13.2.8: Chapter 8

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Try It

8.1 Non-right Triangles: Law of Sines

$\begin{array}{l} α = 98^{\circ} a = 34.6 \\ β = 39^{\circ} b = 22 \\ γ = 43^{\circ} c = 23.8 \end{array}$

Solution 1

$\begin{array}{l} α = 80° & a = 120 \\ β \approx 83.2° & b = 121 \\ γ \approx 16.8° & c \approx 35.2 \end{array}$

Solution 2

$\begin{array}{l} α^{'} = 80° a^{'} = 120 \\ β^{'} \approx 96.8° b^{'} = 121 \\ γ^{'} \approx 3.2° c^{'} \approx 6.8 \end{array}$

$β \approx 5.7°, γ \approx 94.3°, c \approx 101.3$

two

about $8.2$ square feet

161.9 yd.

8.2 Non-right Triangles: Law of Cosines

$a \approx 14.9,$ $β \approx 23.8°,$ $γ \approx 126.2° .$

$α \approx 27.7°,$ $β \approx 40.5°,$ $γ \approx 111.8°$

Area = 552 square feet

about 8.15 square feet

8.3 Polar Coordinates

$Polar grid with point (2, pi/3) plotted.$

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

$(x, y) = (\frac{1}{2}, - \frac{\sqrt{3}}{2})$

$r = \sqrt{3}$

$x^{2} + y^{2} = 2 y$ or, in the standard form for a circle, $x^{2} + {(y - 1)}^{2} = 1$

8.4 Polar Coordinates: Graphs

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

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

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

The graph is a rose curve, $n$ even

$Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4.$

$Graph of rose curve r=3cos(3theta). Three petals equally spaced from origin.$

Rose curve, $n$ odd

$332319af092e1a0f6c3cb6d06848cb5164449fa1$

8.5 Polar Form of Complex Numbers

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

$| z | = \sqrt{50} = 5 \sqrt{2}$

$z = 3 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

$z = 2 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))$

$z = 2 \sqrt{3} - 2 i$

$z_{1} z_{2} = - 4 \sqrt{3}; \frac{z_{1}}{z_{2}} = - \frac{\sqrt{3}}{2} + \frac{3}{2} i$

$z_{0} = 2 (\cos (30°) + i \sin (30°))$

$z_{1} = 2 (\cos (120°) + i \sin (120°))$

$z_{2} = 2 (\cos (210°) + i \sin (210°))$

$z_{3} = 2 (\cos (300°) + i \sin (300°))$

8.6 Parametric Equations


$t$	$x (t)$	$y (t)$
$- 1$	$- 4$	$2$
$0$	$- 3$	$4$
$1$	$- 2$	$6$
$2$	$- 1$	$8$

$7d076e2e41ee124655ea757ed151221ca8490d88$

$\begin{array}{l} x (t) = t^{3} - 2 t \\ y (t) = t \end{array}$

$y = 5 - \sqrt{\frac{1}{2} x - 3}$

$y = \ln \sqrt{x}$

$\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$

$y = x^{2}$

8.7 Parametric Equations: Graphs

$Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.$

$Graph of the given equations - a horizontal ellipse.$

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

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

$3 u = ⟨ 15, 12 ⟩$

$u = 8 i - 11 j$

$v = \sqrt{34} \cos (59°) i + \sqrt{34} \sin (59°) j$

Magnitude = $\sqrt{34}$

$θ = \tan^{- 1} (\frac{5}{3}) = 59.04°$

8.1 Section Exercises

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

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

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

$β = 72°, a \approx 12.0, b \approx 19.9$

$γ = 20°, b \approx 4.5, c \approx 1.6$

11.

$b \approx 3.78$

13.

$c \approx 13.70$

15.

one triangle, $α \approx 50.3°, β \approx 16.7°, a \approx 26.7$

17.

two triangles, $γ \approx 54.3°, β \approx 90.7°, b \approx 20.9$ or $γ^{'} \approx 125.7°, β^{'} \approx 19.3°, b^{'} \approx 6.9$

19.

two triangles, $β \approx 75.7°, γ \approx 61.3°, b \approx 9.9$ or $β^{'} \approx 18.3°, γ^{'} \approx 118.7°, b^{'} \approx 3.2$

21.

two triangles, $α \approx 143.2°, β \approx 26.8°, a \approx 17.3$ or $α^{'} \approx 16.8°, β^{'} \approx 153.2°, a^{'} \approx 8.3$

23.

no triangle possible

25.

$A \approx 47.8°$ or $A^{'} \approx 132.2°$

27.

$8.6$

29.

$370.9$

31.

$12.3$

33.

$12.2$

35.

$16.0$

37.

$29.7°$

39.

$x = 76.9° or x = 103.1°$

41.

$110.6°$

43.

$A \approx 39.4, C \approx 47.6, B C \approx 20.7$

45.

$57.1$

47.

$42.0$

49.

$430.2$

51.

$10.1$

53.

$A D \approx 13.8$

55.

$A B \approx 2.8$

57.

$L \approx 49.7, N \approx 56.3, L N \approx 5.8$

59.

51.4 feet

61.

The distance from the satellite to station $A$ 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

two sides and the angle opposite the missing side.

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

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

11.3

34.7

11.

26.7

13.

257.4

15.

not possible

17.

95.5°

19.

26.9°

21.

$B \approx 45.9°, C \approx 99.1°, a \approx 6.4$

23.

$A \approx 20.6°, B \approx 38.4°, c \approx 51.1$

25.

$A \approx 37.8°, B \approx 43.8, C \approx 98.4°$

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

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.

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

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

$(- 5, 0)$

$(- \frac{3 \sqrt{3}}{2}, - \frac{3}{2})$

11.

$(2 \sqrt{5}, 0.464)$

13.

$(\sqrt{34}, 5.253)$

15.

$(8 \sqrt{2}, \frac{π}{4})$

17.

$r = 4 \csc θ$

19.

$r = \sqrt[3]{\frac{s i n θ}{2 c o s^{4} θ}}$

21.

$r = 3 \cos θ$

23.

$r = \frac{3 \sin θ}{\cos (2 θ)}$

25.

$r = \frac{9 \sin θ}{\cos^{2} θ}$

27.

$r = \sqrt{\frac{1}{9 \cos θ \sin θ}}$

29.

$x^{2} + y^{2} = 4 x$ or $\frac{{(x - 2)}^{2}}{4} + \frac{y^{2}}{4} = 1;$ circle

31.

$3 y + x = 6;$ line

33.

$y = 3;$ line

35.

$x y = 4;$ hyperbola

37.

$x^{2} + y^{2} = 4;$ circle

39.

$x - 5 y = 3;$ line

41.

$(3, \frac{3 π}{4})$

43.

$(5, π)$

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.

$r = \frac{6}{5 \cos θ - \sin θ}$

$Plot of given line in the polar coordinate grid$

57.

$r = 2 \sin θ$

$Plot of given circle in the polar coordinate grid$

59.

$r = \frac{2}{\cos θ}$

$Plot of given circle in the polar coordinate grid$

61.

$r = 3 \cos θ$

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

63.

$x^{2} + y^{2} = 16$

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

65.

$y = x$

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

67.

$x^{2} + {(y + 5)}^{2} = 25$

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

69.

$(1.618, - 1.176)$

71.

$(10.630, 131.186°)$

73.

$(2, 3.14) o r (2, π)$

75.

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

77.

A horizontal line with $a$ 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

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

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 $θ = 0, \frac{π}{2},$ $π$ and $\frac{3 π}{2},$ and sketch the graph.

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

symmetric with respect to the polar axis

symmetric with respect to the polar axis, symmetric with respect to the line $θ = \frac{π}{2},$ 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 $θ$ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to $4 π$ 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.

$(4, \frac{π}{3}), (4, \frac{5 π}{3})$

67.

$(\frac{3}{2}, \frac{π}{3}), (\frac{3}{2}, \frac{5 π}{3})$

69.

$(0, \frac{π}{2}), (0, π), (0, \frac{3 π}{2}), (0, 2 π)$

71.

$(\frac{\sqrt[4]{8}}{2}, \frac{π}{4}), (\frac{\sqrt[4]{8}}{2}, \frac{5 π}{4})$ and at $θ = \frac{3 π}{4},$ $\frac{7 π}{4}$ since $r$ is squared

8.5 Section Exercises

a is the real part, b is the imaginary part, and $i = \sqrt{- 1}$

Polar form converts the real and imaginary part of the complex number in polar form using $x = r \cos θ$ and $y = r \sin θ .$

$z^{n} = r^{n} (\cos (n θ) + i \sin (n θ))$ It is used to simplify polar form when a number has been raised to a power.

$5 \sqrt{2}$

$\sqrt{38}$

11.

$\sqrt{14.45}$

13.

$4 \sqrt{5} cis (333.4°)$

15.

$2 cis (\frac{π}{6})$

17.

$\frac{7 \sqrt{3}}{2} + i \frac{7}{2}$

19.

$- 2 \sqrt{3} - 2 i$

21.

$- 1.5 - i \frac{3 \sqrt{3}}{2}$

23.

$4 \sqrt{3} cis (198°)$

25.

$\frac{3}{4} cis (180°)$

27.

$5 \sqrt{3} cis (\frac{17 π}{24})$

29.

$7 cis (70°)$

31.

$5 cis (80°)$

33.

$5 cis (\frac{π}{3})$

35.

$125 cis (135°)$

37.

$9 cis (240°)$

39.

$cis (\frac{3 π}{4})$

41.

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

43.

$2 \sqrt[3]{4} cis (\frac{2 π}{9}), 2 \sqrt[3]{4} cis (\frac{8 π}{9}), 2 \sqrt[3]{4} cis (\frac{14 π}{9})$

45.

$2 \sqrt{2} cis (\frac{7 π}{8}), 2 \sqrt{2} cis (\frac{15 π}{8})$

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.

$3.61 e^{- 0.59 i}$

59.

$- 2 + 3.46 i$

61.

$- 4.33 - 2.50 i$

8.6 Section Exercises

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

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

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

$y = - 2 + 2 x$

$y = 3 \sqrt{\frac{x - 1}{2}}$

11.

$x = 2 e^{\frac{1 - y}{5}}$ or $y = 1 - 5 l n (\frac{x}{2})$

13.

$x = 4 \log (\frac{y - 3}{2})$

15.

$x = {(\frac{y}{2})}^{3} - \frac{y}{2}$

17.

$y = x^{3}$

19.

${(\frac{x}{4})}^{2} + {(\frac{y}{5})}^{2} = 1$

21.

$y^{2} = 1 - \frac{1}{2} x$

23.

$y = x^{2} + 2 x + 1$

25.

$y = {(\frac{x + 1}{2})}^{3} - 2$

27.

$y = - 3 x + 14$

29.

$y = x + 3$

31.

${\begin{array}{l} x (t) = t \\ y (t) = 2 \sin t + 1 \end{array}$

33.

${\begin{array}{l} x (t) = \sqrt{t} + 2 t \\ y (t) = t \end{array}$

35.

${\begin{array}{l} x (t) = 4 \cos t \\ y (t) = 6 \sin t \end{array};$ Ellipse

37.

${\begin{array}{l} x (t) = \sqrt{10} \cos t \\ y (t) = \sqrt{10} \sin t \end{array};$ Circle

39.

${\begin{array}{l} x (t) = - 1 + 4 t \\ y (t) = - 2 t \end{array}$

41.

${\begin{array}{l} x (t) = 4 + 2 t \\ y (t) = 1 - 3 t \end{array}$

43.

yes, at $t = 2$

45.


$t$	$x$	$y$
1	-3	1
2	0	7
3	5	17

47.

answers may vary: ${\begin{array}{l} x (t) = t - 1 \\ y (t) = t^{2} \end{array} and {\begin{array}{l} x (t) = t + 1 \\ y (t) = {(t + 2)}^{2} \end{array}$

49.

answers may vary: , ${\begin{array}{l} x (t) = t \\ y (t) = t^{2} - 4 t + 4 \end{array} and {\begin{array}{l} x (t) = t + 2 \\ y (t) = t^{2} \end{array}$

8.7 Section Exercises

plotting points with the orientation arrow and a graphing calculator

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

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

$Graph of the given equations - looks like an upward opening parabola.$

$Graph of the given equations - a line, negative slope.$

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 $x (t)$ equation.

43.

The parabola opens up.

45.

${\begin{cases} x (t) = 5 \cos t \\ y (t) = 5 \sin t \end{cases}$

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.

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

55.

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

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 $y$ -intercept changes.

63.

$y (x) = - 16 {(\frac{x}{15})}^{2} + 20 (\frac{x}{15})$

65.

${\begin{cases} x (t) = 64 t \cos (52 °) \\ y (t) = - 16 t^{2} + 64 t \sin (52 °) \end{cases}$

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

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

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

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

$〈 7, - 5 〉$

not equal

11.

equal

13.

equal

15.

$7 i - 3 j$

17.

$- 6 i - 2 j$

19.

$u + v = 〈 - 5, 5 〉, u - v = 〈 - 1, 3 〉, 2 u - 3 v = 〈 0, 5 〉$

21.

$- 10 i - 4 j$

23.

$- \frac{2 \sqrt{29}}{29} i + \frac{5 \sqrt{29}}{29} j$

25.

$- \frac{2 \sqrt{229}}{229} i + \frac{15 \sqrt{229}}{229} j$

27.

$- \frac{7 \sqrt{2}}{10} i + \frac{\sqrt{2}}{10} j$

29.

$| v | = 7.810, θ = 39.806 °$

31.

$| v | = 7.211, θ = 236.310°$

33.

$- 6$

35.

$- 12$

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.

$〈 4, 1 〉$

49.

$v = - 7 i + 3 j$

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

51.

$3 \sqrt{2} i + 3 \sqrt{2} j$

53.

$i - \sqrt{3} j$

55.

ⓐ 58.7
ⓑ 12.5

57.

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

59.

$x = 2.87$ pounds, $y = 4.10$ 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.

$(0.081, 8.602)$

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

Not possible

$C = 120°, a = 23.1, c = 34.1$

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

$b = 71.0°, C = 55.0°, a = 12.8$

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.

$(0, 2)$

15.

$(9.8489, 203.96°)$

17.

$r = 8$

19.

$x^{2} + y^{2} = 7 x$

21.

$y = - x$

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

23.

symmetric with respect to the line $θ = \frac{π}{2}$

25. $Graph of the given polar equation - an inner loop limaçon.$

27. $Graph of the given polar equation - a cardioid.$

29.

31.

$cis (- \frac{π}{3})$

33.

$2.3 + 1.9 i$

35.

$60 cis (\frac{π}{2})$

37.

$3 cis (\frac{4 π}{3})$

39.

$25 cis (\frac{3 π}{2})$

41.

$5 cis (\frac{3 π}{4}), 5 cis (\frac{7 π}{4})$

43.

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

45.

$x^{2} + \frac{1}{2} y = 1$

47.

${\begin{array}{l} x (t) = - 2 + 6 t \\ y (t) = 3 + 4 t \end{array}$

49.

$y = - 2 x^{5}$

$Plot of the given parametric equations.$

51.

${\begin{cases} x (t) = (80 \cos (40°)) t \\ y (t) = - 16 t^{2} + (80 \sin (40°)) t + 4 \end{cases}$
The ball is 14 feet high and 184 feet from where it was launched.
3.3 seconds

53.

not equal

55.

57.

$- \frac{3 \sqrt{10}}{10}$ i $- \frac{\sqrt{10}}{10}$ j

59.

Magnitude: $3 \sqrt{2},$ Direction: $225°$

61.

$16$

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

$α = 67.1°, γ = 44.9°, a = 20.9$

$1712 miles$

$(1, \sqrt{3})$

$y = - 3$

$Plot of the given equation in rectangular form - line y=-3.$

9. $Graph of the given equations - a cardioid.$

11.

$\sqrt{106}$

13.

$\frac{- 5}{2} + i \frac{5 \sqrt{3}}{2}$

15.

$4 cis (21°)$

17.

$2 \sqrt{2} cis (18°), 2 \sqrt{2} cis (198°)$

19.

$y = 2 {(x - 1)}^{2}$

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

23.

−4i − 15j

25.

$\frac{2 \sqrt{13}}{13} i + \frac{3 \sqrt{13}}{13} j$

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8.1 Non-right Triangles: Law of Sines

8.2 Non-right Triangles: Law of Cosines

8.3 Polar Coordinates