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13.2.5: Chapter 5

  • Page ID
    117280
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    Try It

    5.1 Angles

    1.
    Graph of a 240 degree angle.
    2.

    3π 2 3π 2

    3.

    −135°

    4.

    7π 10 7π 10

    5.

    α=150° α=150°

    6.

    β=60° β=60°

    7.

    7π 6 7π 6

    8.

    215π 18 =37.525 units 215π 18 =37.525 units

    9.

    1.88

    10.

    3π 2 3π 2 rad/s

    11.

    1655 kilometers per hour

    5.2 Unit Circle: Sine and Cosine Functions

    1.

    cos(t)= 2 2 ,sin(t)= 2 2 cos(t)= 2 2 ,sin(t)= 2 2

    2.

    cos(π)=1, cos(π)=1, sin(π)=0 sin(π)=0

    3.

    sin(t)= 7 25 sin(t)= 7 25

    4.

    approximately 0.866025403

    5.

    π 3 π 3

    6.
    1. cos(315°)= 2 2 , sin(315°)= 2 2 cos(315°)= 2 2 , sin(315°)= 2 2
    2. cos( π 6 )= 3 2 , sin( π 6 )= 1 2 cos( π 6 )= 3 2 , sin( π 6 )= 1 2
    7.

    ( 1 2 , 3 2 ) ( 1 2 , 3 2 )

    5.3 The Other Trigonometric Functions

    1.

    sint= 2 2 sint= 2 2 , cost= 2 2 cost= 2 2 , tant=1tant=1, sect= 2 sect= 2 , csct= 2 csct= 2 , cott=1 cott=1

    2.

    sin π 3 = 3 2 sin π 3 = 3 2 , cos π 3 = 1 2 cos π 3 = 1 2 , tan π 3 = 3 tan π 3 = 3 , sec π 3 =2sec π 3 =2, csc π 3 = 2 3 3 csc π 3 = 2 3 3 , cot π 3 = 3 3 cot π 3 = 3 3

    3.

    sin( 7π 4 )= 2 2 ,cos( 7π 4 )= 2 2 ,tan( 7π 4 )=1, sin( 7π 4 )= 2 2 ,cos( 7π 4 )= 2 2 ,tan( 7π 4 )=1,
    sec( 7π 4 )= 2 ,csc( 7π 4 )= 2 ,cot( 7π 4 )=1 sec( 7π 4 )= 2 ,csc( 7π 4 )= 2 ,cot( 7π 4 )=1

    4.

    3 3

    5.

    2 2

    6.

    sint sint

    7.

    cost= 8 17 ,sint= 15 17 ,tant= 15 8 cost= 8 17 ,sint= 15 17 ,tant= 15 8
    csct= 17 15 ,cott= 8 15 csct= 17 15 ,cott= 8 15

    8.

    sint=1,cost=0,tant=Undefined sect= Undefined,csct=1,cott=0 sint=1,cost=0,tant=Undefined sect= Undefined,csct=1,cott=0

    9.

    sect= 2 ,csct= 2 ,tant=1,cott=1 sect= 2 ,csct= 2 ,tant=1,cott=1

    10.

    2.414 2.414

    5.4 Right Triangle Trigonometry

    1.

    7 25 7 25

    2.

    sint= 33 65 ,cost= 56 65 ,tant= 33 56 , sect= 65 56 ,csct= 65 33 ,cott= 56 33 sint= 33 65 ,cost= 56 65 ,tant= 33 56 , sect= 65 56 ,csct= 65 33 ,cott= 56 33

    3.

    sin( π 4 )= 1 2 ,cos( π 4 )= 1 2 ,tan( π 4 )=1, sin( π 4 )= 1 2 ,cos( π 4 )= 1 2 ,tan( π 4 )=1,
    sec( π 4 )= 2 ,csc( π 4 )= 2 ,cot( π 4 )=1 sec( π 4 )= 2 ,csc( π 4 )= 2 ,cot( π 4 )=1

    4.

    2

    5.

    adjacent=10; adjacent=10; opposite=10 3 opposite=10 3 ; missing angle is π 6 π 6

    6.

    About 52 ft

    5.1 Section Exercises

    1.
    Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.
    3.

    Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

    5.

    Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

    7.
    Graph of a circle with an angle inscribed.
    9.
    Graph of a circle with an angle inscribed.
    11.
    Graph of a circle with an angle inscribed.
    13.
    Graph of a circle with an angle inscribed.
    15.
    Graph of a circle with an angle inscribed.
    17.

    240°

    Graph of a circle with an angle inscribed.
    19.

    4π 3 4π 3

    Graph of a circle showing the equivalence of two angles.
    21.

    2π 3 2π 3

    Graph of a circle showing the equivalence of two angles.
    23.

    7π 2 11.00 in 2 7π 2 11.00 in 2

    25.

    81π 20 12.72 cm 2 81π 20 12.72 cm 2

    27.

    20°

    29.

    60°

    31.

    −75°

    33.

    π 2 π 2 radians

    35.

    3π 3π radians

    37.

    π π radians

    39.

    5π 6 5π 6 radians

    41.

    5.02π 3 5.26 5.02π 3 5.26 miles

    43.

    25π 9 8.73 25π 9 8.73 centimeters

    45.

    21π 10 6.60 21π 10 6.60 meters

    47.

    104.7198 cm2

    49.

    0.7697 in2

    51.

    250°

    53.

    320°

    55.

    4π 3 4π 3

    57.

    8π 9 8π 9

    59.

    1320 rad 210.085 RPM

    61.

    7 in./s, 4.77 RPM, 28.65 deg/s

    63.

    1,809,557.37 mm/min=30.16 m/s 1,809,557.37 mm/min=30.16 m/s

    65.

    5.76 5.76 miles

    67.

    120° 120°

    69.

    794 miles per hour

    71.

    2,234 miles per hour

    73.

    11.5 inches

    5.2 Section Exercises

    1.

    The unit circle is a circle of radius 1 centered at the origin.

    3.

    Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t, t, formed by the terminal side of the angle t t and the horizontal axis.

    5.

    The sine values are equal.

    7.

    I

    9.

    IV

    11.

    3 2 3 2

    13.

    1 2 1 2

    15.

    2 2 2 2

    17.

    0

    19.

    −1

    21.

    3 2 3 2

    23.

    60° 60°

    25.

    80° 80°

    27.

    45° 45°

    29.

    π 3 π 3

    31.

    π 3 π 3

    33.

    π 8 π 8

    35.

    60°, 60°, Quadrant IV, sin(300°)= 3 2 ,cos(300°)= 1 2 sin(300°)= 3 2 ,cos(300°)= 1 2

    37.

    45°, 45°, Quadrant II, sin(135°)= 2 2 , sin(135°)= 2 2 , cos(135°)= 2 2 cos(135°)= 2 2

    39.

    60°, 60°, Quadrant II, sin(120°)= 3 2 , sin(120°)= 3 2 , cos(120°)= 1 2 cos(120°)= 1 2

    41.

    30°, 30°, Quadrant II, sin(150°)= 1 2 , sin(150°)= 1 2 , cos(150°)= 3 2 cos(150°)= 3 2

    43.

    π 6 , π 6 , Quadrant III, sin( 7π 6 )= 1 2 , sin( 7π 6 )= 1 2 , cos( 7π 6 )= 3 2 cos( 7π 6 )= 3 2

    45.

    π 4 , π 4 , Quadrant II, sin( 3π 4 )= 2 2 , sin( 3π 4 )= 2 2 , cos( 4π 3 )= 2 2 cos( 4π 3 )= 2 2

    47.

    π 3 , π 3 , Quadrant II, sin( 2π 3 )= 3 2 , sin( 2π 3 )= 3 2 , cos( 2π 3 )= 1 2 cos( 2π 3 )= 1 2

    49.

    π 4 , π 4 , Quadrant IV, sin( 7π 4 )= 2 2 , sin( 7π 4 )= 2 2 , cos( 7π 4 )= 2 2 cos( 7π 4 )= 2 2

    51.

    77 9 77 9

    53.

    15 4 15 4

    55.

    ( 10,10 3 ) ( 10,10 3 )

    57.

    ( 2.778,15.757 ) ( 2.778,15.757 )

    59.

    [ 1,1 ] [ 1,1 ]

    61.

    sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2

    63.

    sint= 2 2 ,cost= 2 2 sint= 2 2 ,cost= 2 2

    65.

    sint= 3 2 ,cost= 1 2 sint= 3 2 ,cost= 1 2

    67.

    sint= 2 2 ,cost= 2 2 sint= 2 2 ,cost= 2 2

    69.

    sint=0,cost=1 sint=0,cost=1

    71.

    sint=0.596,cost=0.803 sint=0.596,cost=0.803

    73.

    sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2

    75.

    sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2

    77.

    sint=0.761,cost=0.649 sint=0.761,cost=0.649

    79.

    sint=1,cost=0 sint=1,cost=0

    81.

    −0.1736

    83.

    0.9511

    85.

    −0.7071

    87.

    −0.1392

    89.

    −0.7660

    91.

    2 4 2 4

    93.

    6 4 6 4

    95.

    2 4 2 4

    97.

    2 4 2 4

    99.

    0

    101.

    ( 0,1 ) ( 0,1 )

    103.

    37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

    5.3 Section Exercises

    1.

    Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, at x= π 4 , 5π 4 , x= π 4 , 5π 4 , the sine and cosine values are equal.

    3.

    Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 =1. x 2 + y 2 =1. Solve for x x and take the negative solution.

    5.

    The outputs of tangent and cotangent will repeat every π π units.

    7.

    2 3 3 2 3 3

    9.

    3 3

    11.

    2 2

    13.

    1

    15.

    2

    17.

    3 3 3 3

    19.

    2 3 3 2 3 3

    21.

    3 3

    23.

    2 2

    25.

    −1

    27.

    −2

    29.

    3 3 3 3

    31.

    2

    33.

    3 3 3 3

    35.

    −2

    37.

    −1

    39.

    If sint= 2 2 3 sint= 2 2 3 , sect=3sect=3, csct= 3 2 4 csct= 3 2 4 , tant=2 2 tant=2 2 , cott= 2 4 cott= 2 4

    41.

    sect=2 sect=2, csct= 2 3 3 csct= 2 3 3 , tant= 3 tant= 3 , cott= 3 3 cott= 3 3

    43.

    2 2 2 2

    45.

    3.1

    47.

    1.4

    49.

    sint= 2 2 ,cost= 2 2 ,tant=1,cott=1,sect= 2 ,csct= 2 sint= 2 2 ,cost= 2 2 ,tant=1,cott=1,sect= 2 ,csct= 2

    51.

    sint= 3 2 sint= 3 2 , cost= 1 2 cost= 1 2 , tant= 3 ,cott= 3 3 tant= 3 ,cott= 3 3 , sect=2sect=2, csct= 2 3 3 csct= 2 3 3

    53.

    –0.228

    55.

    –2.414

    57.

    1.414

    59.

    1.540

    61.

    1.556

    63.

    sin( t )0.79 sin( t )0.79

    65.

    csct1.16 csct1.16

    67.

    even

    69.

    even

    71.

    sint cost =tant sint cost =tant

    73.

    13.77 hours, period: 1000π 1000π

    75.

    7.73 inches

    5.4 Section Exercises

    1.
    A right triangle with side opposite, adjacent, and hypotenuse labeled.
    3.

    The tangent of an angle is the ratio of the opposite side to the adjacent side.

    5.

    For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

    7.

    π 6 π 6

    9.

    π 4 π 4

    11.

    b= 20 3 3 ,c= 40 3 3 b= 20 3 3 ,c= 40 3 3

    13.

    a=10,000,c=10,000.5 a=10,000,c=10,000.5

    15.

    b= 5 3 3 ,c= 10 3 3 b= 5 3 3 ,c= 10 3 3

    17.

    5 29 29 5 29 29

    19.

    5 2 5 2

    21.

    29 2 29 2

    23.

    5 41 41 5 41 41

    25.

    5 4 5 4

    27.

    41 4 41 4

    29.

    c=14,b=7 3 c=14,b=7 3

    31.

    a=15,b=15 a=15,b=15

    33.

    b=9.9970,c=12.2041 b=9.9970,c=12.2041

    35.

    a=2.0838,b=11.8177 a=2.0838,b=11.8177

    37.

    a=55.9808,c=57.9555 a=55.9808,c=57.9555

    39.

    a=46.6790,b=17.9184 a=46.6790,b=17.9184

    41.

    a=16.4662,c=16.8341 a=16.4662,c=16.8341

    43.

    188.3159

    45.

    200.6737

    47.

    498.3471 ft

    49.

    1060.09 ft

    51.

    27.372 ft

    53.

    22.6506 ft

    55.

    368.7633 ft

    Review Exercises

    1.

    45° 45°

    3.

    7π 6 7π 6

    5.

    10.385 meters

    7.

    60° 60°

    9.

    2π 11 2π 11

    11.
    A graph of a circle with a negative angle inscribed.
    13.
    A graph of a circle with an angle inscribed.
    15.

    1036.73 miles per hour

    17.

    3 2 3 2

    19.

    –1

    21.

    π 4 π 4

    23.

    2 2 2 2

    25.

    [ 1,1 ] [ 1,1 ]

    27.

    1

    29.

    2 2

    31.

    2 2

    33.

    0.6

    35.

    2 2 2 2 or 2 2 2 2

    37.

    sine, cosecant, tangent, cotangent

    39.

    3 3 3 3

    41.

    0

    43.

    b=8,c=10 b=8,c=10

    45.

    11 157 157 11 157 157

    47.

    a=4,b=4 a=4,b=4

    49.

    14.0954 ft

    Practice Test

    1.

    150° 150°

    3.

    6.283 centimeters

    5.

    15° 15°

    7.
    A graph of a circle with an angle inscribed.
    9.

    3.351 feet per second, 2π 75 2π 75 radians per second

    11.

    3 2 3 2

    13.

    [ 1,1 ] [ 1,1 ]

    15.

    3 3

    17.

    3 3 3 3

    19.

    3 2 3 2

    21.

    π 3 π 3

    23.

    a= 9 2 ,b= 9 3 2 a= 9 2 ,b= 9 3 2


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