College Trigonometry, 2011a

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948 Applications of <strong>Trigonometry</strong> 5 r y π 4 π 2 3π 4 π θ x −5 As θ runs from π 2 to 3π 4 , r becomes negative and ranges from 0 to −5. Since r ≤ 0, the curve pulls away from the negative y-axis into Quadrant IV. 5 r y π 4 π 2 3π 4 π θ x −5 For 3π 4 ≤ θ ≤ π, r increases from −5 to 0, so the curve pulls back to the origin. 5 r y π 4 π 2 3π 4 π θ x −5
11.5 Graphs of Polar Equations 949 Even though we have finished with one complete cycle of r =5sin(2θ), if we continue plotting beyond θ = π, we find that the curve continues into the third quadrant! Below we present a graph of a second cycle of r =5sin(2θ) which continues on from the first. The boxed labels on the θ-axis correspond to the portions with matching labels on the curve in the xy-plane. 5 r 3 y 4 π 1 5π 2 3π 3 7π 4 4 2 4 2π θ 1 x −5 2 We have the final graph below. 5 r 5 y π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π θ −5 5 x −5 −5 r =5sin(2θ) intheθr-plane r =5sin(2θ) inthexy-plane 4. Graphing r 2 =16cos(2θ) is complicated by the r 2 , so we solve to get r = ± √ 16 cos(2θ) = ±4 √ cos(2θ). How do we sketch such a curve? First off, we sketch a fundamental period of r =cos(2θ) which we have dotted in the figure below. When cos(2θ) < 0, √ cos(2θ) is undefined, so we don’t have any values on the interval ( π 4 , 3π ) 4 . On the intervals which remain, cos(2θ) ranges from 0 to 1, inclusive. Hence, √ cos(2θ) ranges from 0 to 1 as well. 7 From this, we know r = ±4 √ cos(2θ) ranges continuously from 0 to ±4, respectively. Below we graph both r =4 √ cos(2θ) andr = −4 √ cos(2θ) ontheθr plane and use them to sketch the corresponding pieces of the curve r 2 =16cos(2θ) inthexy-plane. As we have seen in earlier 7 Owing to the relationship between y = x and y = √ x over [0, 1], we also know √ cos(2θ) ≥ cos(2θ) wherever the former is defined.
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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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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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11.7 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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