What should be labeled in the triangle? How do we remember the formulas? When Solving for a LEG or HYPOTENUSE of the right triangle, use When solving for one of the complementary ANGLES of the right triangle, use Sine Cosine Tangent Inverse Sine Inverse Cosine Inverse Tangent Solve for the indicated variable. 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
x = x = x = 34. 35. 36. x = x = x = 37. 38. x = x = Solve each missing side and angle using trigonometry functions. Round measures to the nearest tenth. 39. a = 16, b = 20, m<b = 40 40. a = 10, b = 15, c = 12 41. a = 42, c = 60, m<b = 58 42. m<a = 60, m<b = 72, c = 9 43. c = 15.6, a = 12.9, b = 18.4 44. m<a = 43, b 23, c = 26 Find each angle measure to the nearest degree.
45. sin B = 0.4848 46. cos A = 0.7431 47. tan W = 19.0811 ANGLE OF DEPRESSION ANLGE OF ELEVEATION Calculate each distance or angle. 48. A roller coaster makes an angle of 52 degrees with the ground. The horizontal distance from the crest of the hill to the bottom of the hill is about 121 feet, as shown. Find the height h of the roller coaster to the nearest foot. 49. The airplane door is 19 feet off the ground and the ram has a 31 degree angle of elevation. What is the length y of the ramp? 50. Find the horizontal distance h the bleachers cover. Round to the nearest foot. 51. A soccer ball is placed 10 feet away from the goal, which is 8 feet high. You kick the ball and it hits the crossbar along the top of the goal. What is the angle of elevation of your kick?
52. You are looking at an eye chart that is 20 feet away. Your eyes are level with the bottom of the E on the chart. To see the top of the E, you look up 1 degree. How tall is the E? 53. Your class is having a class picture taken on the lawn. The photographer is positioned 14 feet away from the center of the class. If she looks toward either end of the class, she turns 50 degrees. What is the distance between the ends of the class? 54. You are standing on a footbridge in a city park that is 12 feet high above a pond. You look down and see a duck in the water 7 feet away from the footbridge. What is the angle of depression? Explain your reasoning. 55. A surveyor is standing 118 feet from the base of the Washington Monument. The surveyor measures the angle between the ground and the top of the monument to be 78 degrees. Find the height h of the Washington Monument to the nearest foot. 56. You are standing on a plateau that is 800 feet above a basin where you can see two hikers. a. If the angle of depression from your line of sight to the hiker at B is 25 degrees, how far is the hiker from the base of the plateau? b. If the angle of depression from your line of sight to the hiker at C is 15 degrees, how far is the hiker from the base of the plateau?
c. How far apart are the two hikers? Explain. 57. You are flying a kite with 20 feet of string extended. The angle of elevation from the spool of string to the kite is 41 degrees. a. Draw and label a diagram to represent the situation. b. How far off the ground is the kite if you hold the spool 5 feet off the ground? Describe how the height were you hold the spool affects the height of the kite. 58. A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What angle does the ladder make with the ground? 59. A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. What angles does the cable form with the vertical tower? 60. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53 degrees. Find the height of the tree. 61. From the top of a lighthouse 210 feet high, the angle of depression to a boat is 27 degrees. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level. (Hint: Use transversal properties) 62. Richard is flying a kite. The kite string makes an angle of 57 degrees with the ground. If Richard is standing 100 feet from the point on the ground directly below the kite, find the length of the kite string. 63. An airplane rises vertically 1000 feet over a horizontal distance of 1 mine. What is the angle of elevation of the airplane s path? (Hint: 1mile = 5280 feet) 64. The angle of elevation from a ship to the top of a 42-meter lighthouse on the shore is 33 degrees. How far is the ship from the shore to the nearest meter?
65. A lighthouse 55 meters above sea level spots a distress signal from a sailboat. The angle of depression to the sailboat is 21 degrees. How far away is the sailboat from the base of the lighthouse to the nearest meter? 66. Ben is flying a kite with 125 meters of kite string out. His kite string makes an angle of 39 degrees with the level ground. How high is his kite to the nearest meter? 67. Chip Woodman, the foreman at the paper plant, must make an estimate of the volume of a conical wood chip pile. The distance from the tip of the cone down to the edge of the base (the slant height) is 304 feet and forms an angle of 54 degrees with the ground. What is the height of the cone to the nearest foot? What is the area of the base of the cone to the nearest thousand square feet? What is the volume of the cone to the nearest hundred thousand cubic feet? Use 3.14 forπ. 68. Ertha Biggs has uncovered the remains of a square-based Egyptian pyramid. The base is intact and measures 130 meters on a side. The top portion of the pyramid eroded away over the centuries, but what remains of each face of the pyramid forms an angle of 65 degrees with the ground (angle of ascent). What was the original height of the pyramid to the nearest meter? 69. A lighthouse is observed by a ship s officer on watch at an angle of 42 degrees to the path of a ship. At the next sighting the lighthouse is observed at an angle of 90 degrees to the path of the ship. The distance traveled between sidings is 1800 meters. To the nearest meter, how far away is the ship to the lighthouse at this second sighting? 70. It is believed that Galileo used the Leaning Tower of Pisa to conduct his experiments on the law of gravity. When he dropped objects from the top of the 55 meter tower (measured length of tower, not height), they landed 4.8 meters from the base of the tower. To the nearest degree, what is the angle that the tower leans off from the vertical? 71. One of the most impressive of the Mayan pyramids is El Castillo in the Yucatan of Mexico. The pyramid has a platform on the top and a flight of 91 steps on each of the four sides. (Four flights of 91 steps make 364 steps in all. With the top platform adding a level, there are 365 levels to represent the 365 days of the Mayan year.) What is the height to the nearest centimeter of the top platform if each of the 91 steps is 30 centimeters deep by 26 centimeters high? What is the angle of ascent to the nearest degree? 72. Sidney Yendis III is the son of NASA scientist and billionaire eccentric Sidney Yendis II. Eager to follow in his father s footsteps, young Sidney launched his sister s pet tribble into space with his homemade rocket, Albatross. He was attempting to set a world record for tribble flight. The record was 412 feet. He selected the backyard football field as his launch site. The launch was perfect. The Albatross soared straight up. Sidney s
father observed the launch through his telescope through the comfort of the announcer s booth 30 feet above the playing field. He measured the angle of depression to the launch-pad and found it to be 21 degrees. He measured the angle of elevation at the moment that the Albatross reached its maximum altitude and found it to be 78 degrees. Did his sister s pet tribble set a record? What was the height reached by the Albatross?