附录 A

Appendix A

Answers and Hints to Selected Exercises

Chapter 1

Section 1.1

1. \(115^\circ\)

3. \(A=52^\circ\), \(B=104^\circ\)

5. \(45^\circ\)

7. \(A=9^\circ\), \(B=81^\circ\)

8. \(0.011^\circ\) and \(89.989^\circ\)

9. \(25\) miles

10. \(111.8\) ft

15. Hint: Are the opposite sides of the four-sided figure inside the circle parallel?

Section 1.2

1.

\(\sin\;A = 5/13\), \(\cos\;A = 12/13\), \(\tan\;A = 5/12\),

\(\csc\;A = 13/5\), \(\sec\;A = 13/12\), \(\cot\;A = 12/5\);

\(\sin\;B = 12/13\), \(\cos\;B = 5/13\), \(\tan\;B = 12/5\),

\(\csc\;B = 13/12\), \(\sec\;B = 13/5\), \(\cot\;B = 5/12\)

3.

\(\sin\;A = 7/25\), \(\cos\;A = 24/25\), \(\tan\;A = 7/24\),

\(\csc\;A = 25/7\), \(\sec\;A = 25/24\), \(\cot\;A = 24/7\);

\(\sin\;B = 24/25\), \(\cos\;B = 7/25\), \(\tan\;B = 24/7\),

\(\csc\;B = 25/24\), \(\sec\;B = 25/7\), \(\cot\;B = 7/24\)

5.

\(\sin\;A = 9/41\), \(\cos\;A = 40/41\), \(\tan\;A = 9/40\),

\(\csc\;A = 41/9\), \(\sec\;A = 41/40\), \(\cot\;A = 40/9\);

\(\sin\;B = 40/41\), \(\cos\;B = 9/41\), \(\tan\;B = 40/9\),

\(\csc\;B = 41/40\), \(\sec\;B = 41/9\), \(\cot\;B = 9/40\)

7.

\(\sin\;A = 1/\sqrt{10}\), \(\cos\;A = 3/\sqrt{10}\), \(\tan\;A = 1/3\),

\(\csc\;A = \sqrt{10}\), \(\sec\;A = \sqrt{10}/3\), \(\cot\;A = 3\);

\(\sin\;B = 3/\sqrt{10}\), \(\cos\;B = 1/\sqrt{10}\), \(\tan\;B = 3\),

\(\csc\;B = \sqrt{10}/3\), \(\sec\;B = \sqrt{10}\), \(\cot\;B = 1/3\)

9.

\(\sin\;A = 5/6\), \(\cos\;A = \sqrt{11}/6\), \(\tan\;A = 5/\sqrt{11}\),

\(\csc\;A = 6/5\), \(\sec\;A = 6/\sqrt{11}\), \(\cot\;A = \sqrt{11}/5\);

\(\sin\;B = \sqrt{11}/6\), \(\cos\;B = 5/6\), \(\tan\;B = \sqrt{11}/5\),

\(\csc\;B = 6/\sqrt{11}\), \(\sec\;B = 6/5\), \(\cot\;B = 5/\sqrt{11}\)

11.

\(\cos\;A = \sqrt{7}/4\), \(\tan\;A = 3/\sqrt{7}\),

\(\csc\;A = 4/3\), \(\sec\;A = 4/\sqrt{7}\), \(\cot\;A = \sqrt{7}/3\)

13.

\(\sin\;A = \sqrt{6}/\sqrt{10}\), \(\tan\;A = \sqrt{6}/2\),

\(\csc\;A = \sqrt{10}/\sqrt{6}\), \(\sec\;A = \sqrt{10}/2\), \(\cot\;A = 2/\sqrt{6}\)

15.

\(\sin\;A = 5/\sqrt{106}\), \(\cos\;A = 9/\sqrt{106}\),

\(\csc\;A = \sqrt{106}/5\), \(\sec\;A = \sqrt{106}/9\), \(\cot\;A = 9/5\)

17.

\(\sin\;A = \sqrt{40}/7\), \(\cos\;A = 3/7\), \(\tan\;A = \sqrt{40}/3\),

\(\csc\;A = 7/\sqrt{40}\), \(\cot\;A = 3/\sqrt{40}\)

19. \(\cos\;3^\circ\)

21. \(\sin\;44^\circ\)

23. \(\csc\;13^\circ\)

25. \(\sin\;77^\circ\)

27. \(\tan\;80^\circ\)

30. Hint: Draw a right triangle with an acute angle \(A\).

33. Hint: Draw two right triangles whose hypotenuses are the same length.

37.

(a) \(\sqrt{13}/4\)

(b) \(4\sqrt{3}/\sqrt{13}\)

(c) \(3/\sqrt{13}\)

Section 1.3

1. \(102.7\) ft

3. \(241.1\) ft

4. \(274\) ft

7. \(1062\) mi

9. \(0.476\) in

11. \(1.955\) in

13. \(0.4866\) in

14. Partial answer: \(DE=a\;\cot\;\theta\;\,\cos^2\,\theta\)

15. \(c=13\), \(A=22.6^\circ\), \(B=67.4^\circ\)

17. \(a=0.28\), \(c=2.02\), \(B=82^\circ\)

19. \(b=6.15\), \(c=6.84\), \(B=64^\circ\)

21. \(a=6.15\), \(c=6.84\), \(A=64^\circ\)

23. \(a=\sqrt{2}\), \(b=\sqrt{2}\), \(B=45^\circ\)

25. (a) \(0.944\) cm (b) \(2.112\) cm

27. (a) \(\sqrt{3}\;a\) (b) \(35.26^\circ\)

29. \(1379.5\) ft \(= 0.2613\) mi

Section 1.4

1. QII

3. QIV

5. negative \(y\)-axis

7. QIII

9. QIV

11. QI, QIII

13. QI, QIV

15. QI, QII

17. \(43^\circ\)

19. \(54^\circ\)

21. \(85^\circ\)

23.

\(\sin\;\theta = \sqrt{3}/2\) and \(\tan\;\theta = -\sqrt{3}\); \(\sin\;\theta = -\sqrt{3}/2\) and \(\tan\;\theta = \sqrt{3}\)

25.

\(\sin\;\theta = \sqrt{21}/5\) and \(\tan\;\theta = \sqrt{21}/2\); \(\sin\;\theta = -\sqrt{21}/5\) and \(\tan\;\theta = -\sqrt{21}/2\)

27.

\(\cos\;\theta = \sqrt{3}/2\) and \(\tan\;\theta = 1/\sqrt{3}\);

\(\cos\;\theta = -\sqrt{3}/2\) and \(\tan\;\theta = -1/\sqrt{3}\)

29. \(\cos\;\theta = \pm 1\) and \(\tan\;\theta = 0\)

31. \(\cos\;\theta = 0\) and \(\tan\;\theta\) is undefined

33.

\(\sin\;\theta = 1/\sqrt{5}\) and \(\cos\;\theta = -2/\sqrt{5}\);

\(\sin\;\theta = -1/\sqrt{5}\) and \(\cos\;\theta = 2/\sqrt{5}\)

35.

\(\sin\;\theta = 5/13\) and \(\cos\;\theta = 12/13\); \(\sin\;\theta = -5/13\) and \(\cos\;\theta = -12/13\)

37. No

39. No

Section 1.5

1. (a) \(328^\circ\) (b) \(148^\circ\) (c) \(212^\circ\)

3. (a) \(248^\circ\) (b) \(68^\circ\) (c) \(292^\circ\)

7. \(25^\circ\), \(155^\circ\)

9. \(65^\circ\), \(295^\circ\)

11. 38^circ, \(218^\circ\)

13. \(169^\circ\), \(191^\circ\)

15. \(D=\left( \frac{ab^2}{a^2 + b^2}, \frac{a^2 b}{a^2 + b^2} \right)\)

Chapter 2

Section 2.1

1. \(b = 7.4\), \(c = 15.1\), \(C = 120^\circ\)

3. \(a = 9.7\), \(b = 10.7\), \(C = 95^\circ\)

5. \(b = 65.1\), \(B = 136.5^\circ\), \(C = 18.5^\circ\)

7. No solution

9. \(b = 24.9\), \(B = 59.9^\circ\), \(C = 70.1^\circ\); \(b = 9.9\), \(B = 20.1^\circ\), \(C = 109.9^\circ\)

11. \(422\) mi/hr

15. \(5.66\) cm and \(12.86\) cm

16. Hint: Think geometrically.

Section 2.2

1. \(a = 10.6\), \(B = 40.9^\circ\), \(C = 79.1\)

3. \(A = 47.9^\circ\), \(b = 8.2\), \(C = 72.1^\circ\)

5. No solution

7. \(4.13\) and \(8.91\) cm

9. \(50.5^\circ\), \(59^\circ\), \(70.5^\circ\)

11. \(7\) cm

15. Hints: One of the angles in the formulas is a right angle; also, use the definition of cosine.

Section 2.3

1. \(A = 79.1^\circ\), \(B = 40.9^\circ\), \(c = 10.6\)

3. \(A = 47.9^\circ\), \(b = 8.2\), \(C = 72.1^\circ\)

5. No

6. Yes

11. Hint: Think of Exercise 10.

Section 2.4

1. \(22.55\)

3. \(9.21\)

5. \(\frac{3}{4}\sqrt{15} \approx 2.905\)

7. \(12.21\)

9. Hints: The diagonals break the quadrilateral into four triangles; also, consider formulas (22)-(24).

Section 2.5

1. \(R = 2.63\), \(r = 0.69\)

3. \(R = 3.51\), \(r = 1.36\)

5. \(R = 24.18\), \(r = 1.12\)

12.

(c) Twice as large

(d) Hint: Bisect each angle.

Chapter 3

Section 3.1

1. \(\theta = 270^\circ\)

3. Hint: See Example 3.7.

19. \(\tan\;\theta = \pm\,\sin\;\theta / \sqrt{1 - \sin^2 \;\theta} = \pm\,\sqrt{1 - \cos^2 \;\theta} / \cos\;\theta\)

Section 3.2

3. \(\sin\;(A+B) = \frac{1020}{1189}\), \(\cos\;(A+B) = -\frac{611}{1189}\), \(\tan\;(A+B) = -\frac{1020}{611}\)

4. \((\sqrt{6} + \sqrt{2})/4\)

5. \(2 - \sqrt{3}\)

15. Hint: For \(a \ne 0\) and \(b \ne 0\), draw a right triangle with legs of lengths \(a\) and \(b\).

Section 3.3

9. Hint: Is \(\sin\;A + \cos\;A\) always positive?

11. \(1/2\)

Section 3.4

13. Hint: One way to do this is with the Law of Tangents. Another way is with the Law of Sines.

Chapter 4

Section 4.1

1. \(\pi/45\)

3. \(13\pi/18\)

5. \(-3\pi/5\)

7. \(36^\circ\)

9. \(174^\circ\)

Section 4.2

1. \(9.6\) cm

3. \(11\pi\) in

5. \(54.94\) in

7. \(12.86\) ft

8. \(34.18\)

9. \(38.26\)

11. \(3.392\) and \(9.174\)

12. \(3.105828541\)

Section 4.3

1. \(1.512~\text{cm}^2\)

3. \(24.5~\text{m}^2\)

5. \(269.1~\text{cm}^2\)

7. \(5~\text{cm}^2\)

9. \(\pi/2~\text{cm}^2\)

11. \(0.017~\text{cm}^2\)

13. \(21.46\)

15. \(48.17\)

17. \(0.522~\text{m}^2\)

19. Sector area is quadrupled, arc length is doubled.

Section 4.4

1. \(\nu=6\) m/sec, \(\omega=1.5\) rad/sec

3. \(\nu=6.6\) m/sec, \(\omega=0.94\) rad/sec

5. \(\nu=3.75\) m/sec, \(\omega=1.875\) rad/sec

7. \(3.375\) rad

9. \(32\) rpm and \(21.33\) rpm

11. \(40.84\) in/sec

Chapter 5

Section 5.1

13. Partial answer: \(\sec\;\theta = OQ\)

Section 5.2

1. amplitude \(= 3\), period \(= 2\), phase shift = \(0\)

3. amplitude \(= 1\), period \(= 2\pi/5\), phase shift = \(-3/5\)

5. amplitude \(= 1\), period \(= 2\pi/5\), phase shift = \(-\pi/5\)

7. amplitude \(= 1\), period \(= \pi\), phase shift = \(3\pi/2\)

9. amplitude undefined, period \(= \pi/2\), phase shift = \(3\pi/2\)

11. amplitude undefined, period \(= \pi\), phase shift = \(1/2\)

13. max. at \(x=\pm\,\sqrt{\pi/2}\), \(\pm\,\sqrt{5\pi/2}\), \(\pm\,\sqrt{9\pi/2}\), \(...\) min. at \(x=\pm\,\sqrt{3\pi/2}\), \(\pm\,\sqrt{7\pi/2}\), \(\pm\,\sqrt{11\pi/2}\), \(...\)

15. amplitude \(= 0.5\), period \(= \pi\)

17. out of phase

18. in phase

19. amplitude \(= \sqrt{34}\), period \(= 2\)

21. amplitude \(= 2\,\sqrt{2}\), period \(= 2\pi\)

23. \(2\pi\)

25. \(6\)

27. amplitude envelope: \(y=\pm\,x^2\)

29. No

Section 5.3

1. \(\pi/4\)

3. \(0\)

5. \(\pi\)

7. \(\pi/2\)

9. \(0\)

11. \(-\pi/3\)

13. \(\pi/7\)

15. \(4\pi/5\)

17. \(\pi/6\)

19. \(-\pi/9\)

21. \(12/13\)

23. \(\pi/2\)

25. \(\pi/2\)

Chapter 6

Section 6.1

1. \(\frac{3\pi}{4} + \pi k\)

3. \(\frac{3\pi}{10} + \frac{2\pi k}{5}\)

5. \(\pm\,\frac{\pi}{6} + \pi k\)

7. \(-0.821 + 2\pi k\), \(3.963 + 2\pi k\)

9. \(\frac{\pi}{4} + \pi k\)

11. \(\frac{2\pi k}{3}\)

Section 6.2

1. \(x=1.89549426703398093962\)

Section 6.3

1. \(-1+i\)

3. \(-13i\)

5. \(-1-i\)

7. \(i\)

9. \(-i\)

11. \(i\)

13. \(-i\)

15. \(i\)

17. Let \(z=a+bi\). Then \(\overline{z}=a-bi\), so \(\overline{\left(\overline{z}\right)} = \overline{a-bi}=a+bi=z\).

23. Hint: Use Exercise 20.

25. \(\sqrt{13}\,\text{cis}\;56.3^\circ\)

27. \(\sqrt{2}\,\text{cis}\;315^\circ\)

29. \(\text{cis}\;0^\circ\)

33. \(81\,\text{cis}\;56^\circ\)

35. \(1.5\,\text{cis}\;253^\circ\)

37. \(\sqrt[6]{2}\,\text{cis}\;15^\circ\), \(\sqrt[6]{2}\,\text{cis}\;135^\circ\), \(\sqrt[6]{2}\,\text{cis}\;255^\circ\)

39. \(\frac{1}{2} + \frac{\sqrt{3}}{2}\,i\), \(-1\), \(\frac{1}{2} - \frac{\sqrt{3}}{2}\,i\)

41. \(\text{cis}\;36^\circ\), \(\text{cis}\;108^\circ\), \(\text{cis}\;180^\circ\), \(\text{cis}\;252^\circ\), \(\text{cis}\;324^\circ\)

Section 6.4

1. \((-3\sqrt{3},-3)\)

3. \((\sqrt{3},-1)\)

5. \((-1/\sqrt{2},-1/\sqrt{2})\)

7. \((\sqrt{10},251.6^\circ)\)

9. \((2\sqrt{5},333.4^\circ)\)

11. \(r = 6\,\cos\;\theta\)

13. \(r^2 \,\cos\;2\theta = 1\)

14. \(r = 3/(2 - \cos\;\theta)\)