College Trigonometry, 2011a

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924 Applications of <strong>Trigonometry</strong> values of θ. Now let’s assume that neither r nor r ′ is zero. If (r, θ) and(r ′ ,θ ′ ) determine the same point P then the (non-zero) distance from P tothepoleineachcasemustbethesame.Sincethis distance is controlled by the first coordinate, we have that either r ′ = r or r ′ = −r. Ifr ′ = r, then when plotting (r, θ) and(r ′ ,θ ′ ), the angles θ and θ ′ have the same initial side. Hence, if (r, θ) and (r ′ ,θ ′ )determinethesamepoint,wemusthavethatθ ′ is coterminal with θ. We know that this means θ ′ = θ +2πk for some integer k, as required. If, on the other hand, r ′ = −r, then when plotting (r, θ) and(r ′ ,θ ′ ), the initial side of θ ′ is rotated π radians away from the initial side of θ. In this case, θ ′ must be coterminal with π + θ. Hence, θ ′ = π + θ +2πk which we rewrite as θ ′ = θ +(2k +1)π for some integer k. Conversely, if r ′ = r and θ ′ = θ +2πk for some integer k, then the points P (r, θ) andP ′ (r ′ ,θ ′ ) lie the same (directed) distance from the pole on the terminal sides of coterminal angles, and hence are the same point. Now suppose r ′ = −r and θ ′ = θ +(2k +1)π for some integer k. To plot P , we first move a directed distance r from the pole; to plot P ′ ,our first step is to move the same distance from the pole as P , but in the opposite direction. At this intermediate stage, we have two points equidistant from the pole rotated exactly π radians apart. Since θ ′ = θ +(2k +1)π =(θ + π)+2πk for some integer k, we see that θ ′ is coterminal to (θ + π) anditisthisextraπ radians of rotation which aligns the points P and P ′ . Next, we marry the polar coordinate system with the Cartesian (rectangular) coordinate system. To do so, we identify the pole and polar axis in the polar system to the origin and positive x-axis, respectively, in the rectangular system. We get the following result. Theorem 11.7. Conversion Between Rectangular and Polar Coordinates: Suppose P is represented in rectangular coordinates as (x, y) and in polar coordinates as (r, θ). Then ˆ x = r cos(θ) andy = r sin(θ) ˆ x 2 + y 2 = r 2 and tan(θ) = y x (provided x ≠0) In the case r>0, Theorem 11.7 is an immediate consequence of Theorem 10.3 along with the quotient identity tan(θ) = sin(θ) cos(θ) . If r
11.4 Polar Coordinates 925 Solution. 1. Even though we are not explicitly told to do so, we can avoid many common mistakes by taking the time to plot the points before we do any calculations. Plotting P ( 2, −2 √ 3 ) shows that it lies in Quadrant IV. With x = 2 and y = −2 √ 3, we get r 2 = x 2 + y 2 =(2) 2 + ( −2 √ 3 ) 2 = 4 + 12 = 16 so r = ±4. Since we are asked for r ≥ 0, we choose r = 4. To find θ, wehave that tan(θ) = y x = −2√ 3 2 = − √ 3. This tells us θ has a reference angle of π 3 , and since P lies in Quadrant IV, we know θ is a Quadrant IV angle. We are asked to have 0 ≤ θ
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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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11.6 Hooked on Conics Again 975 Exa
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11.6 Hooked on Conics Again 981 1.
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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 987 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 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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