Right Triangle Trigonometry

1.

4. The two acute angles are complementary.

7. [latex]\frac{\pi }{6}[/latex]

9. [latex]\frac{\pi }{4}[/latex]

11. [latex]b=\frac{20\sqrt{3}}{3},c=\frac{40\sqrt{3}}{3}[/latex]

13. [latex]a=10,000,c=10,000.5[/latex]

15. [latex]b=\frac{5\sqrt{3}}{3},c=\frac{10\sqrt{3}}{3}[/latex]

17. [latex]\frac{5\sqrt{29}}{29}[/latex]

19. [latex]\frac{5}{2}[/latex]

21. [latex]\frac{\sqrt{29}}{2}[/latex]

23. [latex]\frac{5\sqrt{41}}{41}[/latex]

25. [latex]\frac{5}{4}[/latex]

27. [latex]\frac{\sqrt{41}}{4}[/latex]

29. [latex]c=14, b=7\sqrt{3}[/latex]

31. [latex]a=15, b=15[/latex]

33. [latex]b=9.9970, c=12.2041[/latex]

35. [latex]a=2.0838, b=11.8177[/latex]

37. [latex]a=55.9808,c=57.9555[/latex]

39. [latex]a=46.6790,b=17.9184[/latex]

41. [latex]a=16.4662,c=16.8341[/latex]

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

Inverse Trigonometric Functions

1. The function [latex]y=\sin x[/latex] is one-to-one on [latex]\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right][/latex]; thus, this interval is the range of the inverse function of [latex]y=\sin x\text{, }f\left(x\right)=\sin^{−1}x[/latex]. The function [latex]y=\cos x[/latex] is one-to-one on [0,π]; thus, this interval is the range of the inverse function of [latex]y=\cos x\text{, }f(x)=\cos^{−1}x[/latex].

4. Rewrite the expression as: [latex]\sec^{-1}(2)=\cos^{-1}!\left(\frac{1}{2}\right)[/latex]. Then use the inverse cosine (arccos) function on the calculator: [latex]\theta=\cos^{-1}\left(\frac{1}{2}\right)[/latex]

9. [latex]−\frac{\pi}{6}[/latex]

11. [latex]\frac{3\pi}{4}[/latex]

15. [latex]\frac{\pi}{3}[/latex]

17. 1.98

19. 0.93

21. 1.41

23. 0.56 radians

25. 0

29. −0.71

31. [latex]−\frac{\pi}{4}[/latex]

33. 0.8

35. [latex]\frac{5}{13}[/latex]

43. [latex]\frac{\sqrt{2x+1}}{x+1}[/latex]

45. [latex]\frac{\sqrt{2x+1}}{x+1}[/latex]

47. t

49. domain [−1,1]; range [0,π]

53. 0.395 radians

55. 1.11 radians

57. 1.25 radians

59. 0.405 radians

61. No. The angle the ladder makes with the horizontal is 60 degrees.

Non-Right Triangles: Law of Sines

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]

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

71. 24.1 ft

73. 19,056 ft2

75. 445,624 square miles

Non-Right Triangles: Law of Cosines

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

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

37. 0.5

39. 70.7°

41. 77.4°

43. 25.0

47. 43.52

49. 1.41

51. 0.14

59. 7.62

61. 85.1

63. 24.0 km

65. 99.9 ft

67. 37.3 miles

69. 2371 miles

71.

73. 599.8 miles

75. 65.4 cm2

77. 468 ft2