851 8.7 Polar Equations and Graphs (a) Graph the orbits of the four closest satellites on the same polar grid. Choose a viewing window that results in a graph with nearly circular orbits. (b) Plot the orbits of Earth, Jupiter, Uranus, and Pluto on the same polar grid. How does Earth’s distance from the sun compare to the others’ distances from the sun? (c) Use graphing to determine whether or not Pluto is always farthest from the sun. 90. Radio Towers and Broadcasting Patterns Radio stations do not always broadcast in all directions with the same intensity. To avoid interference with an existing station to the north, a new station may be licensed to broadcast only east and west. To create an east-west signal, two radio towers are sometimes used. See the figure. Locations where the radio signal is received correspond to the interior of the curve r2 = 40,000 cos 2u, where the polar axis (or positive x-axis) points east. E N (a) Graph r2 = 40,000 cos 2u for 0° … u … 360°, where distances are in miles. Assuming the radio towers are located near the pole, use the graph to describe the regions where the signal can be received and where the signal cannot be received. (b) Suppose a radio signal pattern is given by the following equation. Graph this pattern and interpret the results. r2 = 22,500 sin 2u Relating Concepts For individual or collaborative investigation (Exercises 91 – 98) In rectangular coordinates, the graph of ax + by = c is a horizontal line if a = 0 or a vertical line if b = 0. Work Exercises 91 – 98 in order, to determine the general forms of polar equations for horizontal and vertical lines. 91. Begin with the equation y = k, whose graph is a horizontal line. Make a trigonometric substitution for y using r and u. 92. Solve the equation in Exercise 91 for r. 93. Rewrite the equation in Exercise 92 using the appropriate reciprocal function. 94. Sketch the graph of the equation r = 3 csc u. What is the corresponding rectangular equation? 95. Begin with the equation x = k, whose graph is a vertical line. Make a trigonometric substitution for x using r and u. 96. Solve the equation in Exercise 95 for r. 97. Rewrite the equation in Exercise 96 using the appropriate reciprocal function. 98. Sketch the graph of r = 3 sec u. What is the corresponding rectangular equation?
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