College Trigonometry, 2011a

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706 Foundations of Trigonometry 4. Since 117 is positive, the arc corresponding to t = 117 begins at (1, 0) and proceeds counterclockwise. As 117 is much greater than 2π, we wrap around the Unit Circle several times before finally reaching our endpoint. We approximate 117 2π as 18.62 which tells us we complete 18 revolutions counter-clockwise with 0.62, or just shy of 5 8 of a revolution to spare. In other words, the terminal side of the angle which measures 117 radians in standard position is just short of being midway through Quadrant III. y y 1 1 1 x 1 x t = −2 t = 117 10.1.1 Applications of Radian Measure: Circular Motion Now that we have paired angles with real numbers via radian measure, a whole world of applications awaits us. Our first excursion into this realm comes by way of circular motion. Suppose an object is moving as pictured below along a circular path of radius r from the point P to the point Q in an amount of time t. Q r θ s P Here s represents a displacement so that s>0 means the object is traveling in a counter-clockwise direction and s
10.1 Angles and their Measure 707 have v = displacement time = s length . The quantity v has units of t time and conveys two ideas: the direction in which the object is moving and how fast the position of the object is changing. The contribution of direction in the quantity v is either to make it positive (in the case of counter-clockwise motion) or negative (in the case of clockwise motion), so that the quantity |v| quantifies how fast the object is moving - it is the speed of the object. Measuring θ in radians we have θ = s r v = s t = rθ t = r · θ t thus s = rθ and The quantity θ is called the average angular velocity of the object. It is denoted by ω and is t read ‘omega-bar’. The quantity ω is the average rate of change of the angle θ with respect to time and thus has units radians time .Ifω is constant throughout the duration of the motion, then it can be shown 17 that the average velocities involved, namely v and ω, are the same as their instantaneous counterparts, v and ω, respectively. In this case, v is simply called the ‘velocity’ of the object and is the instantaneous rate of change of the position of the object with respect to time. 18 Similarly, ω is called the ‘angular velocity’ and is the instantaneous rate of change of the angle with respect to time. If the path of the object were ‘uncurled’ from a circle to form a line segment, then the velocity of the object on that line segment would be the same as the velocity on the circle. For this reason, the quantity v is often called the linear velocity of the object in order to distinguish it from the angular velocity, ω. Putting together the ideas of the previous paragraph, we get the following. Equation 10.2. Velocity for Circular Motion: For an object moving on a circular path of radius r with constant angular velocity ω, the (linear) velocity of the object is given by v = rω. We need to talk about units here. The units of v are length time ,theunitsofr are length only, and the units of ω are radians length time . Thus the left hand side of the equation v = rω has units time ,whereas the right hand side has units length · radians time = length·radians time . The supposed contradiction in units is resolved by remembering that radians are a dimensionless quantity and angles in radian measure are identified with real numbers so that the units length·radians time reduce to the units length time . We are long overdue for an example. Example 10.1.5. Assuming that the surface of the Earth is a sphere, any point on the Earth can be thought of as an object traveling on a circle which completes one revolution in (approximately) 24 hours. The path traced out by the point during this 24 hour period is the Latitude of that point. Lakeland Community College is at 41.628 ◦ north latitude, and it can be shown 19 that the radius of the earth at this Latitude is approximately 2960 miles. Find the linear velocity, in miles per hour, of Lakeland Community College as the world turns. Solution. To use the formula v = rω, we first need to compute the angular velocity ω. Theearth 2π radians π makes one revolution in 24 hours, and one revolution is 2π radians, so ω = 24 hours = 12 hours , 17 You guessed it, using Calculus . . . 18 See the discussion on Page 161 for more details on the idea of an ‘instantaneous’ rate of change. 19 We will discuss how we arrived at this approximation in Example 10.2.6.
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 and 14: 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
Page 21: 10.1 Angles and their Measure 705 y
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 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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10.7 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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11.7 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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College Trigonometry, 2011a

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