# Mid-Chapter Quiz: Lessons 9-1 through 9-3

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1 Graph each point on a polar grid. 1. A( 2, 45 ) 3. Because = 45, locate the terminal side of a 45 angle with the polar axis as its initial side. Because r = 2, plot a point 2 units from the pole in the opposite direction of the terminal side of the angle. Because =, locate the terminal side of a - angle with the polar axis as its initial side. Because r = 1.5, plot a point 1.5 units from the pole in the opposite direction of the terminal side of the angle. 2. D(1, 315 ) Because = 315, locate the terminal side of a angle with the polar axis as its initial side. Because r = 1, plot a point 1 unit from the pole along the terminal side of the angle. esolutions Manual - Powered by Cognero Page 1

2 4. 6. Because =, locate the terminal side of a The solutions of = are ordered pairs of the - angle with the polar axis as its initial side. form, where r is any real number. The graph consists of all points on the line that make an angle of with the positive polar axis. Because r = 3, plot a point 3 units from the pole along the terminal side of the angle. 7. = 60 Graph each polar equation. 5. r = 3 The solutions of r = 3 are ordered pairs of the form (3, ), where is any real number. The graph consists of all points that are 3 units from the pole, so the graph is a circle centered at the origin with radius 3. The solutions of = 60 are ordered pairs of the form (r, 60 ), where r is any real number. The graph consists of all points on the line that make an angle of 60 with the positive polar axis. esolutions Manual - Powered by Cognero Page 2

4 Graph each equation. 10. r = sec Make a table of values to find the r-values corresponding to various values of on the interval [0, 2π]. Round each r-value to the nearest tenth. r = π π 0.3 sec 11. r = cos Because the polar equation is a function of the cosine function, it is symmetric with respect to the polar axis. Therefore, make a table and calculate the values of r on [0, π]. r = π 0.3 cos Use these points and polar axis symmetry to graph the function. Graph the ordered pairs (r, ) and connect them with a line. esolutions Manual - Powered by Cognero Page 4

5 12. r = 3 csc Make a table of values to find the r-values corresponding to various values of on the interval [0, 2π]. Round each r-value to the nearest tenth. r = 3 csc 0 π r = 4 sin Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = 4 sin Use these points and symmetry with respect to the line = to graph the function. Graph the ordered pairs (r, ) and connect them with a line. 14. STAINED GLASS A rose window is a circular window seen in gothic architecture. The pattern of the window radiates from the center. The window shown can be approximated by the equation r = 3 sin 6. Use symmetry, zeros, and maximum r-values of the function to graph the function. Refer to the image on Page 560. esolutions Manual - Powered by Cognero Page 5

6 Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Sketch the graph of the rectangular function y = 3 sin 6x on the interval. From the graph, you Use these and a few additional points to sketch the graph of the function. can see that = 3 when and y = 0 when Interpreting these results in terms of the polar equation r = 3 sin 6, we can say that has a maximum value of 3 when and r = 0 when Since the function is symmetric with respect to the line =, make a table and calculate the values of r on. r = 3 sin esolutions Manual - Powered by Cognero Page 6

7 Identify and graph each classic curve. 15. r = sin The equation is of the form r = a sin, so its graph is a circle. Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = sin r = + 3, 0 The equation is of the form r = a + b, so its graph is a spiral of Archimedes. Use points on the interval [0, 2π] to sketch the graph of the function. r = π π 5.1 Use these points and symmetry with respect to the line = to graph the function. esolutions Manual - Powered by Cognero Page 7

8 17. r = cos The equation is of the form r = a + b cos, so its graph is a limacon. Since a < b, the graph with have an inner loop. Because this polar equation is a function of the cosine function, it is symmetric with respect to the polar axis. Therefore, make a table and calculate the values of r on. r = cos 0 3 π Use these points and polar axis symmetry to graph the function. 18. r = 5 sin 3 The equation is of the form r = a sin n, so its graph is a rose. Because this polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = 5 sin Use these points and symmetry with respect to the line = to graph the function. esolutions Manual - Powered by Cognero Page 8

9 19. MULTIPLE CHOICE Identify the polar graph of y 2 = x. 20. Find the rectangular coordinates for each point with the given polar coordinates. For, r = 4 and =. Write the rectangular equation y 2 = x in polar The rectangular coordinates of. are form. 21. For, r = 2 and =. Graph r = cos csc 2 using a graphing calculator. Let = and solve for r. The rectangular coordinates of. are The point corresponds to graph B. The correct answer is B. esolutions Manual - Powered by Cognero Page 9

10 22. ( 1, 210 ) For ( 1, 210 ), r = 1 and = 210. Find two pairs of polar coordinates for each point with the given rectangular coordinates if 0 2π. Round to the nearest hundredth. 24. ( 3, 5) For ( 3, 5), x = 3 and y = 5. Since x < 0, use to find. The rectangular coordinates of ( 1, 210 ) are. 23. (3, 30 ) For (3, 30 ), r = 3 and = 30. One set of polar coordinates is (5.83, 2.11). Another representation that uses a negative r-value is ( 5.83, π) or ( 5.83, 5.25). 25. (8, 1) The rectangular coordinates of (3, 30 ) are. For (8, 1), x = 8 and y = 1. Since x > 0, use to find. One set of polar coordinates is (8.06, 0.12). Another representation that uses a negative r-value is ( 8.06, π) or ( 8.06, 3.27). esolutions Manual - Powered by Cognero Page 10

11 26. (7, 6) Write a rectangular equation for each graph. For (7, 6), x = 7 and y = 6. Since x > 0, use to find. 28. One set of polar coordinates is (9.22, 0.71). Since this set is not in the required domain, two more sets have to be found. A representation that uses a positive r-value is (9.22, π) or (9.22, 5.57). A representation that uses a negative r-value is ( 9.22, π) or ( 9.22, 2.43). 27. ( 4, 10) For ( 4, 10), x = 4 and y = 10. Since x < 0, use tan 1 + π to find. 29. One set of polar coordinates is (10.77, 4.33). Another representation that uses a negative r-value is ( 10.77, 4.33 π) or ( 10.77, 1.19). esolutions Manual - Powered by Cognero Page 11

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