Trigonometry. This booklet belongs to: Period. HW Mark: RE-Submit. Questions that I find difficult LESSON # DATE QUESTIONS FROM NOTES

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1 HW Mark: RE-Submit Trigonometry This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher has important instructions for you to write down below. Page 1 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

2 Trigonometry STRAND DAILY TOPIC EXAMPLE Measurement 4. Develop and apply the primary trigonometric ratios 4.1 Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios. (sine, cosine, tangent) to solve problems that involve 4.2 Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given acute angle in the triangle. right triangles. 4.3 Solve right triangles, with or without technology. 4.4 Solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem. 4.5 Solve a problem that involves indirect and direct measurement, using the trigonometric ratios, the Pythagorean theorem and measurement instruments such as a clinometer or metre stick. [C] Communication [PS] Problem Solving, [CN] Connections [R] Reasoning, [ME] Mental Mathematics [T] Technology, and Estimation, [V] Visualizatio Page 2 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

3 Key Terms Term Definition Example Triangle Similar Triangles Theta Acute angle Obtuse angle Right Triangle Oblique Triangle Legs Hypotenuse Trigonometry Opposite side Adjacent side Sine ratio Cosine ratio Tangent ratio Theta () Page 3 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

4 Why Trigonometry? There is an application of trigonometry that could solve each problem below. A student approaches a large Sequoia tree outside the entrance to the school and wonders how tall the tree is. A homeowner wants to cut a new board to replace a decaying roof truss. He can measure the horizontal distance and the angle of inclination but needs to know how long to cut the board. An engineer is constructing a Ferris wheel for a downtown park. There are 16 passenger carts. Page 4 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

5 Similar Triangles: To understand what trigonometry try is, we need to understand the properties of similar triangles. Two triangles are similar if They have the same angles. Ratios of corresponding sides are equal. Δ Determine if each of the following pairs of triangles are similar. Explain why or why or not Page 5 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

6 The Right Triangle A right triangle is a triangle with one right angle (90 ). The side opposite the right angle is called the hypotenuse. The other two sides are called legs. The sides of the right triangle form a Pythagorean Triple. That is, they satisfy the Pythagorean Theorem:. Some Pythagorean Triples: 3,4,5,5,12,13,7,24,25,8,15,17,9,40,41,11,60,61,, Find the indicated side length (nearest tenth) in the following right triangles Page 6 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

7 One acute angle is indicated on each of the following triangles. If possible, label each triangle with: opposite, adjacent, and hypotenuse in respect to that angle. Remember, only right triangles can be labeled this way Page 7 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

8 Observe the three embedded similar triangles below. Find the missing information on the right. 17. What are the names of the three triangles? ABC Based on the work you ve done, how do you know all three triangles are similar? 21. Earlier, we saw that similar triangles have the following similar ratios: (using the two smaller triangles embedded above) 1 2 In the questions above we see that we could write different equivalent ratios: [TANGENT RATIO] Write two more sets of equivalent ratios that would be true for the similar right triangles above [SINE RATIO] [COSINE RATIO] Page 8 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

9 Trigonometry of Right Triangles Since similar right triangles have equivalent ratios for corresponding angles, we can use those ratios to find unknown angles and/or side lengths. These ratios have been calculated and stored in our calculator for many angles to help us solve problems. We will use the three primary trigonometric ratios: Tangent ratio Sine ratio Cosine ratio The Tangent Ratio For an acute angle in a right triangle, the ratio of is called the TANGENT RATIO. We have seen previously that in similar right triangles, the ratios of the legs of a triangle remain constant despite reducing or enlarging the triangle. These ratios have been calculated and stored in our calculator for many angles to help us solve problems. 22. Challenge Question Find the tangent ratio for angle C in the triangle to the right. 23. Challenge Question What is for the triangle to the right? Page 9 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

10 The Tangent Ratio Remember, the tangent ratio is a ratio involving the legs of the right triangle. From the diagram we see that Tan C = The ratio of one leg to the other is 3:4 or Find the tangent ratio for the indicated angles below. Answer as a fraction AND as a decimal to 4 places. 24. Find the tangent ratio for 25. Find the tangent ratio angle C in the triangle for angle in the below. triangle below. 26. Find the tangent ratio for angle in the triangle below = = = Page 10 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

11 Find the tangent ratio for the indicated angles below. Answer as a fraction AND as a decimal to 4 places. 27. What is for the triangle below? 28. What is for the triangle below? 29. What is for the triangle below? = = Skill Reminder: Solve the following equations. Answer to the nearest hundredth if necessary Page 11 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

12 36. Challenge Question Given that the ratio of for in is , find the length of the missing leg. 37. Challenge Question Given that the tangent ratio of Angle F is , find the length of the missing leg. 38. Challenge Question Use your calculator to find tan30 (round to 4 decimal places). Use tan30 to find the length of the missing leg. Page 12 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

13 Finding Missing Sides Lengths Using the Tangent Ratio 39. Given that the ratio of 40. Given that the tangent for in is ratio of Angle F is , find the length of the missing leg , find the length of the missing leg. 41. Use your calculator to find tan30 (round to 4 decimal places). Use tan30 to find the length of the missing leg. 42. Draw a diagram illustrating the tangent ratio for in right. 43. The tangent ratio is a ratio of what two sides in a right triangle? 44. Can you use the tangent ratio to find the hypotenuse of a right triangle? Use the tangent ratio to find the missing side lengths to the nearest tenth. 45. The tangent ratio for the triangle below is Calculate x. 46. The tangent ratio for the triangle below is Calculate x. 47. The tangent ratio for the triangle below is Calculate x Page 13 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

14 Use your calculator to find the following ratios to 4 decimal places, then solve for x. 48. tan tan30 Find the length of the indicated side to the nearest tenth Page 14 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

15 Solving Triangles: To solve a triangle means to find the length of all unknown sides and measure of unknown angles. 56. Explain the steps you would take to solve the following triangle. Solve the following triangles. Answer to tenths The dotted line is an altitude. Page 15 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

16 Solve each of the following word problems. Include a diagram in your solution. 61. From a point 220 m from the Empire State 62. A radio tower is 396 feet tall. How far Building, a tourist measures the angle of from the base of the tower is a technician inclination to the top to be 60. Calculate if the angle of inclination to the top of the height of the building to the nearest the tower is 27? Answer to the nearest metre.. foot. 63. An airplane approaches a control tower. The angle of depression from the pilot to the tower is 12. If the plane is flying at an altitude of 1500 m, how far is the plane from being directly above the tower (to the nearest kilometer)? 64. At 11:00 in the morning, the angle of elevation to the sun 58. A tree in the school yard casts a shadow of 56 m. How tall is the tree to the nearest metre? 65. A student crossing to the west building casts a shadow on the path. She is 165 cm tall and the angle to the sun is 25. How long is the shadow on the path to the nearest centimetre? Page 16 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

17 The Sine Ratio The sine ratio is a ratio involving the hypotenuse and one leg of the right triangle. From the diagram to the right we see that sin C = The ratio of the opposite leg to the hypotenuse is 3:5 or Find the sine ratio for the indicated angles below. Answer as a fraction AND as a decimal to 4 places. 66. Find the sine ratio for 67. Find the sine ratio for angle in the triangle angle C in the triangle below. below. 68. Find the sine ratio for angle C in the triangle below = = = = 69. What is for the triangle below? 70. What is for the triangle below? 71. What is for the triangle below? = = = Page 17 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

18 Finding Missing Sides Lengths Using the Sine Ratio 72. Draw a diagram illustrating 73. The sine ratio is a the sine ratio for in ratio of what two right. sides in a right triangle? 74. Can you use the sine ratio to find the hypotenuse of a right triangle? Use the sine ratio to find the missing side lengths to the nearest tenth. 75. The sine ratio for the triangle below is Find the length of side x. 76. The sine ratio for the triangle below is Find the 77. The sine ratio for the triangle below is Find the length of side t. length of side y. 78. The sine ratio of a right triangle is If the hypotenuse is 20 cm long, what are the lengths of the other two sides? 79. The sine ratio of a right triangle is If the hypotenuse is 8 m long, what are the lengths of the other two sides? 80. The sine ratio for a right triangle is Find the opposite side if the hypotenuse is 4 mm. 81. The sine ratio for a right triangle is Find the hypotenuse if the opposite side is 17 cm. Page 18 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

19 82. Explain why the previous questions have no solutions. What do you notice about the value of ratio where this happens? Interpret that kind of ratio in the sense of a right triangle. Use your calculator to find the following ratios to 4 decimal places. 83. sin sin sin Find the length of side x. 87. Find the length of side x. 88. Find the length of side x. Find the length of the indicated side to the nearest tenth Page 19 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

20 Find the length of the indicated side to the nearest tenth A hiker loses track of her direction and wanders 22 degrees off course. If she continues to walk for 12 km to the river destination, how far away from her actual destination will she be? (nearest tenth of a kilometer) 96. A lighthouse attendant has a range of visibility of 24 km. A ship on the horizon passes by the lighthouse. The attendant sees the ship for a total of 120 degrees. For how many kilometers was the ship within the attendant s range of sight? (nearest tenth) Page 20 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

21 Draw a scale diagram that would represent each of the following. 97. Draw a triangle that has a the following:, hypotenuse is 10 cm long. 98. Draw a triangle that has a the following:, hypotenuse is 26 cm long. 99. Solve the triangle Solve the triangle Find the perimeter of the following rectangle Find the total area. Page 21 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

22 103. While golfing with his father-in-law, Mr. J hits a shot short of a pond. He walks 20 m to his left to a point directly across the pond from the hole. The angle between the two lines of sight is 22. Find the distance from his ball to the hole to the nearest tenth of a metre Find the area of the circle that is not covered by the shadedd triangles. Answer to the nearest tenth Anya lets out 125 feet of kite string at Clover Point. The wind pulls the kite string tight at an angle of 55 to the ground. Approximate the height of the kite to the nearest foot The radius of a circular tunnel in Shanghai is m. During a flood, a worker in the water at the side of the tunnel measured an angle to the centre to be 37. Find the depth of the water at its deepest point. (The water surface forms a chord across the tunnel.) What assumptions did you make? Page 22 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

23 The Cosine Ratio The cosine ratio is a ratio involving the hypotenuse and one leg (adjacent to angle) of the right triangle. From the diagram to the right we see that cos C = This means the ratio of the adjacent leg to the hypotenuse is 4:5 or Find the cosine ratio for the indicated angles below. Answer as a fraction AND as a decimal to 4 places Find the cosine ratio for 108. Find the cosine ratio for angle in the triangle below. angle C in the triangle below = = 109. Find the cosine ratio for angle C in the triangle below = 110. What is for the triangle below? 111. What is for the triangle below? 112. What is for the triangle below? = = = Page 23 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

24 Finding Missing Sides Lengths Using the Cosine Ratio 113. Draw a diagram illustrating the cosine ratio for in right The cosine ratio is a ratio of what two sides in a right triangle? 115. Can you use the cosine ratio to find the hypotenuse of a right triangle? Use the cosine ratio to find the missing side lengths to the nearest tenth The cosine ratio for the triangle below is Find x The cosine ratio for the triangle below is Find x The cosine ratio for angle Φ in a right triangle is Find the length of the hypotenuse if the side adjacent to Φ is 3.5 cm long The cosine ratio for the triangle below is Find x Draw a right triangle with a cosine ratio of Label all side lengths Explain why you cannot have a right triangle with a cosine ratio of Page 24 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

25 122. Draw a right triangle with an acute angle that has an adjacent side equal in length to the opposite side. Find the cosine ratio for that angle Draw a right triangle with an acute angle that has a hypotenuse 50% longer than the adjacent side. Find the cosine ratio for that angle Use a protractor to measure the indicated angle. Then determine the length of side x using the cosine ratio. x 125. Use a protractor to measure the indicated angle. Then determine the length of side x using the cosine ratio. Page 25 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

26 Use your calculator to find the following ratios to 4 decimal places, then find the indicated side length to 2 decimal places cos cos cos Find the length of side x Find the length of side x Find the length of side x. Find the length of the indicated side to the nearest tenth Find the length of each leg. Page 26 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

27 Solve the following problems involving triangles Find the area of the triangle below Find the area of a rectangle with a diagonal of 20 m if the angle between the diagonal and longer side is 25 degrees. (nearest tenth) 140. Tucker has two choices to get his ball to the hole at the 17 th at Cordova Bay, go around the lake or go over it. He decides to go around the lake as shown on the diagram. How much farther does he have to hit the ball going around the lake instead of going straight over it? (Careful, not a right triangle shown.) 141. Find the area of the circle that is not covered by the shaded rectangle to the nearest tenth of a square unit. Page 27 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

28 142. Find the area of the circle not covered by the shaded rectangle Find the area of the circle not covered by the shaded rectangle Find the length of AG to the nearest tenth of a millimetre Find the area of rectangle WXYZ to the nearest square unit. Page 28 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

29 Finding Angles Using the Three Ratios Recall: The three primary trig. ratios: Tangent Ratio: tan Unless otherwise stated, calculate the measure of angles to the nearest tenth of a degree. Eg Sine Ratio: Cosine Ratio: The stored values in your calculator allow you to find angles using the ratios. The magic of,, and. The inverse trigonometric functions. These functions convert the stored ratios in your calculator to the angle. Challenge 146. Find the measure of angle A in a right triangle if tana = Challenge 147. Find the measure angle B in a right triangle if sinb = Challenge 148. What ratio would you use to find the measure of the indicated angle? Challenge 149. Find the measure of the indicated angle. Page 29 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

30 Use the Inverse functions to find the indicated angle to the nearest tenth Find the measure angle A in a right triangle if tana= Use the tan -1 button. Type: tan -1 (1.000)=A A = 45 On your calculator, find the inverse functions. Usually 2 nd function required. These give us the angle from the ratio! 151. Find the measure angle B in a right triangle if sinb= Use the sin -1 button. Type: sin -1 (0.5000)=B B = What ratio would you use to find the measure of the indicated angle? Use the tangent ratio: Find the measure of the indicated angle. Type: tan -1 ( )= = Find angle A, if sina= Find angle B, if cosb= Find angle Q, if tanq= Find angle T, if sint= Find angle D, if cosd= Find angle U, if tanu= In a right triangle, one acute angle has sine ratio of 0.5. Find the sine ratio of the other acute angle In a right triangle, one acute angle has cosine ratio of Find the sine ratio of the other acute angle In a right triangle, one acute angle has cosine ratio of 0.5. Find the tangent ratio of the other acute angle Which of the three trigonometric ratios (sine, cosine, tangent) can have a value greater than 1? 163. Draw a right triangle and use it to explain your answer to the previous question. Page 30 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

31 Working with the ratios to find angles. Have a plan 1. Choose the correct ratio {sine, cosine, or tangent}. 2. Fill in the known side lengths into your chosen ratio. 3. Use the inverse trig. function to convert ratio angle What ratio do the given sides form for the indicated angle? 165. What ratio do the given sides form for the indicated angle? 166. What ratio do the given sides form for the indicated angle? Sine Cosine Tangent Sine Cosine Tangent Sine Cosine Tangent 167. Calculate the ratio above. Round to 4 decimal places Calculate the ratio above. Round to 4 decimal places Calculate the ratio above. Round to 4 decimal places Calculate the measure of angle to the nearest tenth of a degree Calculate the measure of angle to the nearest tenth of a degree Calculate the measure of angle to the nearest tenth of a degree. Page 31 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

32 Working with the ratios to find angles What ratio do the given 174. What ratio do the given sides form for the sides form for the indicated indicated angle? angle? 175. What ratio do the given sides form for the indicated angle? Sine Cosine Tangent Sine Cosine Tangent Sine Cosine Tangent 176. Calculate the ratio above. Round to 4 decimal places Calculate the ratio above. Round to 4 decimal places Calculate the ratio above. Round to 4 decimal places Calculate the measure of angle to the nearest tenth of a degree Calculate the measure of angle to the nearest tenth of a degree Calculate the measure of angle to the nearest tenth of a degree. Page 32 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

33 Find the measure of the indicated angle. Round answers to the nearest tenth of a degree Find the measure of angle x to the nearest tenth of a degree Find the measure of angle to the nearest degree. Page 33 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

34 Solve the following triangles. Calculate answers to the nearest tenth Page 34 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

35 Find the area of the following triangles to the nearest square unit Area = 194. Area = 1.2 Find base: Area = A triangle has side lengths of 8 cm, 7cm and 12 cm. Find the area of the triangle if the angle between the 8 cm and 12 cm side is A triangle has side lengths of 10 km, 23 km and 32 km. The angle opposite the 10 km side is 9.2 Find the area of the triangle. Page 35 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

36 Applications of trigonometry A kite stuck in a nearby tree. A child standing 25 m from the base of a tree pulls the string tight. If the tree is 30 m tall, approximately how far is the kite from the child to the nearest metre? 200. A surveyor measures the angle of elevation to the top of a building to be 23. If the surveyor is 1345 feet from the base of the building, how tall is the building to the nearest foot? 201. From the top of a 20 m cliff above a road, the angle of depression to two approaching cars is 25 and 40 respectively. How far apart are the cars to the nearest metre? 202. Two hot air balloons float above the ocean at a height of 1000 feet. From a sailboat, the angle of elevation to one balloon is 60 and to the other balloon is 50. How far apart are the balloons to the nearest foot? 203. Two boys on opposite sides of the tree below measure the angle of elevation to the top of the tree. If the tree is 175 feet tall, how many feet apart are the boys? 204. Highway sign shows that the road descends at a rate of 8%. Draw a diagram that shows what this means. If a 3 km section of straight road descends at this grade, what is the drop in elevation? Page 36 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

37 205. While golfing with his father-in-law, Mr. J hits a shot short of a pond. The flag (hole) is directly across the pond from his ball. He paces 20 m to the right of his ball and measures the angle back to the hole to be 76. How far is the ball from the hole to the nearest metre? 206. A hiker leaves base camp travelling due north at 5 km/h. After two hours, she turns and travels east. Three hours later, she sprains her ankle. At what bearing would a rescue team need to travel to reach the injured hiker? How far away is she from base camp? (nearest tenth) 207. A student approaches a large Sequoia tree outside the entrance to the school and wonders how tall the tree is. He paces 150 metres from the base of the tree and measures the angle of elevation to the top of the tree to be 35. Find the height of the tree to the nearest metre. Page 37 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

38 208. A homeowner wants to cut a new board to replace a decaying roof truss. He can measure the horizontal distance and the angle of inclination but needs to know how long to cut the board. The horizontal distance is 14 feet and the angle of inclination is 24. Find the distance to the nearest tenth of a foot An engineer is constructing a Ferris wheel for a downtown park. There are 16 passenger carts and the radius of the wheel is 10 metres. How far apart are the passenger carts to the nearest hundredth of a metre? 210. Find the area of the circle to the nearest square centimetre. [ ] 211. Find the perimeter of the octagon inscribed in a circle of radius 8 cm. (Nearest cm) Page 38 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

39 212. Find the length of the 25 line of latitude. The Earth s radius is 6380 km. Answer to the nearest km Find the length of the 45 line of latitude. The Earth s radius is 6380 km. Answer to the nearest km. 25 N 0 45 N Mr. Teespré s backyard slopes away from his house towards the beach. The instructions for his new lawnmower state that the mower should not be used if the slope is greater than 15. Being a trigonometry specialist, he extends a level string 125 feet from the base of his house. From that point, he measures that the distance along the ground back to his house is 130 m. Is his yard too steep for this mower? Page 39 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

40 215. A regular pentagon is inscribed in a circle of radius 10 cm. Calculate the perimeter of the pentagon. Answer to the nearest cm A regular decagon (10 sides) is inscribed inside a circle of radius 8 cm. Find the perimeter of the decagon. Answer to the nearest cm Find the area of the octagon inscribed in a circle of radius 8 cm. Answer to the nearest square cm A regular hexagon is inscribed in a circle with a radius 18 cm. What would be the side length of the hexagon? Answer to the nearest cm From a point 15 m from the base of a tree, a woman found the angle of inclination to the top of the tree to be 45. Her sister found the angle to be 18 from a point farther away from the base of the tree. How far away are the two women away from each other? (nearest tenth of a metre) Page 40 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

41 More word problems using right triangles: Draw a diagram. Fill in known values. Let a variable represent unknown(s) Choose an appropriate strategy to solve for the unknown(s). Interpret the problem Solve the triangle given the following A firefighter is walking along the river at point C when she spots two fires on the opposite river bank. She measures the angles below and paces a distance of 300 m from point C to point D. Point D is directly across the river from one of the fires. How far apart are the fires to the nearest metre? Page 41 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

42 222. Anya stands on top of a building in downtown Victoria. From her position, the angle of elevation to the top of an adjacent building is 47. The angle of depression to the base of the building is 62. She is told that the buildings are 45 m apart. Based on this information, what is the height of the taller building to the nearest metre? 223. Find the length of diagonal BG in the rectangular prism. Answer to the nearest tenth of a millimeter The line of sight from an inflatable boat to the top of an oil derrick is 24 degrees. If the derrick is 45 m tall, how far is the boat from its base? (nearest tenth) Page 42 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

43 225. A pilot on a level path knows she should descend at an angle of 3 degrees to maintain comfort and safety. If she is flying at an altitude of feet, how many miles from the runway should she begin her descent? 226. An aircraft ascends after takeoff at an angle of 22 degrees. What will be the altitude of the aircraft after it flies at that angle for 1200 m? (nearest metre) 227. A hamster scurries up a ramp at a speed of 1.5 m/s. The ramp is inclined at an angle of 18 degrees. How many metres above the ground will the hamster be after 30 seconds? 228. Anya travels down a zip line at 25 km/h. The angle of descent of the zip line is 11 degrees. How many vertical metres has she fallen after 3 minutes? 229. The Earth s radius is 6380 km. Find the length of the 35 latitude to the nearest 10 km. What assumptions did you make? 230. Find the angle of inclination at the back of the roof. The rise of the roof is 0.9 m. (nearest tenth) Page 43 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

44 231. A ladder should make an angle of 72 with the ground for maximum safety. If the ladder is 4 m long, how far should it reach up the wall? (nearest tenth) 232. The angle of elevation to the top of a tree, measured on a 1.5 m transit from a distance of 30 m, is 15. Find the height of the tree. (nearest tenth) 233. Find the value of x. 18 cm x cm 234. Mr. J has developed the ideal ice cream cone. The cone has a slant height of 13 cm and a diameter of 7.8 cm. Find the angle that the curved surface makes with the diameter Mr. J continues to work on his isolated surf hut. Below is two-thirds of a roof truss he wants to complete. Find the length of wood he must cut (nearest tenth) to complete the truss. The long side is 8.2 m and the short side is 6.8 m. The angle between them is 35. Page 44 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

45 236. Both triangles (large and smaller inset) are isosceles. Find the area of the shaded trapezoid to the nearest tenth of a square unit From a fire station in central BC, Georgia travels on a bearing of 37 at 6 km/h. Shelby leaves the station at the same time travelling due east at 5 km/h. How far apart are they after 4.5 hours? (Nearest tenth) 238. Find the measure of angle x to the nearest tenth of a degree At 9:00 am, a ship leaves port traveling at 30 km/h on a bearing of 63. At the same time, another ship leaves port on a bearing of 315 at a speed of 19 km/h. When the boats stop after two hours, how far east is the boat at point C? Page 45 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011

46 Draw an accurate diagram to answer each of the following questions In, 90, 12 and 10. Find the measure of 241. In, 90, 115 and 99. Find the measure of 242. In, 90, 12 and 30. Find the length of In, 90, 5 and 12. Find the length of. Page 46 Trigonometry Copyright Mathbeacon.com. Use with permission. Do not use after June 2011