College Trigonometry, 2011a

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698 Foundations of <strong>Trigonometry</strong> angle, the direction of the rotation is important. We imagine the angle being swept out starting from an initial side and ending at a terminal side, as shown below. When the rotation is counter-clockwise 9 from initial side to terminal side, we say that the angle is positive; when the rotation is clockwise, we say that the angle is negative. Terminal Side Initial Side Terminal Side Initial Side A positive angle, 45 ◦ A negative angle, −45 ◦ At this point, we also extend our allowable rotations to include angles which encompass more than one revolution. For example, to sketch an angle with measure 450 ◦ we start with an initial side, rotate counter-clockwise one complete revolution (to take care of the ‘first’ 360 ◦ ) then continue with an additional 90 ◦ counter-clockwise rotation, as seen below. 450 ◦ To further connect angles with the Algebra which has come before, we shall often overlay an angle diagram on the coordinate plane. An angle is said to be in standard position if its vertex is the origin and its initial side coincides with the positive x-axis. Angles in standard position are classified according to where their terminal side lies. For instance, an angle in standard position whose terminal side lies in Quadrant I is called a ‘Quadrant I angle’. If the terminal side of an angle lies on one of the coordinate axes, it is called a quadrantal angle. Two angles in standard position are called coterminal if they share the same terminal side. 10 In the figure below, α = 120 ◦ and β = −240 ◦ are two coterminal Quadrant II angles drawn in standard position. Note that α = β + 360 ◦ , or equivalently, β = α − 360 ◦ . We leave it as an exercise to the reader to verify that coterminal angles always differ by a multiple of 360 ◦ . 11 More precisely, if α and β are coterminal angles, then β = α + 360 ◦ · k where k is an integer. 12 9 ‘widdershins’ 10 Note that by being in standard position they automatically share the same initial side which is the positive x-axis. 11 It is worth noting that all of the pathologies of Analytic <strong>Trigonometry</strong> result from this innocuous fact. 12 Recall that this means k =0, ±1, ±2,....
10.1 Angles and their Measure 699 4 3 2 1 y α = 120 ◦ −4 −3 −2 −1 1 2 3 4 −1 β = −240 ◦ −2 −3 −4 x Two coterminal angles, α = 120 ◦ and β = −240 ◦ ,instandardposition. Example 10.1.2. Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three coterminal angles, at least one of which is positive and one of which is negative. 1. α =60 ◦ 2. β = −225 ◦ 3. γ = 540 ◦ 4. φ = −750 ◦ Solution. 1. To graph α =60 ◦ , we draw an angle with its initial side on the positive x-axis and rotate counter-clockwise 60◦ 360 = 1 ◦ 6 of a revolution. We see that α is a Quadrant I angle. To find angles which are coterminal, we look for angles θ of the form θ = α + 360 ◦ · k, for some integer k. When k =1,wegetθ =60 ◦ +360 ◦ = 420 ◦ . Substituting k = −1givesθ =60 ◦ −360 ◦ = −300 ◦ . Finally, if we let k =2,wegetθ =60 ◦ + 720 ◦ = 780 ◦ . 2. Since β = −225 ◦ is negative, we start at the positive x-axis and rotate clockwise 225◦ 360 ◦ = 5 8 of a revolution. We see that β is a Quadrant II angle. To find coterminal angles, we proceed as before and compute θ = −225 ◦ + 360 ◦ · k for integer values of k. We find 135 ◦ , −585 ◦ and 495 ◦ are all coterminal with −225 ◦ . y y 4 4 3 3 2 1 α =60 ◦ 2 1 −4 −3 −2 −1 1 2 3 4 −1 x −4 −3 −2 −1 1 2 3 4 −1 x −2 −3 β = −225 ◦ −2 −3 −4 −4 α =60 ◦ in standard position. β = −225 ◦ in standard position.
Page 1 and 2: College Trigonometry Version ⌊π
Page 3 and 4: Table of Contents vii 10 Foundation
Page 5 and 6: Preface Thank you for your interest
Page 7 and 8: xi text and we have gone to great l
Page 9 and 10: Chapter 10 Foundations of Trigonome
Page 11 and 12: 10.1 Angles and their Measure 695 k
Page 13: 10.1 Angles and their Measure 697 2
Page 17 and 18: 10.1 Angles and their Measure 701 T
Page 19 and 20: 10.1 Angles and their Measure 703 y
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Page 23 and 24: 10.1 Angles and their Measure 707 h
Page 25 and 26: 10.1 Angles and their Measure 709 1
Page 27 and 28: 10.1 Angles and their Measure 711 I
Page 33 and 34: 10.2 The Unit Circle: Cosine and Si
Page 53 and 54: 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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10.5 Graphs of the Trigonometric Fu
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10.6 The Inverse Trigonometric Func
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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 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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