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13.2.8: Chapter 8

  • Page ID
    117283
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    Try It

    8.1 Non-right Triangles: Law of Sines

    1.

    α= 98 a=34.6 β= 39 b=22 γ= 43 c=23.8 α= 98 a=34.6 β= 39 b=22 γ= 43 c=23.8

    2.

    Solution 1

    α=80° a=120 β83.2° b=121 γ16.8° c35.2 α=80° a=120 β83.2° b=121 γ16.8° c35.2

    Solution 2

    α =80° a =120 β 96.8° b =121 γ 3.2° c 6.8 α =80° a =120 β 96.8° b =121 γ 3.2° c 6.8

    3.

    β5.7°,γ94.3°,c101.3 β5.7°,γ94.3°,c101.3

    4.

    two

    5.

    about 8.2 8.2 square feet

    6.

    161.9 yd.

    8.2 Non-right Triangles: Law of Cosines

    1.

    a14.9, a14.9, β23.8°,β23.8°, γ126.2°. γ126.2°.

    2.

    α27.7°, α27.7°, β40.5°,β40.5°, γ111.8° γ111.8°

    3.

    Area = 552 square feet

    4.

    about 8.15 square feet

    8.3 Polar Coordinates

    1.
    Polar grid with point (2, pi/3) plotted.
    2.
    Points (2, 9pi/4) and (3, -pi/6) are plotted in the polar grid.
    3.

    ( x,y )=( 1 2 , 3 2 ) ( x,y )=( 1 2 , 3 2 )

    4.

    r= 3 r= 3

    5.

    x 2 + y 2 =2y x 2 + y 2 =2y or, in the standard form for a circle, x 2 + ( y1 ) 2 =1 x 2 + ( y1 ) 2 =1

    8.4 Polar Coordinates: Graphs

    1.

    The equation fails the symmetry test with respect to the line θ= π 2 θ= π 2 and with respect to the pole. It passes the polar axis symmetry test.

    2.

    Tests will reveal symmetry about the polar axis. The zero is ( 0, π 2 ), ( 0, π 2 ), and the maximum value is (3,0). (3,0).

    3.
    Graph of the limaçon r=3-2cos(theta). Extending to the left.
    4.

    The graph is a rose curve, n n even

    Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4.
    5.
    Graph of rose curve r=3cos(3theta). Three petals equally spaced from origin.

    Rose curve, n n odd

    6.
    332319af092e1a0f6c3cb6d06848cb5164449fa1

    8.5 Polar Form of Complex Numbers

    1.
    Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).
    2.

    13

    3.

    | z |= 50 =5 2 | z |= 50 =5 2

    4.

    z=3( cos( π 2 )+isin( π 2 ) ) z=3( cos( π 2 )+isin( π 2 ) )

    5.

    z=2( cos( π 6 )+isin( π 6 ) ) z=2( cos( π 6 )+isin( π 6 ) )

    6.

    z=2 3 2i z=2 3 2i

    7.

    z 1 z 2 =4 3 ; z 1 z 2 = 3 2 + 3 2 i z 1 z 2 =4 3 ; z 1 z 2 = 3 2 + 3 2 i

    8.

    z 0 =2(cos(30°)+isin(30°)) z 0 =2(cos(30°)+isin(30°))

    z 1 =2(cos(120°)+isin(120°)) z 1 =2(cos(120°)+isin(120°))

    z 2 =2(cos(210°)+isin(210°)) z 2 =2(cos(210°)+isin(210°))

    z 3 =2(cos(300°)+isin(300°)) z 3 =2(cos(300°)+isin(300°))

    8.6 Parametric Equations

    1.
    t t x( t ) x( t ) y( t ) y( t )
    1 1 4 4 2 2
    0 0 3 3 4 4
    1 1 2 2 6 6
    2 2 1 1 8 8
    7d076e2e41ee124655ea757ed151221ca8490d88
    2.

    x(t)= t 3 2t y(t)=t x(t)= t 3 2t y(t)=t

    3.

    y=5 1 2 x3 y=5 1 2 x3

    4.

    y=ln x y=ln x

    5.

    x 2 4 + y 2 9 =1 x 2 4 + y 2 9 =1

    6.

    y= x 2 y= x 2

    8.7 Parametric Equations: Graphs

    1.
    Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.
    2.
    Graph of the given equations - a horizontal ellipse.
    3.

    The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.

    Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.

    8.8 Vectors

    1.
    A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.
    2.

    3u= 15,12 3u= 15,12

    3.

    u=8i11j u=8i11j

    4.

    v= 34 cos(59°)i+ 34 sin(59°)j v= 34 cos(59°)i+ 34 sin(59°)j

    Magnitude = 34 34

    θ= tan 1 ( 5 3 )=59.04° θ= tan 1 ( 5 3 )=59.04°

    8.1 Section Exercises

    1.

    The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.

    3.

    When the known values are the side opposite the missing angle and another side and its opposite angle.

    5.

    A triangle with two given sides and a non-included angle.

    7.

    β=72°,a12.0,b19.9 β=72°,a12.0,b19.9

    9.

    γ=20°,b4.5,c1.6 γ=20°,b4.5,c1.6

    11.

    b3.78 b3.78

    13.

    c13.70 c13.70

    15.

    one triangle, α50.3°,β16.7°,a26.7 α50.3°,β16.7°,a26.7

    17.

    two triangles, γ54.3°,β90.7°,b20.9 γ54.3°,β90.7°,b20.9 or γ 125.7°, β 19.3°, b 6.9 γ 125.7°, β 19.3°, b 6.9

    19.

    two triangles, β75.7°,γ61.3°,b9.9 β75.7°,γ61.3°,b9.9 or β 18.3°, γ 118.7°, b 3.2 β 18.3°, γ 118.7°, b 3.2

    21.

    two triangles, α143.2°,β26.8°,a17.3 α143.2°,β26.8°,a17.3 or α 16.8°, β 153.2°, a 8.3 α 16.8°, β 153.2°, a 8.3

    23.

    no triangle possible

    25.

    A47.8° A47.8° or A 132.2° A 132.2°

    27.

    8.6 8.6

    29.

    370.9 370.9

    31.

    12.3 12.3

    33.

    12.2 12.2

    35.

    16.0 16.0

    37.

    29.7° 29.7°

    39.

    x=76.9°orx=103.1° x=76.9°orx=103.1°

    41.

    110.6° 110.6°

    43.

    A39.4,C47.6,BC20.7 A39.4,C47.6,BC20.7

    45.

    57.1 57.1

    47.

    42.0 42.0

    49.

    430.2 430.2

    51.

    10.1 10.1

    53.

    AD13.8 AD13.8

    55.

    AB2.8 AB2.8

    57.

    L49.7,N56.3,LN5.8 L49.7,N56.3,LN5.8

    59.

    51.4 feet

    61.

    The distance from the satellite to station A A is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.

    63.

    2.6 ft

    65.

    5.6 km

    67.

    371 ft

    69.

    5936 ft

    71.

    24.1 ft

    73.

    19,056 ft2

    75.

    445,624 square miles

    77.

    8.65 ft2

    8.2 Section Exercises

    1.

    two sides and the angle opposite the missing side.

    3.

    s s is the semi-perimeter, which is half the perimeter of the triangle.

    5.

    The Law of Cosines must be used for any oblique (non-right) triangle.

    7.

    11.3

    9.

    34.7

    11.

    26.7

    13.

    257.4

    15.

    not possible

    17.

    95.5°

    19.

    26.9°

    21.

    B45.9°,C99.1°,a6.4 B45.9°,C99.1°,a6.4

    23.

    A20.6°,B38.4°,c51.1 A20.6°,B38.4°,c51.1

    25.

    A37.8°,B43.8,C98.4° A37.8°,B43.8,C98.4°

    27.

    177.56 in2

    29.

    0.04 m2

    31.

    0.91 yd2

    33.

    3.0

    35.

    29.1

    37.

    0.5

    39.

    70.7°

    41.

    77.4°

    43.

    25.0

    45.

    9.3

    47.

    43.52

    49.

    1.41

    51.

    0.14

    53.

    18.3

    55.

    48.98

    57.
    A triangle. One angle is 52 degrees with opposite side = x. The other two sides are 5 and 6.
    59.

    7.62

    61.

    85.1

    63.

    24.0 km

    65.

    99.9 ft

    67.

    37.3 miles

    69.

    2371 miles

    71.
    Angle BO is 9.1 degrees, angle PH is 150.2 degrees, and angle DC is 20.7 degrees.
    73.

    292.4 miles

    75.

    65.4 cm2

    77.

    468 ft2

    8.3 Section Exercises

    1.

    For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.

    3.

    Determine θ θ for the point, then move r r units from the pole to plot the point. If r r is negative, move r r units from the pole in the opposite direction but along the same angle. The point is a distance of r r away from the origin at an angle of θ θ from the polar axis.

    5.

    The point ( 3, π 2 ) ( 3, π 2 ) has a positive angle but a negative radius and is plotted by moving to an angle of π 2 π 2 and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point ( 3, π 2 ) ( 3, π 2 ) has a negative angle and a positive radius and is plotted by first moving to an angle of π 2 π 2 and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.

    7.

    ( 5,0 ) ( 5,0 )

    9.

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

    11.

    ( 2 5 ,0.464 ) ( 2 5 ,0.464 )

    13.

    ( 34 ,5.253 ) ( 34 ,5.253 )

    15.

    ( 8 2 , π 4 ) ( 8 2 , π 4 )

    17.

    r=4cscθ r=4cscθ

    19.

    r= sinθ 2co s 4 θ 3 r= sinθ 2co s 4 θ 3

    21.

    r=3cosθ r=3cosθ

    23.

    r= 3sinθ cos( 2θ ) r= 3sinθ cos( 2θ )

    25.

    r= 9sinθ cos 2 θ r= 9sinθ cos 2 θ

    27.

    r= 1 9cosθsinθ r= 1 9cosθsinθ

    29.

    x 2 + y 2 =4x x 2 + y 2 =4x or ( x2 ) 2 4 + y 2 4 =1; ( x2 ) 2 4 + y 2 4 =1; circle

    31.

    3y+x=6; 3y+x=6; line

    33.

    y=3; y=3; line

    35.

    xy=4; xy=4; hyperbola

    37.

    x 2 + y 2 =4; x 2 + y 2 =4; circle

    39.

    x5y=3; x5y=3; line

    41.

    ( 3, 3π 4 ) ( 3, 3π 4 )

    43.

    ( 5,π ) ( 5,π )

    45.
    Polar coordinate system with a point located on the second concentric circle and two-thirds of the way between pi and 3pi/2 (closer to 3pi/2).
    47.
    Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi/2 and 2pi.
    49.
    Polar coordinate system with a point located on the fifth concentric circle and pi/2.
    51.
    Polar coordinate system with a point located on the third concentric circle and 2/3 of the way between pi/2 and pi (closer to pi).
    53.
    Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi/2.
    55.

    r= 6 5cosθsinθ r= 6 5cosθsinθ

    Plot of given line in the polar coordinate grid
    57.

    r=2sinθ r=2sinθ

    Plot of given circle in the polar coordinate grid
    59.

    r= 2 cosθ r= 2 cosθ

    Plot of given circle in the polar coordinate grid
    61.

    r=3cosθ r=3cosθ

    Plot of given circle in the polar coordinate grid.
    63.

    x 2 + y 2 =16 x 2 + y 2 =16

    Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.
    65.

    y=x y=x

    Plot of line y=x in the rectangular coordinates grid.
    67.

    x 2 + ( y+5 ) 2 =25 x 2 + ( y+5 ) 2 =25

    Plot of circle with radius 5 centered at (0,-5).
    69.

    ( 1.618,1.176 ) ( 1.618,1.176 )

    71.

    ( 10.630,131.186° ) ( 10.630,131.186° )

    73.

    ( 2,3.14 )or( 2,π ) ( 2,3.14 )or( 2,π )

    75.

    A vertical line with a a units left of the y-axis.

    77.

    A horizontal line with a a units below the x-axis.

    79.
    Graph of shaded circle of radius 4 with the edge not included (dotted line) - polar coordinate grid.
    81.
    Graph of ray starting at (2, pi/4) and extending in a positive direction along pi/4 - polar coordinate grid.
    83.
    Graph of the shaded region 0 to pi/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid

    8.4 Section Exercises

    1.

    Symmetry with respect to the polar axis is similar to symmetry about the x x -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line θ= π 2 θ= π 2 is similar to symmetry about the y y -axis.

    3.

    Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at θ=0, π 2 , θ=0, π 2 , ππ and 3π 2 , 3π 2 , and sketch the graph.

    5.

    The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

    7.

    symmetric with respect to the polar axis

    9.

    symmetric with respect to the polar axis, symmetric with respect to the line θ= π 2 , θ= π 2 , symmetric with respect to the pole

    11.

    no symmetry

    13.

    no symmetry

    15.

    symmetric with respect to the pole

    17.

    circle

    Graph of given circle.
    19.

    cardioid

    Graph of given cardioid.
    21.

    cardioid

    Graph of given cardioid.
    23.

    one-loop/dimpled limaçon

    Graph of given one-loop/dimpled limaçon
    25.

    one-loop/dimpled limaçon

    Graph of given one-loop/dimpled limaçon
    27.

    inner loop/two-loop limaçon

    Graph of given inner loop/two-loop limaçon
    29.

    inner loop/two-loop limaçon

    Graph of given inner loop/two-loop limaçon
    31.

    inner loop/two-loop limaçon

    Graph of given inner loop/two-loop limaçon
    33.

    lemniscate

    Graph of given lemniscate (along horizontal axis)
    35.

    lemniscate

    Graph of given lemniscate (along y=x)
    37.

    rose curve

    Graph of given rose curve - four petals.
    39.

    rose curve

    Graph of given rose curve - eight petals.
    41.

    Archimedes’ spiral

    Graph of given Archimedes' spiral
    43.

    Archimedes’ spiral

    Graph of given Archimedes' spiral
    45.
    Graph of given equation.
    47.
    Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)
    49.
    Graph of given equation.
    51.
    Graph of given equation. Similar to original Archimedes' spiral.
    53.
    Graph of given equation.
    55.

    They are both spirals, but not quite the same.

    57.

    Both graphs are curves with 2 loops. The equation with a coefficient of θ θ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to 4π 4π to get a better picture.

    59.

    When the width of the domain is increased, more petals of the flower are visible.

    61.

    The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

    63.

    The graphs are spirals. The smaller the coefficient, the tighter the spiral.

    65.

    ( 4, π 3 ),( 4, 5π 3 ) ( 4, π 3 ),( 4, 5π 3 )

    67.

    ( 3 2 , π 3 ),( 3 2 , 5π 3 ) ( 3 2 , π 3 ),( 3 2 , 5π 3 )

    69.

    ( 0, π 2 ),( 0,π ),( 0, 3π 2 ),( 0,2π ) ( 0, π 2 ),( 0,π ),( 0, 3π 2 ),( 0,2π )

    71.

    ( 8 4 2 , π 4 ),( 8 4 2 , 5π 4 ) ( 8 4 2 , π 4 ),( 8 4 2 , 5π 4 ) and at θ= 3π 4 , θ= 3π 4 , 7π 4 7π 4 since r r is squared

    8.5 Section Exercises

    1.

    a is the real part, b is the imaginary part, and i= 1 i= 1

    3.

    Polar form converts the real and imaginary part of the complex number in polar form using x=rcosθ x=rcosθ and y=rsinθ. y=rsinθ.

    5.

    z n = r n ( cos( nθ )+isin( nθ ) ) z n = r n ( cos( nθ )+isin( nθ ) ) It is used to simplify polar form when a number has been raised to a power.

    7.

    5 2 5 2

    9.

    38 38

    11.

    14.45 14.45

    13.

    4 5 cis( 333.4° ) 4 5 cis( 333.4° )

    15.

    2cis( π 6 ) 2cis( π 6 )

    17.

    7 3 2 +i 7 2 7 3 2 +i 7 2

    19.

    2 3 2i 2 3 2i

    21.

    1.5i 3 3 2 1.5i 3 3 2

    23.

    4 3 cis( 198° ) 4 3 cis( 198° )

    25.

    3 4 cis( 180° ) 3 4 cis( 180° )

    27.

    5 3 cis( 17π 24 ) 5 3 cis( 17π 24 )

    29.

    7cis( 70° ) 7cis( 70° )

    31.

    5cis( 80° ) 5cis( 80° )

    33.

    5cis( π 3 ) 5cis( π 3 )

    35.

    125cis( 135° ) 125cis( 135° )

    37.

    9cis( 240° ) 9cis( 240° )

    39.

    cis( 3π 4 ) cis( 3π 4 )

    41.

    3cis( 80° ),3cis( 200° ),3cis( 320° ) 3cis( 80° ),3cis( 200° ),3cis( 320° )

    43.

    2 4 3 cis( 2π 9 ),2 4 3 cis( 8π 9 ),2 4 3 cis( 14π 9 ) 2 4 3 cis( 2π 9 ),2 4 3 cis( 8π 9 ),2 4 3 cis( 14π 9 )

    45.

    2 2 cis( 7π 8 ),2 2 cis( 15π 8 ) 2 2 cis( 7π 8 ),2 2 cis( 15π 8 )

    47.
    Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).
    49.
    Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).
    51.
    Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).
    53.
    Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).
    55.
    Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).
    57.

    3.61 e 0.59i 3.61 e 0.59i

    59.

    2+3.46i 2+3.46i

    61.

    4.332.50i 4.332.50i

    8.6 Section Exercises

    1.

    A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, x=f( t ) x=f( t ) and y=f( t ). y=f( t ).

    3.

    Choose one equation to solve for t, t, substitute into the other equation and simplify.

    5.

    Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.

    7.

    y=2+2x y=2+2x

    9.

    y=3 x1 2 y=3 x1 2

    11.

    x=2 e 1y 5 x=2 e 1y 5 or y=15ln( x 2 ) y=15ln( x 2 )

    13.

    x=4log( y3 2 ) x=4log( y3 2 )

    15.

    x= ( y 2 ) 3 y 2 x= ( y 2 ) 3 y 2

    17.

    y= x 3 y= x 3

    19.

    ( x 4 ) 2 + ( y 5 ) 2 =1 ( x 4 ) 2 + ( y 5 ) 2 =1

    21.

    y 2 =1 1 2 x y 2 =1 1 2 x

    23.

    y= x 2 +2x+1 y= x 2 +2x+1

    25.

    y= ( x+1 2 ) 3 2 y= ( x+1 2 ) 3 2

    27.

    y=3x+14 y=3x+14

    29.

    y=x+3 y=x+3

    31.

    { x( t )=t y( t )=2sint+1 { x( t )=t y( t )=2sint+1

    33.

    { x( t )= t +2t y( t )=t { x( t )= t +2t y( t )=t

    35.

    { x( t )=4cost y( t )=6sint ; { x( t )=4cost y( t )=6sint ; Ellipse

    37.

    { x( t )= 10 cost y( t )= 10 sint ; { x( t )= 10 cost y( t )= 10 sint ; Circle

    39.

    { x( t )=1+4t y( t )=2t { x( t )=1+4t y( t )=2t

    41.

    { x( t )=4+2t y( t )=13t { x( t )=4+2t y( t )=13t

    43.

    yes, at t=2 t=2

    45.
    t t x x y y
    1 -3 1
    2 0 7
    3 5 17
    47.

    answers may vary: { x( t )=t1 y( t )= t 2 and { x( t )=t+1 y( t )= ( t+2 ) 2 { x( t )=t1 y( t )= t 2 and { x( t )=t+1 y( t )= ( t+2 ) 2

    49.

    answers may vary: , { x( t )=t y( t )= t 2 4t+4 and { x( t )=t+2 y( t )= t 2 { x( t )=t y( t )= t 2 4t+4 and { x( t )=t+2 y( t )= t 2

    8.7 Section Exercises

    1.

    plotting points with the orientation arrow and a graphing calculator

    3.

    The arrows show the orientation, the direction of motion according to increasing values of t. t.

    5.

    The parametric equations show the different vertical and horizontal motions over time.

    7.
    Graph of the given equations - looks like an upward opening parabola.
    9.
    Graph of the given equations - a line, negative slope.
    11.
    Graph of the given equations - looks like a sideways parabola, opening to the right.
    13.
    Graph of the given equations - looks like the left half of an upward opening parabola.
    15.
    Graph of the given equations - looks like a downward opening absolute value function.
    17.
    Graph of the given equations - a vertical ellipse.
    19.
    Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.
    21.
    Graph of the given equations- looks like an upward opening parabola.
    23.
    Graph of the given equations- looks like a downward opening parabola.
    25.
    Graph of the given equations- horizontal ellipse.
    27.
    Graph of the given equations- looks like the lower half of a sideways parabola opening to the right
    29.
    Graph of the given equations- looks like an upwards opening parabola
    31.
    Graph of the given equations- looks like the upper half of a sideways parabola opening to the left
    33.
    Graph of the given equations- the left half of a hyperbola with diagonal asymptotes
    35.
    Graph of the given equations - vertical periodic trajectory
    37.
    Graph of the given equations - vertical periodic trajectory
    39.

    There will be 100 back-and-forth motions.

    41.

    Take the opposite of the x( t ) x( t ) equation.

    43.

    The parabola opens up.

    45.

    { x( t )=5cost y( t )=5sint { x( t )=5cost y( t )=5sint

    47.
    Graph of the given equations
    49.
    Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 1 unit.
    51.
    Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 3 units.
    53.

    a=4, a=4, b=3,b=3, c=6,c=6, d=1 d=1

    55.

    a=4, a=4, b=2,b=2, c=3,c=3, d=3 d=3

    57.
    Graph of the given equations Graph of the given equations Graph of the given equations
    59.
    Graph of the given equations Graph of the given equations Graph of the given equations
    61.

    The y y -intercept changes.

    63.

    y( x )=16 ( x 15 ) 2 +20( x 15 ) y( x )=16 ( x 15 ) 2 +20( x 15 )

    65.

    { x(t)=64tcos( 52° ) y(t)=16 t 2 +64tsin( 52° ) { x(t)=64tcos( 52° ) y(t)=16 t 2 +64tsin( 52° )

    67.

    approximately 3.2 seconds

    69.

    1.6 seconds

    71.
    Graph of the given equations - a hypocycloid
    73.
    Graph of the given equations - a four petal rose

    8.8 Section Exercises

    1.

    lowercase, bold letter, usually u,v,w u,v,w

    3.

    They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

    5.

    The first number always represents the coefficient of the i, i, and the second represents the j. j.

    7.

    7,5 7,5

    9.

    not equal

    11.

    equal

    13.

    equal

    15.

    7i3j 7i3j

    17.

    6i2j 6i2j

    19.

    u+v= 5,5 ,uv= 1,3 ,2u3v= 0,5 u+v= 5,5 ,uv= 1,3 ,2u3v= 0,5

    21.

    10i4j 10i4j

    23.

    2 29 29 i+ 5 29 29 j 2 29 29 i+ 5 29 29 j

    25.

    2 229 229 i+ 15 229 229 j 2 229 229 i+ 15 229 229 j

    27.

    7 2 10 i+ 2 10 j 7 2 10 i+ 2 10 j

    29.

    | v |=7.810,θ=39.806° | v |=7.810,θ=39.806°

    31.

    | v |=7.211,θ=236.310° | v |=7.211,θ=236.310°

    33.

    6 6

    35.

    12 12

    37.
    86d1b5b8e793785f00658728abc6aeed57d92964
    39.
    Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).
    41.
    Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).
    43.
    Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).
    45.
    Vector extending from the origin to (7,5), taking the base as the origin.
    47.

    4,1 4,1

    49.

    v=7i+3j v=7i+3j

    Vector going from (4,-1) to (-3,2).
    51.

    3 2 i+3 2 j 3 2 i+3 2 j

    53.

    i 3 j i 3 j

    55.
    1. 58.7
    2. 12.5
    57.

    x=7.13 x=7.13 pounds, y=3.63 y=3.63 pounds

    59.

    x=2.87 x=2.87 pounds, y=4.10 y=4.10 pounds

    61.

    4.635 miles, 17.764° N of E

    63.

    17 miles. 10.318 miles

    65.

    Distance: 2.868. Direction: 86.474° North of West, or 3.526° West of North

    67.

    4.924°. 659 km/hr

    69.

    4.424°

    71.

    ( 0.081,8.602 ) ( 0.081,8.602 )

    73.

    21.801°, relative to the car’s forward direction

    75.

    parallel: 16.28, perpendicular: 47.28 pounds

    77.

    19.35 pounds, 231.54° from the horizontal

    79.

    5.1583 pounds, 75.8° from the horizontal

    Review Exercises

    1.

    Not possible

    3.

    C=120°,a=23.1,c=34.1 C=120°,a=23.1,c=34.1

    5.

    distance of the plane from point A: A: 2.2 km, elevation of the plane: 1.6 km

    7.

    b=71.0°,C=55.0°,a=12.8 b=71.0°,C=55.0°,a=12.8

    9.

    40.6 km

    11. Polar coordinate grid with a point plotted on the fifth concentric circle 2/3 the way between pi and 3pi/2 (closer to 3pi/2).
    13.

    ( 0,2 ) ( 0,2 )

    15.

    ( 9.8489,203.96° ) ( 9.8489,203.96° )

    17.

    r=8 r=8

    19.

    x 2 + y 2 =7x x 2 + y 2 =7x

    21.

    y=x y=x

    Plot of the function y=-x in rectangular coordinates.
    23.

    symmetric with respect to the line θ= π 2 θ= π 2

    25. Graph of the given polar equation - an inner loop limaçon.
    27. Graph of the given polar equation - a cardioid.
    29.

    5

    31.

    cis( π 3 ) cis( π 3 )

    33.

    2.3+1.9i 2.3+1.9i

    35.

    60cis( π 2 ) 60cis( π 2 )

    37.

    3cis( 4π 3 ) 3cis( 4π 3 )

    39.

    25cis( 3π 2 ) 25cis( 3π 2 )

    41.

    5cis( 3π 4 ),5cis( 7π 4 ) 5cis( 3π 4 ),5cis( 7π 4 )

    43.
    Plot of -1 + 3i in the complex plane (-1 along the real axis, 3 along the imaginary).
    45.

    x 2 + 1 2 y=1 x 2 + 1 2 y=1

    47.

    { x( t )=2+6t y( t )=3+4t { x( t )=2+6t y( t )=3+4t

    49.

    y=2 x 5 y=2 x 5

    Plot of the given parametric equations.
    51.
    1. { x( t )=( 80cos( 40° ) )t y( t )=16 t 2 +( 80sin( 40° ) )t+4 { x( t )=( 80cos( 40° ) )t y( t )=16 t 2 +( 80sin( 40° ) )t+4
    2. The ball is 14 feet high and 184 feet from where it was launched.
    3. 3.3 seconds
    53.

    not equal

    55.

    4i

    57.

    3 10 10 3 10 10 i 10 10 10 10 j

    59.

    Magnitude: 3 2 , 3 2 , Direction: 225° 225°

    61.

    16 16

    63. Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).

    Practice Test

    1.

    α=67.1°,γ=44.9°,a=20.9 α=67.1°,γ=44.9°,a=20.9

    3.

    1712 miles 1712 miles

    5.

    ( 1, 3 ) ( 1, 3 )

    7.

    y=3 y=3

    Plot of the given equation in rectangular form - line y=-3.
    9. Graph of the given equations - a cardioid.
    11.

    106 106

    13.

    5 2 +i 5 3 2 5 2 +i 5 3 2

    15.

    4cis( 21° ) 4cis( 21° )

    17.

    2 2 cis( 18° ),2 2 cis( 198° ) 2 2 cis( 18° ),2 2 cis( 198° )

    19.

    y=2 ( x1 ) 2 y=2 ( x1 ) 2

    21. Graph of the given equations - a vertical ellipse.
    23.

    −4i − 15j

    25.

    2 13 13 i+ 3 13 13 j 2 13 13 i+ 3 13 13 j


    13.2.8: Chapter 8 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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