College Trigonometry, 2011a

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754 Foundations of <strong>Trigonometry</strong> object θ Example 10.3.5. ‘base line’ The angle of inclination from the base line to the object is θ 1. The angle of inclination from a point on the ground 30 feet away to the top of Lakeland’s Armington Clocktower 7 is 60 ◦ . Find the height of the Clocktower to the nearest foot. 2. In order to determine the height of a California Redwood tree, two sightings from the ground, one 200 feet directly behind the other, are made. If the angles of inclination were 45 ◦ and 30 ◦ , respectively, how tall is the tree to the nearest foot? Solution. 1. We can represent the problem situation using a right triangle as shown below. If we let h denote the height of the tower, then Theorem 10.10 gives tan (60 ◦ )= h 30 . Fromthisweget h = 30 tan (60 ◦ )=30 √ 3 ≈ 51.96. Hence, the Clocktower is approximately 52 feet tall. h ft. 60 ◦ 30 ft. Finding the height of the Clocktower 2. Sketching the problem situation below, we find ourselves with two unknowns: the height h of the tree and the distance x from the base of the tree to the first observation point. 7 Named in honor of Raymond Q. Armington, Lakeland’s Clocktower has been a part of campus since 1972.
10.3 The Six Circular Functions and Fundamental Identities 755 h ft. 30 ◦ 45 ◦ 200 ft. x ft. Finding the height of a California Redwood Using Theorem 10.10, we get a pair of equations: tan (45 ◦ )= h x and tan (30◦ )= x+200 .Since tan (45 ◦ ) = 1, the first equation gives h x =1,orx = h. Substituting this into the second √ h equation gives h+200 = tan (30◦ )= 3 3 . Clearing fractions, we get 3h =(h + 200)√ 3. The result is a linear equation for h, so we proceed to expand the right hand side and gather all the terms involving h to one side. Hence, the tree is approximately 273 feet tall. 3h = (h + 200) √ 3 3h = h √ 3 + 200 √ 3 3h − h √ 3 = 200 √ 3 (3 − √ 3)h = 200 √ 3 h = 200√ 3 3 − √ 3 ≈ 273.20 As we did in Section 10.2.1, we may consider all six circular functions as functions of real numbers. At this stage, there are three equivalent ways to define the functions sec(t), csc(t), tan(t) and cot(t) for real numbers t. First, we could go through the formality of the wrapping function on page 704 and define these functions as the appropriate ratios of x and y coordinates of points on the Unit Circle; second, we could define them by associating the real number t with the angle θ = t radians so that the value of the trigonometric function of t coincides with that of θ; lastly, we could simply define them using the Reciprocal and Quotient Identities as combinations of the functions f(t) =cos(t) andg(t) =sin(t). Presently, we adopt the last approach. We now set about determining the domains and ranges of the remaining four circular functions. Consider the function F (t) = sec(t) defined as F (t) = sec(t) = 1 cos(t) .WeknowF is undefined whenever cos(t) = 0. From Example 10.2.5 number 3, we know cos(t) = 0 whenever t = π 2 + πk for integers k. Hence, our domain for F (t) = sec(t), in set builder notation is {t : t ≠ π 2 + πk, for integers k}. Togetabetter understanding what set of real numbers we’re dealing with, it pays to write out and graph this set. Running through a few values of k, we find the domain to be {t : t ≠ ± π 2 , ± 3π 2 , ± 5π 2 , ...}. Graphing this set on the number line we get h
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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 699 4
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10.1 Angles and their Measure 701 T
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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 975 Exa
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11.6 Hooked on Conics Again 983 The
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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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