College Trigonometry, 2011a

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810 Foundations of <strong>Trigonometry</strong> 33. f(x) = 5√ 2 2 sin(x) − 5√ 2 x 2 cos(x) ) 34. f(x) =3sin( − 3 √ ( x ) 3cos 6 6 35. In Exercises 25 - 34, you should have noticed a relationship between the phases φ for the S(x) and C(x). Show that if f(x) =A sin(ωx + α) +B, thenf(x) =A cos(ωx + β) +B where β = α − π 2 . 36. Let φ be an angle measured in radians and let P (a, b) beapointontheterminalsideofφ when it is drawn in standard position. Use Theorem 10.3 and the sum identity for sine in Theorem 10.15 to show that f(x) =a sin(ωx)+b cos(ωx)+B (with ω>0) can be rewritten as f(x) = √ a 2 + b 2 sin(ωx + φ)+B. 37. With the help of your classmates, express the domains of the functions in Examples 10.5.4 and 10.5.5 using extended interval notation. (We will revisit this in Section 10.7.) In Exercises 38 - 43, verify the identity by graphing the right and left hand sides on a calculator. ( π ) 38. sin 2 (x)+cos 2 (x) = 1 39. sec 2 (x) − tan 2 (x) = 1 40. cos(x) =sin 2 − x ( x ) 41. tan(x + π) =tan(x) 42. sin(2x) =2sin(x) cos(x) 43. tan = sin(x) 2 1+cos(x) In Exercises 44 - 50, graph the function with the help of your calculator and discuss the given questions with your classmates. 44. f(x) = cos(3x)+sin(x). Is this function periodic? If so, what is the period? 45. f(x) = sin(x) x . What appears to be the horizontal asymptote of the graph? 46. f(x) =x sin(x). Graph y = ±x on the same set of axes and describe the behavior of f. 47. f(x) =sin ( 1 x) . What’s happening as x → 0? 48. f(x) =x − tan(x). Graph y = x on the same set of axes and describe the behavior of f. 49. f(x) =e −0.1x (cos(2x)+sin(2x)). Graph y = ±e −0.1x on the same set of axes and describe the behavior of f. 50. f(x) =e −0.1x (cos(2x)+2sin(x)). Graph y = ±e −0.1x on the same set of axes and describe the behavior of f. 51. Show that a constant function f is periodic by showing that f(x + 117) = f(x) for all real numbers x. Then show that f has no period by showing that you cannot find a smallest number p such that f(x + p) =f(x) for all real numbers x. Said another way, show that f(x + p) =f(x) for all real numbers x for ALL values of p>0, so no smallest value exists to satisfy the definition of ‘period’.
10.5 Graphs of the Trigonometric Functions 811 10.5.5 Answers 1. y =3sin(x) Period: 2π Amplitude: 3 Phase Shift: 0 Vertical Shift: 0 3 y π 2 π 3π 2 2π x −3 2. y =sin(3x) 2π Period: 3 Amplitude: 1 Phase Shift: 0 Vertical Shift: 0 1 y π 6 π 3 π 2 2π 3 x −1 3. y = −2 cos(x) Period: 2π Amplitude: 2 Phase Shift: 0 Vertical Shift: 0 2 y π 2 π 3π 2 2π x −2 ( 4. y =cos x − π ) 2 Period: 2π Amplitude: 1 π Phase Shift: 2 Vertical Shift: 0 1 y π 2 π 3π 2 2π 5π 2 x −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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10.7 Trigonometric Equations and In
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Chapter 11 Applications of Trigonom
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11.1 Applications of Sinusoids 883
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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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College Trigonometry, 2011a

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