Answers and Hints to Selected Exercises
Chapter 1
Section 1.1 (p. § 1.1)
1. \(115\Degrees\) 3. \(A=52\Degrees\), \(B=104\Degrees\)
5. \(45\Degrees\)
7. \(A=9\Degrees\), \(B=81\Degrees\)
8. \(0.011\Degrees\) and \(89.989\Degrees\)
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 (p. § 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. \(\cos\;3\Degrees\) 21. \(\sin\;44\Degrees\)
23. \(\csc\;13\Degrees\)
25. \(\sin\;77\Degrees\)
27. \(\tan\;80\Degrees\) 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 (p. § 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. \(c=13\), \(A=22.6\Degrees\), \(B=67.4\Degrees\) 17. \(a=0.28\), \(c=2.02\),
\(B=82\Degrees\) 19. \(b=6.15\), \(c=6.84\), \(B=64\Degrees\)
21. \(a=6.15\), \(c=6.84\), \(A=64\Degrees\) 23. \(a=\sqrt{2}\), \(b=\sqrt{2}\),
\(B=45\Degrees\) 25. (a) \(0.944\) cm
(b) \(2.112\) cm
27. (a) \(\sqrt{3}\;a\) (b) \(35.26\Degrees\)
29. \(1379.5\) ft \(= 0.2613\) mi
Section 1.4 (p. § 1.4)
1. QII 3. QIV 5. negative \(y\)-axis
7. QIII
9. QIV 11. QI, QIII 13. QI, QIV
15. QI, QII 17. \(43\Degrees\) 19. \(54\Degrees\)
21. \(85\Degrees\) 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 (p. § 1.5)
1. (a) \(328\Degrees\) (b) \(148\Degrees\) (c) \(212\Degrees\)
3. (a) \(248\Degrees\) (b) \(68\Degrees\) (c) \(292\Degrees\)
7. \(25\Degrees\), \(155\Degrees\) 9. \(65\Degrees\), \(295\Degrees\)
11. \(38\Degrees\), \(218\Degrees\) 13. \(169\Degrees\),
\(191\Degrees\)
15. \(D=\left( \frac{ab^2}{a^2 + b^2}, \frac{a^2 b}{a^2 + b^2} \right)\)
Chapter 2
Section 2.1 (p. § 2.1)
1. \(b = 7.4\), \(c = 15.1\), \(C = 120\Degrees\) 3. \(a = 9.7\), \(b = 10.7\), \(C = 95\Degrees\) 5. \(b = 65.1\), \(B = 136.5\Degrees\), \(C = 18.5\Degrees\) 7. No solution 9. \(b = 24.9\), \(B = 59.9\Degrees\), \(C = 70.1\Degrees\); \(b = 9.9\), \(B = 20.1\Degrees\), \(C = 109.9\Degrees\) 11. \(422\) mi/hr 15. \(5.66\) cm and \(12.86\) cm 16. Hint: Think geometrically.
Section 2.2 (p. § 2.2)
1. \(a = 10.6\), \(B = 40.9\Degrees\), \(C = 79.1\) 3. \(A = 47.9\Degrees\),
\(b = 8.2\), \(C = 72.1\Degrees\) 5. No solution 7. \(4.13\) and \(8.91\) cm
9. \(50.5\Degrees\), \(59\Degrees\), \(70.5\Degrees\)
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 (p. § 2.3)
1. \(A = 79.1\Degrees\), \(B = 40.9\Degrees\), \(c = 10.6\) 3. \(A = 47.9\Degrees\), \(b = 8.2\), \(C = 72.1\Degrees\) 5. No 6. Yes 11. Hint: Think of Exercise 10.
Section 2.4 (p. § 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 (2.23)-(2.25).
Section 2.5 (p. § 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 (p. § 3.1)
1. \(\theta = 270\Degrees\) 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 (p. § 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 (p. § 3.3)
9. Hint: Is \(\sin\;A + \cos\;A\) always positive? 11. \(1/2\)
Section 3.4 (p. § 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 (p. § 4.1)
1. \(\pi/45\) 3. \(13\pi/18\) 5. \(-3\pi/5\) 7. \(36\Degrees\) 9. \(174\Degrees\)
Section 4.2 (p. § 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 (p. § 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 (p. § 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 (p. § 5.1)
13. Partial answer: \(\sec\;\theta = OQ\)
Section 5.2 (p. § 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 (p. § 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 (p. § 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 (p. § 6.2)
1. \(x=1.89549426703398093962\)
Section 6.3 (p. § 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. \(\sqrt{13}\,\text{cis}\;56.3\Degrees\)
27. \(\sqrt{2}\,\text{cis}\;315\Degrees\) 29. \(\text{cis}\;0\Degrees\)
33. \(81\,\text{cis}\;56\Degrees\) 35. \(1.5\,\text{cis}\;253\Degrees\)
37. \(\sqrt[6]{2}\,\text{cis}\;15\Degrees\), \(\sqrt[6]{2}\,\text{cis}\;135\Degrees\),
\(\sqrt[6]{2}\,\text{cis}\;255\Degrees\)
39. \(\frac{1}{2} + \frac{\sqrt{3}}{2}\,i\),
\(-1\), \(\frac{1}{2} - \frac{\sqrt{3}}{2}\,i\) 41. \(\text{cis}\;36\Degrees\),
\(\text{cis}\;108\Degrees\), \(\text{cis}\;180\Degrees\), \(\text{cis}\;252\Degrees\),
\(\text{cis}\;324\Degrees\)
Section 6.4 (p. § 6.4)
1. \((-3\sqrt{3},-3)\) 3. \((\sqrt{3},-1)\)
5. \((-1/\sqrt{2},-1/\sqrt{2})\) 7. \((\sqrt{10},251.6\Degrees)\)
9. \((2\sqrt{5},333.4\Degrees)\) 11. \(r = 6\,\cos\;\theta\)
13. \(r^2 \,\cos\;2\theta = 1\) 14. \(r = 3/(2 - \cos\;\theta)\)