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

3 8. r = 1.5 The solutions of r = 1.5 are ordered pairs of the form ( 1.5, ), where is any real number. The graph consists of all points that are 1.5 units from the pole, so the graph is a circle centered at the origin with radius or 72. The length of each blade is 11.5 feet. So, r = 11.5 for each blade. The angle blade A makes with the polar axis is 3, therefore the tip of blade A can be represented by the polar coordinates (11.5, 3 ). The angle that blade B makes with the polar axis is or 75. Thus, the tip of blade B can be represented by the polar coordinates (11.5, 75 ). By adding 72 to 75, the polar coordinates for blade C can be found as (11.5, 147 ). By adding 72 to 147, the polar coordinates for blade D can be found as (11.5, 219 ). By adding 72 to 219, the polar coordinates for blade E can be found as (11.5, 291 ). 9. HELICOPTERS A helicopter rotor consists of five equally spaced blades. Each blade is 11.5 feet long. b. Blade B has the polar coordinates (11.5, 75 ) and blade C has the polar coordinates (11.5, 147 ). Use the Polar Distance Formula to find the distance between them. The distance between the tips of the two blades is approximately 13.5 feet. a. If the angle blade A makes with the polar axis is 3, write an ordered pair to represent the tip of each blade on a polar grid. Assume that the rotor is centered at the pole. b. What is the distance d between the tips of the helicopter blades to the nearest tenth of a foot? a. Sample answer: Sketch a diagram of the situation. Since the blades of the rotor create 5 angles, the angle between each pair of adjacent blades is 360 esolutions Manual - Powered by Cognero Page 3

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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