College Trigonometry, 2011a

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902 Applications of <strong>Trigonometry</strong> tell us what kind and how many triangles to expect in each case. If a = h, thena = c sin(α) and the Law of Sines gives sin(α) a = sin(γ) c so that sin(γ) = c sin(α) a = a a = 1. Here, γ =90◦ as required. Moving along, now suppose h
11.2 The Law of Sines 903 Let x denote the distance from the second observation point Q to the point C and let y denote the distance from C to the island. ( Using Theorem 10.4, we get sin (45 ◦ )= y d . After some rearranging, √2 ) we find y = d sin (45 ◦ ) ≈ 9.66 2 ≈ 6.83 miles. Hence, the island is approximately 6.83 miles from the coast. To find the distance from Q to C, wenotethatβ = 180 ◦ − 90 ◦ − 45 ◦ =45 ◦ so by symmetry, 12 we get x = y ≈ 6.83 miles. Hence, the point on the shore closest to the island is approximately 6.83 miles down the coast from the second observation point. Sasquatch Island Sasquatch Island P 30 ◦ γ Q 45 ◦ β d ≈ 9.66 miles Shoreline β d ≈ 9.66 miles 45 ◦ Q C y miles 5 miles x miles We close this section with a new formula to compute the area enclosed by a triangle. Its proof uses the same cases and diagrams as the proof of the Law of Sines and is left as an exercise. Theorem 11.4. Suppose (α, a), (β,b) and(γ,c) are the angle-side opposite pairs of a triangle. Then the area A enclosed by the triangle is given by A = 1 2 bc sin(α) =1 2 ac sin(β) =1 ab sin(γ) 2 Example 11.2.4. Find the area of the triangle in Example 11.2.2 number 1. Solution. From our work in Example 11.2.2 number 1, wehaveallthreeanglesandallthreesides to work with. However, to minimize propagated error, we choose A = 1 2ac sin(β) fromTheorem 11.4 because it uses the most pieces of given information. We are ( given a ) = 7 and β =45 ◦ ,andwe calculated c = 7 sin(15◦ ) sin(120 ◦ ) . Using these values, we find A = 1 2 (7) 7 sin(15 ◦ ) sin(120 ◦ ) sin (45 ◦ )=≈ 5.18 square units. The reader is encouraged to check this answer against the results obtained using the other formulas in Theorem 11.4. 12 Or by Theorem 10.4 again . . .
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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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11.5 Graphs of Polar Equations 953
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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 977 The
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11.6 Hooked on Conics Again 979 We
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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 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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