College Trigonometry, 2011a

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840 Foundations of <strong>Trigonometry</strong> to tan(θ) =−2 are coterminal with β. Moreover, the solutions from Quadrant II differ by exactly π units from the solutions in Quadrant IV, so all the solutions to tan(θ) =−2 areof the form θ = β + πk =arctan(−2) + πk for some integer k. Switching back to the variable t, we record our final answer to tan(t) =−2 ast =arctan(−2) + πk for integers k. 3. The last equation we are asked to solve, sec(x) =− 5 3 , poses two immediate problems. First, we are not told whether or not x represents an angle or a real number. We assume the latter, but note that we will use angles and the Unit Circle to solve the equation regardless. Second, as we have mentioned, there is no universally accepted range of the arcsecant function. For that reason, we adopt the advice given in Section 10.3 and convert this to the cosine problem cos(x) =− 3 5 . Adopting an angle approach, we consider the equation cos(θ) =− 3 5 and note that our solutions lie in Quadrants II and III. Since − 3 5 isn’t the cosine of any of the ‘common angles’, we’ll need to express our solutions in terms of the arccosine function. The real number t = arccos ( − 3 5) is defined so that π 2
10.6 The Inverse Trigonometric Functions 841 10.6.5 Exercises In Exercises 1 - 40, find the exact value. ( √ ) 3 1. arcsin (−1) 2. arcsin − 2 ( 1 5. arcsin (0) 6. arcsin 2) ( √ ) 2 3. arcsin − 2 (√ ) 2 7. arcsin 2 ( √ ) 3 9. arcsin (1) 10. arccos (−1) 11. arccos − 2 ( 13. arccos − 1 ) 2 17. arccos (√ ) 3 2 ( √ ) 3 21. arctan − 3 ( 1 14. arccos (0) 15. arccos 2) ( 4. arcsin − 1 ) 2 (√ ) 3 8. arcsin 2 ( √ ) 2 12. arccos − 2 (√ ) 2 16. arccos 2 18. arccos (1) 19. arctan ( − √ 3 ) 20. arctan (−1) (√ ) 3 22. arctan (0) 23. arctan 3 24. arctan (1) 25. arctan (√ 3 ) 26. arccot ( − √ 3 ) ( √ ) 3 27. arccot (−1) 28. arccot − 3 29. arccot (0) 30. arccot (√ ) 3 3 31. arccot (1) 32. arccot (√ 3 ) 33. arcsec (2) 34. arccsc (2) 35. arcsec (√ 2 ) 36. arccsc (√ 2 ) ( 2 √ ) ( 3 2 √ ) 3 37. arcsec 38. arccsc 39. arcsec (1) 40. arccsc (1) 3 3 In Exercises 41 - 48, assume that the range of arcsecant is [ 0, π ) [ ) 2 ∪ π, 3π 2 and that the range of arccosecant is ( 0, π ] ( ] 2 ∪ π, 3π 2 when finding the exact value. 41. arcsec (−2) 42. arcsec ( − √ 2 ) ( ) 43. arcsec − 2√ 3 44. arcsec (−1) 3 45. arccsc (−2) 46. arccsc ( − √ 2 ) ( ) 47. arccsc − 2√ 3 3 48. arccsc (−1)
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College Trigonometry Version ⌊π
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Table of Contents vii 10 Foundation
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Preface Thank you for your interest
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xi text and we have gone to great l
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Chapter 10 Foundations of Trigonome
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10.1 Angles and their Measure 695 k
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10.1 Angles and their Measure 697 2
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10.1 Angles and their Measure 701 T
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10.1 Angles and their Measure 703 y
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10.1 Angles and their Measure 705 y
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10.1 Angles and their Measure 707 h
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10.1 Angles and their Measure 711 I
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10.2 The Unit Circle: Cosine and Si
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30 ◦ 1 10.2 The Unit Circle: Cosi
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10.3 The Six Circular Functions and
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10.4 Trigonometric Identities 771 f
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10.4 Trigonometric Identities 773 2
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10.4 Trigonometric Identities 779 T
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10.4 Trigonometric Identities 781 R
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10.4 Trigonometric Identities 785 I
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11.1 Applications of Sinusoids 891
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11.2 The Law of Sines 897 minimizes
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11.2 The Law of Sines 899 a angle-s
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11.2 The Law of Sines 901 determine
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11.2 The Law of Sines 903 Let x den
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11.2 The Law of Sines 905 24. Along
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11.2 The Law of Sines 907 33. The a
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11.2 The Law of Sines 909 25. (a)
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11.3 The Law of Cosines 911 of B ar
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11.3 The Law of Cosines 913 the pre
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11.3 The Law of Cosines 915 A 2 = =
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11.3 The Law of Cosines 917 22. A n
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11.4 Polar Coordinates 919 11.4 Pol
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11.4 Polar Coordinates 921 The poin
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11.4 Polar Coordinates 923 4. We mo
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11.4 Polar Coordinates 925 Solution
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11.4 Polar Coordinates 927 Solution
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11.4 Polar Coordinates 929 algebrai
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11.4 Polar Coordinates 931 45. ( )
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11.4 Polar Coordinates 933 5. ( 12,
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11.4 Polar Coordinates 935 13. (
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11.4 Polar Coordinates 937 73. r =7
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11.5 Graphs of Polar Equations 939
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11.6 Hooked on Conics Again 973 11.
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11.6 Hooked on Conics Again 975 Exa
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11.6 Hooked on Conics Again 977 The
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11.6 Hooked on Conics Again 979 We
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11.6 Hooked on Conics Again 983 The
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11.6 Hooked on Conics Again 985 In
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11.6 Hooked on Conics Again 989 2 9
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11.7 Polar Form of Complex Numbers
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11.8 Vectors 1013 Q ′ (1, 7) up 4
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11.8 Vectors 1015 ⃗v + ⃗w = 〈
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11.8 Vectors 1017 ⃗v − ⃗w =
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11.8 Vectors 1019 The remaining pro
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11.8 Vectors 1021 ‖k⃗v‖ = ‖
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11.8 Vectors 1023 at the origin, th
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11.8 Vectors 1025 Geometrically, th
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11.8 Vectors 1027 11.8.1 Exercises
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11.8 Vectors 1029 44. ⃗v = 〈−
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11.8 Vectors 1031 11.8.2 Answers 1.
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11.8 Vectors 1033 47. ‖⃗v‖ =2
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11.9 The Dot Product and Projection
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11.10 Parametric Equations 1049 obj
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11.10 Parametric Equations 1051 2.
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11.10 Parametric Equations 1053 Now
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11.10 Parametric Equations 1057 Our
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11.10 Parametric Equations 1061 Sup
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Index n th root of a complex number
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Index 1071 central angle, 701 chang
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Index 1073 reflective property, 523
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Index 1075 matrix, multiplicative,
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Index 1077 midpoint definition of,
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Index 1079 instantaneous, 161, 472
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Index 1081 inconsistent, 553 indepe
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