 12

    radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 2

    2. 40

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 40, obtaining

    40 = 40180

    radians = 29

    radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 3

    3. 45

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 45, obtaining

    45 = 45180

    radians = 4

    radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 4

    4. 60

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 60, obtaining

    60 = 60180

    radians = 3

    radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 5

    5. 270

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 270, obtaining

    270 = 270180

    radians = 32

    radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 6

    6. 240

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 240, obtaining

    240 = 240180

    radians = 43

    radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 7

    7. 1080

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 1080, obtaining

    1080 = 1080180

    radians = 6 radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 8

    8. 1440

    solution Start with the equation

    360 = 2 radians.

    Divide both sides by 360 to obtain

    1 = 180

    radians.

    Now multiply both sides by 1440, obtaining

    1440 = 1440180

    radians = 8 radians.

  • Instructors Solutions Manual, Section 5.2 Exercise 9

    In Exercises 916, convert each angle to degrees.

    9. 4 radians

    solution Start with the equation

    2 radians = 360.

    Multiply both sides by 2, obtaining

    4 radians = 2 360 = 720.

  • Instructors Solutions Manual, Section 5.2 Exercise 10

    10. 6 radians

    solution Start with the equation

    2 radians = 360.

    Multiply both sides by 3, obtaining

    6 radians = 3 360 = 1080.

  • Instructors Solutions Manual, Section 5.2 Exercise 11

    11. 9 radians

    solution Start with the equation

    2 radians = 360.

    Divide both sides by 2 to obtain

    radians = 180.

    Now divide both sides by 9, obtaining

    9

    radians = 1809

    = 20.

  • Instructors Solutions Manual, Section 5.2 Exercise 12

    12. 10 radians

    solution Start with the equation

    2 radians = 360.

    Divide both sides by 2 to obtain

    radians = 180.

    Now divide both sides by 10, obtaining

    10

    radians = 18010

    = 18.

  • Instructors Solutions Manual, Section 5.2 Exercise 13

    13. 3 radians

    solution Start with the equation

    2 radians = 360.

    Divide both sides by 2 to obtain

    1 radian = 180

    .

    Now multiply both sides by 3, obtaining

    3 radians = 3 180

    = 540

    .

  • Instructors Solutions Manual, Section 5.2 Exercise 14

    14. 5 radians

    solution Start with the equation

    2 radians = 360.

    Divide both sides by 2 to obtain

    1 radian = 180

    .

    Now multiply both sides by 5, obtaining

    5 radians = 5 180

    = 900

    .

  • Instructors Solutions Manual, Section 5.2 Exercise 15

    15. 23 radianssolution Start with the equation

    2 radians = 360.

    Divide both sides by 2 to obtain

    radians = 180.

    Now multiply both sides by 23 , obtaining

    23

    radians = 23 180 = 120.

  • Instructors Solutions Manual, Section 5.2 Exercise 16

    16. 34 radianssolution Start with the equation

    2 radians = 360.

    Divide both sides by 2 to obtain

    radians = 180.

    Now multiply both sides by 34 , obtaining

    34

    radians = 34 180 = 135.

  • Instructors Solutions Manual, Section 5.2 Exercise 17

    17. Suppose an ant walks counterclockwise on the unit circle from thepoint (0,1) to the endpoint of the radius that forms an angle of 54radians with the positive horizontal axis. How far has the ant walked?

    solution The radius whose endpoint equals (0,1) makes an angle of2 radians with the positive horizontal axis. This radius corresponds tothe smaller angle shown below.

    Because 54 = + 4 , the radius that forms an angle of 54 radians withthe positive horizontal axis lies 4 radians beyond the negativehorizontal axis (half-way between the negative horizontal axis and thenegative vertical axis). Thus the ant ends its walk at the endpoint of theradius corresponding to the larger angle shown below:

    1

    The ant walks along the thickened circular arc shown above. Thiscircular arc corresponds to an angle of 54 2 radians, which equals 34radians. Thus the distance walked by the ant is 34 .

  • Instructors Solutions Manual, Section 5.2 Exercise 18

    18. Suppose an ant walks counterclockwise on the unit circle from thepoint (1,0) to the endpoint of the radius that forms an angle of 6radians with the positive horizontal axis. How far has the ant walked?

    solution The radius whose endpoint equals (1,0) makes an angleof radians with the positive horizontal axis. This radius lies on thenegative horizontal axis.

    Because 2 6.28, the radius that forms an angle of 6 radians with thepositive horizontal axis lies slightly below the positive horizontal axis,as shown below:

    1

    The ant walks along the thickened circular arc shown above. Thiscircular arc corresponds to an angle of 6 radians. Thus the distancewalked by the ant is 6 .

  • Instructors Solutions Manual, Section 5.2 Exercise 19

    19. Find the lengths of both circular arcs of the unit circle connectingthe point (1,0) and the endpoint of the radius that makes an angle of 3radians with the positive horizontal axis.

    solution Because 3 is a bit less than , the radius that makes anangle of 3 radians with the positive horizontal axis lies a bit above thenegative horizontal axis, as shown below. The thickened circular arccorresponds to an angle of 3 radians and thus has length 3. The entireunit circle has length 2 . Thus the length of the other circular arc is2 3, which is approximately 3.28.

    1

  • Instructors Solutions Manual, Section 5.2 Exercise 20

    20. Find the lengths of both circular arcs of the unit circle connectingthe point (1,0) and the endpoint of the radius that makes an angle of 4radians with the positive horizontal axis.

    solution The radius that makes an angle of 4 radians with thepositive horizontal axis is shown below. The thickened circular arccorresponds to an angle of 4 radians and thus has length 4. The entireunit circle has length 2 . Thus the length of the other circular arc is2 4, which is approximately 2.28.

    1

  • Instructors Solutions Manual, Section 5.2 Exercise 21

    21. Find the lengths of both circular arcs of the unit circle connectingthe point

    (22 ,

    2

    2

    )and the point whose radius makes an angle of 1

    radian with the positive horizontal axis.

    solution The radius of the unit circle whose endpoint equals(22 ,

    2

    2

    )makes an angle of 4 radians with the positive horizontal

    axis, as shown with the clockwise arrow below. The radius that makesan angle of 1 radian with the positive horizontal axis is shown with acounterclockwise arrow.

    1

    Thus the thickened circular arc above corresponds to an angle of 1+ 4and thus has length 1+ 4 , which is approximately 1.79. The entire unitcircle has length 2 . Thus the length of the other circular arc below is2 (1+ 4 ), which equals 74 1, which is approximately 4.50.

  • Instructors Solutions Manual, Section 5.2 Exercise 22

    22. Find the lengths of both circular arcs of the unit circle connectingthe point

    (22 , 22 ) and the point whose radius makes an angle of 2radians with the positive horizontal axis.

    solution The radius of the unit circle whose endpoint equals(22 , 22 ) makes an angle of 34 radians with the positive horizontalaxis, as shown with the longer arrow below. The radius that makes anangle of 2 radian with the positive horizontal axis is shown with theshorter arrow below.

    1

    Thus the thickened circular arc above corresponds to an angle of 34 2and thus has length 34 2, which is approximately 0.356. The entireunit circle has length 2 . Thus the length of the other circular arcbelow is 2 (34 2), which equals 54 + 2, which is approximately5.927.

  • Instructors Solutions Manual, Section 5.2 Exercise 23

    23. For a 16-inch pizza, find the area of a slice with angle 34 radians.

    solution Pizzas are measured by their diameters; thus this pizza hasa radius of 8 inches. Thus the area of the slice is 12 34 82, which equals24 square inches.

  • Instructors Solutions Manual, Sec