Right Triangle Trigonometry
1.
2. The two acute angles are complementary.
3. [latex]\frac{\pi }{6}[/latex]
4. [latex]\frac{\pi }{4}[/latex]
5. [latex]b=\frac{20\sqrt{3}}{3},c=\frac{40\sqrt{3}}{3}[/latex]
6. [latex]a=10,000,c=10,000.5[/latex]
7. [latex]b=\frac{5\sqrt{3}}{3},c=\frac{10\sqrt{3}}{3}[/latex]
8. [latex]\frac{5\sqrt{29}}{29}[/latex]
9. [latex]\frac{5}{2}[/latex]
10. [latex]\frac{\sqrt{29}}{2}[/latex]
11. [latex]\frac{5\sqrt{41}}{41}[/latex]
12. [latex]\frac{5}{4}[/latex]
13. [latex]\frac{\sqrt{41}}{4}[/latex]
14. [latex]c=14, b=7\sqrt{3}[/latex]
15. [latex]a=15, b=15[/latex]
16. [latex]b=9.9970, c=12.2041[/latex]
17. [latex]a=2.0838, b=11.8177[/latex]
18. [latex]a=55.9808,c=57.9555[/latex]
19. [latex]a=46.6790,b=17.9184[/latex]
20. [latex]a=16.4662,c=16.8341[/latex]
21. 188.3159
22. 200.6737
23. 498.3471 ft
24. 1060.09 ft
25. 27.372 ft
26. 22.6506 ft
27. 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].
2. 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]
3. [latex]−\frac{\pi}{6}[/latex]
4. [latex]\frac{3\pi}{4}[/latex]
5. [latex]\frac{\pi}{3}[/latex]
6. 1.98
7. 0.93
8. 1.41
9. 0.56 radians
10. 0
11. −0.71
12. [latex]−\frac{\pi}{4}[/latex]
13. 0.8
14. [latex]\frac{5}{13}[/latex]
15. [latex]\frac{\sqrt{2x+1}}{x+1}[/latex]
16. [latex]\frac{\sqrt{2x+1}}{x+1}[/latex]
17. t
18. domain [−1,1]; range [0,π]
19. 0.395 radians
20. 1.11 radians
21. 1.25 radians
22. 0.405 radians
Non-Right Triangles: Law of Sines
1. When the known values are the side opposite the missing angle and another side and its opposite angle.
2. A triangle with two given sides and a non-included angle.
3. [latex]\beta =72^\circ ,a\approx 12.0,b\approx 19.9[/latex]
4. [latex]\gamma =20^\circ ,b\approx 4.5,c\approx 1.6[/latex]
5. [latex]b\approx 3.78[/latex]
6. [latex]c\approx 13.70[/latex]
7. one triangle, [latex]\alpha \approx 50.3^\circ ,\beta \approx 16.7^\circ ,a\approx 26.7[/latex]
8. 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]
9. 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]
10. 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]
11. no triangle possible
12. [latex]A\approx 47.8^\circ[/latex] or [latex]{A}^{\prime }\approx 132.2^\circ[/latex]
13. [latex]8.6[/latex]
14. [latex]370.9[/latex]
15. [latex]12.3[/latex]
16. [latex]12.2[/latex]
17. [latex]16.0[/latex]
18. [latex]29.7^\circ[/latex]
19. [latex]x=76.9^\circ \text{or }x=103.1^\circ[/latex]
20. [latex]110.6^\circ[/latex]
21. [latex]A\approx 39.4,\text{ }C\approx 47.6,\text{ }BC\approx 20.7[/latex]
22. [latex]57.1[/latex]
23. [latex]42.0[/latex]
24. [latex]430.2[/latex]
25. 51.4 feet
26. The distance from the satellite to station [latex]A[/latex] is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.
27. 2.6 ft
28. 5.6 km
29. 371 ft
30. 24.1 ft
31. 19,056 ft2
32. 445,624 square miles
Non-Right Triangles: Law of Cosines
1. two sides and the angle opposite the missing side
[new item 2, “what do you need to know for a missing angle,” has no matching answer in this key]
2. [latex]s[/latex] is the semi-perimeter, which is half the perimeter of the triangle.
[unmatched: key answer “The Law of Cosines must be used for any oblique (non-right) triangle” doesn’t correspond to any problem in the exercise list]
3. 11.3
4. 257.4
5. not possible
6. 95.5°
7. 26.9°
8. [latex]B\approx 45.9^\circ ,C\approx 99.1^\circ ,a\approx 6.4[/latex]
9. [latex]A\approx 20.6^\circ ,B\approx 38.4^\circ ,c\approx 51.1[/latex]
10. [latex]A\approx 37.8^\circ ,B\approx 43.8,C\approx 98.4^\circ[/latex]
11. 177.56 in2
12. 0.04 m2
13. 0.91 yd2
14. 3.0
15. 0.5
16. 70.7°
17. 77.4°
18. 25.0
19. 43.52
20. 1.41
21. 0.14
22. 7.62
23. 85.1
24. 24.0 km
25. 99.9 ft
26. 37.3 miles
27. 2371 miles
28.
29. 599.8 miles
30. 65.4 cm2
31. 468 ft2