Chapter 6 Solutions to Odd-Numbered Problems

Section 6.1 Solutions

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. [latex]\beta =72^\circ ,a\approx 12.0,b\approx 19.9[/latex]

9. [latex]\gamma =20^\circ ,b\approx 4.5,c\approx 1.6[/latex]

11. [latex]b\approx 3.78[/latex]

13. [latex]c\approx 13.70[/latex]

15. one triangle, [latex]\alpha \approx 50.3^\circ ,\beta \approx 16.7^\circ ,a\approx 26.7[/latex]

17. two triangles, [latex]\gamma \approx 54.3^\circ ,\beta \approx 90.7^\circ ,b\approx 20.9[/latex] or [latex]{\gamma }^{\prime }\approx 125.7^\circ ,{\beta }^{\prime }\approx 19.3^\circ ,{b}^{\prime }\approx 6.9[/latex]

19. two triangles, [latex]\beta \approx 75.7^\circ , \gamma \approx 61.3^\circ ,b\approx 9.9[/latex] or [latex]{\beta }^{\prime }\approx 18.3^\circ ,{\gamma }^{\prime }\approx 118.7^\circ ,{b}^{\prime }\approx 3.2[/latex]

21. two triangles, [latex]\alpha \approx 143.2^\circ ,\beta \approx 26.8^\circ ,a\approx 17.3[/latex] or [latex]{\alpha }^{\prime }\approx 16.8^\circ ,{\beta }^{\prime }\approx 153.2^\circ ,{a}^{\prime }\approx 8.3[/latex]

23. no triangle possible

25. [latex]A\approx 47.8^\circ[/latex] or [latex]{A}^{\prime }\approx 132.2^\circ[/latex]

27. [latex]8.6[/latex]

29. [latex]370.9[/latex]

31. [latex]12.3[/latex]

33. [latex]12.2[/latex]

35. [latex]16.0[/latex]

37. [latex]29.7^\circ[/latex]

39. [latex]x=76.9^\circ \text{or }x=103.1^\circ[/latex]

41. [latex]110.6^\circ[/latex]

43. [latex]A\approx 39.4,\text{ }C\approx 47.6,\text{ }BC\approx 20.7[/latex]

45. [latex]57.1[/latex]

47. [latex]42.0[/latex]

49. [latex]430.2[/latex]

51. [latex]10.1[/latex]

53. [latex]AD\approx \text{ }13.8[/latex]

55. [latex]AB\approx 2.8[/latex]

57. [latex]L\approx 49.7,\text{ }N\approx 56.3,\text{ }LN\approx 5.8[/latex]

59. 51.4 feet

61. The distance from the satellite to station [latex]A[/latex] 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

Section 6.2 Solutions

1. two sides and the angle opposite the missing side

3. [latex]s[/latex] 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. [latex]B\approx 45.9^\circ ,C\approx 99.1^\circ ,a\approx 6.4[/latex]

23. [latex]A\approx 20.6^\circ ,B\approx 38.4^\circ ,c\approx 51.1[/latex]

25. [latex]A\approx 37.8^\circ ,B\approx 43.8,C\approx 98.4^\circ[/latex]

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. 599.8 miles

75. 65.4 cm2

77. 468 ft2

Section 6.3 Solutions

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

5. The point [latex]\left(-3,\frac{\pi }{2}\right)[/latex] has a positive angle but a negative radius and is plotted by moving to an angle of [latex]\frac{\pi }{2}[/latex] and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point [latex]\left(3,-\frac{\pi }{2}\right)[/latex] has a negative angle and a positive radius and is plotted by first moving to an angle of [latex]-\frac{\pi }{2}[/latex] 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.
a) [latex]\left(5,-\frac{4\pi}{3}\right)[/latex]
b) [latex]\left(-5,\frac{5\pi}{3}\right)[/latex]
c) [latex]\left(5,\frac{8\pi}{3}\right)[/latex]

9.
a) [latex]\left(3,-\frac{5\pi}{4}\right)[/latex]
b) [latex]\left(-3,\frac{7\pi}{4}\right)[/latex]
c) [latex]\left(3,\frac{11\pi}{4}\right)[/latex]

11.
a) [latex]\left(4,-135^\circ\right)[/latex]
b) [latex]\left(-4,45^\circ\right)[/latex]
c) [latex]\left(4,585^\circ\right)[/latex]

13.
a) [latex]\left(5,-60^\circ\right)[/latex]
b) [latex]\left(-5,120^\circ\right)[/latex]
c) [latex]\left(5,480^\circ\right)[/latex]

15. [latex]\left(-5,0\right)[/latex]

17. [latex]\left(-\frac{3\sqrt{3}}{2},-\frac{3}{2}\right)[/latex]

19. [latex]\left(2\sqrt{5}, 0.464\right)[/latex]

21. [latex]\left(\sqrt{34},5.253\right)[/latex]

23. [latex]\left(8\sqrt{2},\frac{\pi }{4}\right)[/latex]

25. [latex]r=4\csc \theta[/latex]

27. [latex]r=\sqrt[3]{\frac{sin\theta }{2co{s}^{4}\theta }}[/latex]

29. [latex]r=3\cos \theta[/latex]

31. [latex]r=\frac{3\sin \theta }{\cos \left(2\theta \right)}[/latex]

33. [latex]r=\frac{9\sin \theta }{{\cos }^{2}\theta }[/latex]

35. [latex]r=\sqrt{\frac{1}{9\cos \theta \sin \theta }}[/latex]

37. [latex]{x}^{2}+{y}^{2}=4x[/latex] or [latex]\frac{{\left(x - 2\right)}^{2}}{4}+\frac{{y}^{2}}{4}=1[/latex]; circle

39. [latex]3y+x=6[/latex]; line

41. [latex]y=3[/latex]; line

43. [latex]xy=4[/latex]; hyperbola

45. [latex]{x}^{2}+{y}^{2}=4[/latex]; circle

47. [latex]x - 5y=3[/latex]; line

49. [latex]\left(3,\frac{3\pi }{4}\right)[/latex]

51. [latex]\left(5,\pi \right)[/latex]

53.
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).

55.
Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi/2 and 2pi.

57.
Polar coordinate system with a point located on the fifth concentric circle and pi/2.

59.
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).

61.
Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi/2.

63. [latex]r=\frac{6}{5\cos \theta -\sin \theta }[/latex]
Plot of given line in the polar coordinate grid

65. [latex]r=2\sin \theta[/latex]
Plot of given circle in the polar coordinate grid

67. [latex]r=\frac{2}{\cos \theta }[/latex]
Plot of given circle in the polar coordinate grid

69. [latex]r=3\cos \theta[/latex]
Plot of given circle in the polar coordinate grid.

71. [latex]{x}^{2}+{y}^{2}=16[/latex]
Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.

73. [latex]y=x[/latex]
Plot of line y=x in the rectangular coordinates grid.

75. [latex]{x}^{2}+{\left(y+5\right)}^{2}=25[/latex]
Plot of circle with radius 5 centered at (0,-5).

77. A vertical line with [latex]a[/latex] units left of the y-axis.

79. A horizontal line with [latex]a[/latex] units below the x-axis.

Section 6.4 Solutions

1. Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] is similar to symmetry about the [latex]y[/latex] -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 [latex]\theta =0,\frac{\pi }{2},\pi \text{and }\frac{3\pi }{2}[/latex], 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 [latex]\theta =\frac{\pi }{2}[/latex], 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 [latex]\theta[/latex] has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to [latex]4\pi[/latex] 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. [latex]\left(4,\frac{\pi }{3}\right),\left(4,\frac{5\pi }{3}\right)[/latex]

67. [latex]\left(\frac{3}{2},\frac{\pi }{3}\right),\left(\frac{3}{2},\frac{5\pi }{3}\right)[/latex]

69. [latex]\left(0,\frac{\pi }{2}\right),\left(0,\pi \right),\left(0,\frac{3\pi }{2}\right),\left(0,2\pi \right)[/latex]

71. [latex]\left(\frac{\sqrt[4]{8}}{2},\frac{\pi }{4}\right),\left(\frac{\sqrt[4]{8}}{2},\frac{5\pi }{4}\right)[/latex]
and at [latex]\theta =\frac{3\pi }{4},\frac{7\pi }{4}[/latex] since [latex]r[/latex] is squared

Section 6.5 Solutions

1. a is the real part, b is the imaginary part, and [latex]i=\sqrt{−1}[/latex]

3. Polar form converts the real and imaginary part of the complex number in polar form using [latex]x=r\cos\theta[/latex] and [latex]y=r\sin\theta[/latex]

5. [latex]z^{n}=r^{n}\left(\cos\left(n\theta\right)+i\sin\left(n\theta\right)\right)[/latex]. It is used to simplify polar form when a number has been raised to a power.

7. [latex]5\sqrt{2}[/latex]

9. [latex]\sqrt{38}[/latex]

11. [latex]\sqrt{14.45}[/latex]

13. [latex]4\sqrt{5}\text{cis}\left(333.4^{\circ}\right)[/latex]

15. [latex]2\text{cis}\left(\frac{\pi}{6}\right)[/latex]

17. [latex]\frac{7\sqrt{3}}{2}+i\frac{7}{2}[/latex]

19. [latex]−2\sqrt{3}−2i[/latex]

21. [latex]−1.5−i\frac{3\sqrt{3}}{2}[/latex]

23. [latex]4\sqrt{3}\text{cis}\left(198^{\circ}\right)[/latex]

25. [latex]\frac{3}{4}\text{cis}\left(180^{\circ}\right)[/latex]

27. [latex]5\sqrt{3}\text{cis}\left(\frac{17\pi}{24}\right)[/latex]

29. [latex]7\text{cis}\left(70^{\circ}\right)[/latex]

31. [latex]5\text{cis}\left(80^{\circ}\right)[/latex]

33. [latex]5\text{cis}\left(\frac{\pi}{3}\right)[/latex]

35. [latex]125\text{cis}\left(135^{\circ}\right)[/latex]

37. [latex]9\text{cis}\left(240^{\circ}\right)[/latex]

39. [latex]\text{cis}\left(\frac{3\pi}{4}\right)[/latex]

41. [latex]3\text{cis}\left(80^{\circ}\right)\text{, }3\text{cis}\left(200^{\circ}\right)\text{, }3\text{cis}\left(320^{\circ}\right)[/latex]

43. [latex]2\sqrt[3]{4}\text{cis}\left(\frac{2\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{8\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{14\pi}{9}\right)[/latex]

45. [latex]2\sqrt{2}\text{cis}\left(\frac{7\pi}{8}\right)\text{, }2\sqrt{2}\text{cis}\left(\frac{15\pi}{8}\right)[/latex]

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. [latex]3.61e^{−0.59i}[/latex]

59. [latex]−2+3.46i[/latex]

61. [latex]−4.33−2.50i[/latex]

Section 6.6 Solutions

1. lowercase, bold letter, usually 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, and the second represents the j.

7. [latex]\langle 7,−5\rangle[/latex]

9. not equal

11. equal

13. equal

15. [latex]7\boldsymbol{i}−3\boldsymbol{j}[/latex]

17. [latex]−6\boldsymbol{i}−2\boldsymbol{j}[/latex]

19. [latex]\boldsymbol{u}+\boldsymbol{v}=\langle−5,5\rangle,\boldsymbol{u}−\boldsymbol{v}=\langle−1,3\rangle,2\boldsymbol{u}−3\boldsymbol{v}=\langle 0,5\rangle[/latex]

21. [latex]−10\boldsymbol{i}–4\boldsymbol{j}[/latex]

23. [latex]−\frac{2\sqrt{29}}{29}\boldsymbol{i}+\frac{5\sqrt{29}}{29}\boldsymbol{j}[/latex]

25. [latex]–\frac{2\sqrt{229}}{229}\boldsymbol{i}+\frac{15\sqrt{229}}{229}\boldsymbol{j}[/latex]

27. [latex]–\frac{7\sqrt{2}}{10}\boldsymbol{i}+\frac{\sqrt{2}}{10}\boldsymbol{j}[/latex]

29. [latex]|\boldsymbol{v}|=7.810,\theta=39.806^{\circ}[/latex]

31. [latex]|\boldsymbol{v}|=7.211,\theta=236.310^{\circ}[/latex]

33. −6

35. −12

37.

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).