850 CHAPTER 8 Applications of Trigonometry 80. Based on the results in Exercise 79, fill in the blank(s) to correctly complete each sentence. (a) The graph of r = ƒ1u2 is symmetric with respect to the polar axis if substitution of for u leads to an equivalent equation. (b) The graph of r = ƒ1u2 is symmetric with respect to the vertical line u = p 2 if substitution of for u leads to an equivalent equation. (c) Alternatively, the graph of r = ƒ1u2 is symmetric with respect to the vertical line u = p 2 if substitution of for r and for u leads to an equivalent equation. (d) The graph of r = ƒ1u2 is symmetric with respect to the pole if substitution of for r leads to an equivalent equation. (e) Alternatively, the graph of r = ƒ1u2 is symmetric with respect to the pole if substitution of for u leads to an equivalent equation. (f) In general, the completed statements in parts (a)–(e) mean that the graphs of polar equations of the form r = a{bcos u (where a may be 0) are symmetric with respect to . (g) In general, the completed statements in parts (a)–(e) mean that the graphs of polar equations of the form r = a{bsinu (where a may be 0) are symmetric with respect to . (Modeling) Solve each problem. 89. Orbits of Satellites The polar equation r = a11 - e 22 1 + e cos u can be used to graph the orbits of the satellites of our sun, where a is the average distance in astronomical units from the sun and e is a constant called the eccentricity. The sun will be located at the pole. The table lists the values of a and e. Satellite a e Mercury 0.39 0.206 Venus 0.78 0.007 Earth 1.00 0.017 Mars 1.52 0.093 Jupiter 5.20 0.048 Saturn 9.54 0.056 Uranus 19.20 0.047 Neptune 30.10 0.009 Pluto 39.40 0.249 Data from Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners (Editors), Fundamental Astronomy, 4th edition, Springer-Verlag; Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Saunders College Publishers. Spirals of Archimedes The graph of r = au in polar coordinates is an example of a spiral of Archimedes. With a calculator set to radian mode, use the given value of a and interval of u to graph the spiral in the window specified. 81. a = 1, 0 … u … 4p, 3-15, 154 by 3-15, 154 82. a = 2, -4p … u … 4p, 3-30, 304 by 3-30, 304 83. a = 1.5, -4p … u … 4p, 3-20, 204 by 3-20, 204 84. a = -1, 0 … u … 12p, 3-40, 404 by 3-40, 404 Intersection of Polar Curves Find the polar coordinates of the points of intersection of the given curves for the specified interval of u. 85. r = 4 sinu, r = 1 + 2 sinu; 0 … u 62p 86. r = 3, r = 2 + 2 cos u; 0° … u 6360° 87. r = 2 + sinu, r = 2 + cos u; 0 … u 62p 88. r = sin2u, r = 22 cos u; 0 … u 6p
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