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rt:VU1ve 10 • .JOV ~.J 1.2 ~ = I • •• UV 111L, secuuus. Exercises 1. Explain the difference between a segment and a ray. 3. Concept Check What angle is its own complement? 2. What part of a complete revolution is an angle of 45°? 4. Concept Check What angle is its own supplement? Find the measure of the smaller angle formed by the hands of a clock at the following times. ~ & Find the measure of each angle in Exercises 7-12. See Example 1. ~ \ ( (7rl ---- & (llxt ) 9. 22 The Trigonometric Functions Chapter1 + 10. supplementary angles with measures 10m 11. supplementary angles with measures 6x - 4 and 8x - 12. complementary angles with measures 9z Concept Check Use the concepts presented tion to answer each question. 13. If an angle measures x O , + 7 and 7m + 3 degrees 12 degrees 6 and 3z degrees 15. If a positive angle has measure in this sec- how can we represent its 16. If a negative angle has measure X between 0° and - 60°, how can we represent the first positive angle complement? O 14. If an angle measures x O , how can we represent its coterminal supplement? Perform each calculation. 17.62° 18' with it? See Example 2. + 21° 41' 18. 75° 15' + 83° 32' 19.71° 18' - 47° 29' 22. 180° - 124° 51' 20. 47° 23' - 73° 48' 21. 90° - 51° 28' 23. 90° - 72° 58' II" 24. 90° - 36° 18' 47" Convert each angle measure Example 3. to decimal degrees. Use a calculator, and round to the nearest thousandth 25. 20° 54' 26. 38° 42' 27. 91° 35' 54" 28. 34° 51' 35" 29. 274° 18' 59" 30. 165° 51' 9" Convert each angle measure to degrees, -minutes, and seconds. Use a calculator as necessary. 32. 59.0854° 33. 89.9004° 34. 102.3771° 35. 178.5994° 36. 122.6853° about the degree symbol (0) in the manual for your graphing calculator. How is it used? Find the angle of smallest positive measure coterminal ~38. of a degree. See See Example 3. 31. 31.4296° ru 37. Read ~ between 0° and 60°, the first negative angle cotermi- how can we represent nal with it? XO Show that 1.21 hours is the same as 1 hour, 12 minutes, and 36 seconds. Discuss the similarity between converting hours, minutes, and seconds to decimal hours and converting degrees, minutes, and seconds to decimal degrees. with each angle. See Example 4. ~ 39. -40° 40. -98° 41. -125° 42. -203° 'C'> 43. 539° 44. 699° 45. 850° 46. 1000° \.() :::l' Give an expression -...J 47. 30° that generates all angles coterminal 48. 45° with each angle. Let n represent any integer. 49. 135° 50. 270° 51. -90° ~ 53. Explain why the answers to Exercises 50 and 51 give the same set of angles. 54. Concept Check A. 360° + r" Which two of the following B. r" - 360° are not coterminal C. 360° - r" ru Consider the function Y1 = 360((Xj360) - int(Xj360)) specified on a graphing calculator. (Note: The value of int(x) is the largest integer less than or equal to x. With some calculators, int is found in the MATH menu.} The screen here shows that for X = 908 and X = -75, the junction returns the smallest possible positive measure coterminal with the angle. See Example 4. Use Y1 to do the following. 55. Rework Exercise 39 with a graphing calculator. 56. Rework Exercise 40 with a graphing calculator. D. r" with rO? + 180° V1(908) V1 (-75) 188 285 52. -135° 1.2 Angles 23 Concept Check Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle. 57. 75° 5S. 89° 59. 174° 60. 234° 61. 300° 62. 512° 63. -61° 64. -159° Concept Check Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having smallest positive measure. Then find the distance rfrom the origin to the point, using the distance formula of Section 1.1. 65. (-3, -3) 6S. 66. (-5,2) (\13,1) 69. (-2,2\13) Solve each problem. See Example 5. 71. Revolutions of a Windmill A windmill makes 90 revolutions per minute. How many revolutions does it make per second? 72. Revolutions of a Turntable A turntable in a shop makes 45 revolutions per minute. How many revolutions does it make per second? 73. Rotating Tire A tire is rotating 600 times per minute. Through how many degrees does a point on the edge of the tire move in 1/2 second? 67. (-3, -5) 70. (4\13, -4) grees that a point on the edge of the propeller will rotate in I second. 75. Rotating Pulley A pulley rotates through 75° in one minute. How many rotations does the pulley make in an hour? 76. Surveying One student in a surveying class measures an angle as 74.25°, while another student measures the same angle as 74° 20'. Find the difference between these measurements, both to the nearest minute and to the nearest hundredth of a degree. 74. Rotating Airplane Propeller An airplane-propeller rotates 1000 times per minute. Find the number of deAngle of a Star Refer to thefigure and table given in the chapter introduction. For each star in Exercises 77 and 78, find the measure of the other acute angle in thefigure, 90° - e, using the values from the table. 77. Alpha Centauri 79. 7S.61 Cygni Viewing Field of a Telescope Due to Earth's rotation, celestial objects like the moon and the stars appear to move across the sky, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 minute in a direction perpendicular to Earth's axis. SO. Angle Measure of a Star on the American Flag Determine the measure of the angle in each point of the five-pointed star appearing on the American flag. (Hint: Inscribe the star in a circle, and use the following theorem from geometry: An angle whose vertex lies on the circumference of a circle is equal to half the central angle that cuts off the same arc. See the figure.)