SECTION 9.1 Polar Coordinates 609 In Problems 79–86, the letters r and θ represent polar coordinates. Write each equation using rectangular coordinates ( ) x y , . 79. θ = r cos 80. θ = + r sin 1 81. θ θ = + r 2 sin cos 82. θ θ = − r sin cos 83. = r 2 84. = r 4 85. θ = − r 4 1 cos 86. θ = − r 3 3 cos 87. Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive x -axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express θ in degrees. (c) Guaranteed Rate Field, home of the White Sox, is located at 35th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of Guaranteed Rate Field using rectangular coordinates. (d) Write the location of Guaranteed Rate Field using polar coordinates. Use the east direction for the polar axis. Express θ in degrees. 88. Show that the formula for the distance d between two points θ ( ) = P r , 1 1 1 and θ ( ) = P r , 2 2 2 is θ θ ( ) = + − − d r r r r 2 cos 1 2 2 2 1 2 2 1 Applications and Extensions Explaining Concepts 91. In converting from polar coordinates to rectangular coordinates, what equations will you use? 92. Explain how to convert from rectangular coordinates to polar coordinates. 93. Is the street system in your town based on a rectangular coordinate system, a polar coordinate system, or some other system? Explain. 94. Solve: ( ) ( ) + − − = x x log 3 log 1 2 4 4 95. Use Descartes’ Rule of Signs to determine the possible number of positive or negative real zeros for the function ( ) = − + − − f x x x x 2 6 7 8 3 2 96. Find the midpoint of the line segment connecting the points ( ) −3, 7 and ( ) 1 2 , 2 . 97. Given that the point ( ) 3, 8 is on the graph of ( ) = y f x , what is the corresponding point on the graph of ( ) = − + + y f x 2 3 5? Retain Your Knowledge Problems 94–103 are based on previously learned material.The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus.. City of Chicago, Illinois 1 mile 1 km N Madison Street 35th Street Addison Street State Street Addison Street 35th Street Wrigley Field 1060 West Addison Guaranteed Rate Field 35th and Princeton 89. Challenge Problem Radar Detection At 10:15 am, a radar station detects an aircraft at a point 80 miles away and 25 degrees north of due east.At 10:25 am, the aircraft is 110 miles away and 5 degrees south of due east. (a) Using the radar station as the pole and due east as the polar axis, write the two locations of the aircraft in polar coordinates. (b) Write the two locations of the aircraft in rectangular coordinates. Round answers to two decimal places. (c) What is the speed of the aircraft in miles per hour? Round the answer to one decimal place. 90. Challenge Problem Radar station A uses a coordinate system where A is located at the pole and due east is the polar axis. On this system, two other radar stations, B and C , are located at coordinates ( ) − ° 150, 24 and ( )° 100, 32 , respectively. If radar station B uses a coordinate system where B is located at the pole and due east is the polar axis, then what are the coordinates of radar stations A and C on this second system? Round answers to one decimal place.
RkJQdWJsaXNoZXIy NjM5ODQ=