626 CHAPTER 9 Polar Coordinates; Vectors In Problems 67–72, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. 67. θ θ = = r r 8cos ; 2 sec 68. θ θ = = r r 8 sin ; 4 csc 69. r r sin ; 1 cos θ θ = = + 70. θ = = + r r 3; 2 2 cos 71. r r 1 sin ; 1 cos θ θ = + = + 72. θ θ = + = r r 1 cos ; 3cos In Problems 73–82, graph each polar equation. 73. r parabola 2 1 cosθ ( ) = − 74. r hyperbola 2 1 2cosθ ( ) = − 75. r ellipse 1 3 2cosθ ( ) = − 76. r parabola 1 1 cosθ ( ) = − 77. r , 0 θ θ = ≥ (spiral of Archimedes) 78. r 3 θ = (reciprocal spiral) 79. r csc 2, 0 θ θ π = − < < (conchoid) 80. r sin tan θ θ = (cissoid) 81. θ π θ π ( ) = − < < r kappa curve tan , 2 2 82. r cos 2 θ = 83. Show that the graph of the equation θ = r a sin is a horizontal line a units above the pole if a 0 ≥ and a units below the pole if a 0. < 84. Show that the graph of the equation θ = r a cos is a vertical line a units to the right of the pole if a 0 ≥ and a units to the left of the pole if a 0. < 85. Show that the graph of the equation r a a 2 sin , 0, θ = > is a circle of radius a with center a 0, ( ) in rectangular coordinates. 86. Show that the graph of the equation r a a 2 sin , 0, θ = − > is a circle of radius a with center a 0, ( ) − in rectangular coordinates. 87. Show that the graph of the equation r a a 2 cos , 0, θ = > is a circle of radius a with center a, 0 ( ) in rectangular coordinates. 88. Show that the graph of the equation r a a 2 cos , 0, θ = − > is a circle of radius a with center a, 0 ( ) − in rectangular coordinates. 89. Sailing Polar plots provide attainable speeds of a specific sailboat sailing at different angles to a wind of given speed. See the figure to the right. Use the plot to approximate the attainable speed of the sailboat for the given conditions. (a) Sailing at a 140° angle to a 6-knot wind. (b) Sailing at a 160° angle to a 10-knot wind. (c) Sailing at a 80° angle to a 20-knot wind. (d) If the wind blows at 20 knots, for what angles will the sailboat attain a speed of 10 knots or faster? (e) If the wind blows at 10 knots, what is the maximum attainable speed of the sailboat? For what angle(s) does this speed occur? Source: myhanse.com 5 6 7 8 9 10 11 Boat Speed (knots) 20° 30° 40° 50° 60° 70° 60° 50° 40° 30° 10° Wind Speed 6 knots 10 knots 20 knots Wind Direction 0 1 2 3 4 12 70° 80° 90° 100° 110° 120° 130° 140° 150° 170° 160° 170° 180° 160° 150° 140° 130° 120° 110° 100° 90° 80° 20° 10° 90. Challenge Problem Cardioid Microphones Cardioid microphones are often used to suppress background noise. Podcasters as well as sound engineers use them when recording live performances on a stage. In a live setting, a cardioid pickup pattern suppresses noise from the audience. Suppose the microphone is placed at the front of the stage (as in the figure) and the boundary of the optimal pickup region is given by the cardioid θ = + r 8 8sin , where r is measured in meters and the microphone is at the pole. An audio engineer wants to place a large speaker near the front of the stage but does not want it to be inside the optimal pickup range (shaded area in the figure); otherwise, the speaker will cause feedback if too close to the microphone. If the speaker is 10 meters to the right of the microphone (when looking at the stage) and 6 meters back from the edge, will the speaker be outside the optimal pickup range? Stage Audience Speaker Microphone 10 6 91. Challenge Problem Express r cos 2 2 θ ( ) = in rectangular coordinates free of radicals. 92. Challenge Problem Prove that the area of the triangle with vertices 0, 0 , ( ) r , , 1 1θ ( ) and r , , 0 , 2 2 1 2 θ θ θ π ( ) ≤ < ≤ is θ θ ( ) = − K r r 1 2 sin 1 2 2 1
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