College Trigonometry, 2011a

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982 Applications of <strong>Trigonometry</strong> Using a polar coordinate representation P (r, θ) for a point on the conic with r>0, we get e = the distance from P to F the distance from P to L = r d + r cos(θ) so that r = e(d + r cos(θ)). Solving this equation for r, yields ed r = 1 − e cos(θ) At this point, we convert the equation r = e(d + r cos(θ)) back into a rectangular equation in the variables x and y. Ife>0, but e ≠ 1, the usual conversion process outlined in Section 11.4 gives 6 (( 1 − e 2 ) 2 e 2 d 2 ) ( x − e2 d 1 − e 2 ) 2 + ( 1 − e 2 e 2 d 2 ) y 2 =1 ( ) We leave it to the reader to show if 0 0. We could have just as easily chosen the directrix to be x = d, y = −d or y = d. As the reader can verify, in these cases ed we obtain the forms r = 1+e cos(θ) , r = ed 1−e sin(θ) and r = ed 1+e sin(θ) , respectively. The key thing to remember is that in any of these cases, the directrix is always perpendicular to the major axis of an ellipse and it is always perpendicular to the transverse axis of the hyperbola. For parabolas, knowing the focus is (0, 0) and the directrix also tells us which way the parabola opens. We have established the following theorem. 6 Turn r = e(d + r cos(θ)) into r = e(d + x) and square both sides to get r 2 = e 2 (d + x) 2 .Replacer 2 with x 2 + y 2 , expand (d + x) 2 , combine like terms, complete the square on x and clean things up. 7 Since e>1inthiscase,1− e 2 < 0. Hence, we rewrite ( 1 − e 2)2 = ( e 2 − 1 )2 to help simplify things later on.
11.6 Hooked on Conics Again 983 Theorem 11.12. Suppose e and d are positive numbers. Then ˆ the graph of r = ˆ the graph of r = ˆ the graph of r = ˆ the graph of r = ed 1−e cos(θ) ed 1+e cos(θ) ed 1−e sin(θ) ed 1+e sin(θ) is the graph of a conic section with directrix x = −d. is the graph of a conic section with directrix x = d. is the graph of a conic section with directrix y = −d. is the graph of a conic section with directrix y = d. In each case above, (0, 0) is a focus of the conic and the number e is the eccentricity of the conic. ˆ If 0
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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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11.2 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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11.8 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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