Prerequisite Skills BLM 1 1... Solve Equations 1. Solve. 2x + 5 = 11 x 5 + 6 = 7 x 2 = 225 d) x 2 = 24 2 + 32 2 e) 60 2 + x 2 = 61 2 f) 13 2 12 2 = x 2 The Pythagorean Theorem 2. Find the measure of the unknown side. 3. A post is to be temporarily braced until the cement that holds the post perpendicular to the ground can dry. The 10-ft support is to be attached 6 ft off the ground. Draw a diagram to model the problem. How far from the base of the post will the support be attached to the ground? Ratios 4. Express each ratio in lowest terms. 5:15 10:35 12:30 d) 52:14 6. A 450-g mixture of peanuts and cashews contains 150 g of cashews. Write the ratio of the mass of peanuts to the mass of cashews in lowest terms. Proportions 7. Solve for each unknown. x 5 = 6 27 15 30 = 9 x 11:3 = x:15 d) 7:2 = 42:y e) 12:b = 108:36 f) x = 27 = y 6 18 54 8. The ratio of chocolate chips to walnuts in a cookie recipe is 8:3. If the recipe calls for 24 g of chocolate chips, how many grams of walnuts would be needed? Rounding 9. Evaluate to one decimal place. 56 200 12 d) 417 Angle Sum of a Triangle 10. Determine the measures of the missing angles. 5. A mixture of oil to gas in a small engine is in the ratio of 1:7. Write each component as a ratio to the total mix. BLM 1 1 Prerequisite Skills
Section 1.1 Revisit the Primary Trigonometric Ratios BLM 1 3... 1. Name the sides that form the indicated angle. 5. Find the measure of side q to the nearest tenth of a metre. Find the measure of P. Find the measure of Q. 6. Solve the right triangle. 2. Evaluate. Round your answers to three decimal places. sin 45 cos 60 tan 30 3. Find the measure of each angle to the nearest hundredth of a degree. sin A = 0.8356 cos B = 0.2457 tan C = 1.8659 7. A ladder is resting against a tree. The foot of the ladder is 2 m from the base of the tree. The ladder forms a 48 angle with the ground. How far up the tree does the ladder reach, to the nearest tenth of a metre? 8. Find the measures of sides x and y to the nearest tenth of a metre. 4. Solve the right triangle. BLM 1 3 Section 1.1 Revisit the Primary Trigonometric Ratios
BLM 1 4... Section 1.2 Solve Problems Using Trigonometric Ratios 1. Explain each term in your own words. angle of elevation angle of depression 2. A 5-m ladder is resting against a wall. The base of the ladder is 2 m along the ground from the base of the wall. What angle does the base of the ladder make with the ground? Express your answer to the nearest tenth of a degree. 3. An 80-m tower is supported by a guy wire attached to the top of the tower. If the wire forms an angle of elevation of 79, how long is it? Express your answer to the nearest tenth of a metre. 4. Jason is flying his kite. He lets out 63 m of string and the wind takes his kite up to a point where the angle of elevation of the kite is 58. Find the altitude of the kite to the nearest metre. 5. The ancient Greek mathematician Talis used trigonometry to find the slant side length of the face of the Great Pyramid of Giza. An archeologist wants to replicate Talis's calculations. She measures the base length along one side of the pyramid to be 230 m and the angle of elevation of the side to be 52. Calculate the slant side length. 6. The highest point along a cliff is 80 m above the lakeshore. A surveyor stands on the top of the cliff, looking through a 1.5 m tall transit instrument. He spots a boat out on the lake, at an angle of depression of 38. How far, to the nearest tenth of a metre, is it from the boat to the base of the cliff? 7. Michael stands 10.0 m from the base of a building. He measures the angle of elevation to the top of the building to be 65.0. Michael s measurement was made from 1.5 m above the ground. Determine the height of the building to the nearest metre. 8. A search and rescue helicopter is flying at an altitude of 500 m. As it passes over a field, the pilot spots a campfire at an angle of depression of 23.5. If the helicopter were to land in the field directly below it, how far would the crew have to travel to reach the campfire? 9. Two buildings are 60 m apart. The angle of depression from the top of the taller building to the top of the shorter building is 15. The height of the shorter building is 30.4 m. What is the height of taller building? Express your answer to the nearest tenth of a metre. BLM 1 4 Section 1.2 Solve Problems Using Trigonometric Ratios
Section 1.3 The Sine Law BLM 1 7... (page 1) 1. State the sine law in two different forms. 2. Find the measure of C, to the nearest tenth of a degree. 4. Explain why the triangle cannot be solved using the sine law. 3. Find the measure of the indicated side, to the nearest tenth. 5. Solve each triangle. Round your answers to the nearest unit, if necessary. ABC, given B = 57, C = 72, and b = 18 m. DEF, given D = 66, F = 39, and e = 10 ft. GHI, given G = 72, g = 15 cm, and h = 8 cm. BLM 1 7 Section 1.3 The Sine Law
6. Find the perimeter of isosceles ABC, to the nearest inch. BLM 1 7... (page 2) 9. The longest side of a triangle is 33 ft. Find the lengths of the other two sides to the nearest foot. 7. Find the length of side AB to the nearest tenth of a metre. 10. 8. Two guy wires 27 m and 15 m in length are to be fastened to the top of a TV tower from two points B and C as shown. The angle of elevation to the top of the tower of the longer wire is 32. Use the sine ratio to find the value of x, to the nearest tenth. Use the sine law to find the value of x, to the nearest tenth. Explain why the two methods are equivalent for a right triangle. How far apart are points B and C? How tall is the tower? BLM 1 7 Section 1.3 The Sine Law
Section 1.4 The Cosine Law BLM 1 9... 1. Given ABC, write the cosine law for each side in the triangle. 2. Given ABC, write the cosine law for each angle in the triangle. 3. Find the measure of the marked angle. Express your answer to the nearest degree. 5. Solve ABC given A = 52, AC = 26. 2 cm, and AB = 18.8 cm. 6. Solve ABC given a = 9 cm, b = 7 cm, and c = 8 cm. 7. A radar station located at point A is tracking ships at points B and C. How far apart are the two ships, to the nearest tenth of a kilometre? 4. Find the measure of the unknown side, to the nearest tenth of a unit. 8. In parallelogram ABCD, the length of AB is 4 cm and the length of BC is 9 cm. If B is 54, how long is each diagonal, to the nearest tenth of a centimetre? BLM 1 9 Section 1.4 The Cosine Law
BLM 1 11... Section 1.5 Make Decisions Using Trigonometry 1. Choose the best formula to solve each triangle. 2. A ferry is used to transport guests from the dock to two hotels across a large lake. The hotels are located 550 m apart. The first hotel is at a 49 angle between the dock and the second hotel. The second hotel is at a 56 angle between the dock and the first hotel. How far is each hotel from the dock? 4. From one end of a bridge above a railroad track, the angle of depression to the tracks is 37. If that point is 112 m from the track and the bridge is 122 m long, how far from the other end of the bridge is the track, to the nearest metre? 5. A funnel used to pour oil into an engine is in the shape of a cone. The sides of the cone are 15 cm long and the angle between the sides is 17.9. What is the diameter of the cone? 6. Jesse is in a hot air balloon 6500 m above a lake. She measures the angle of depression to the far side of a lake to be 32 and the angle of depression to the near side of the lake to be 45. Determine the distance across the lake. 7. A hydro pole on the side of a hill casts a 36 m long shadow up the hill. The hill has a 13 angle of elevation to the horizontal and the sun has an angle of elevation of 43. How tall is the hydro pole? 3. Jayveer and Seema are standing 325 m apart, watching a hot air balloon above them. Jayveer measures the angle of elevation to the balloon to be 54. Seema measures the angle of elevation to the balloon to be 38. How far is each person from the balloon, to the nearest metre? What is the height of the balloon, to the nearest metre? BLM 1 11 Section 1.5 Make Decisions Using Trigonometry
Chapter 1 Review BLM 1 14... 1.1 Revisit the Primary Trigonometric Ratios, pages 6-15 1. Solve each right triangle. 8. In a triangle, the sides have lengths of 14 cm, 15 cm, and 16 cm. What are the angle measures, to the nearest tenth? 9. A surveyor in a canyon takes measurements and draws the diagram shown. Determine the length of a bridge that would stretch across the canyon. 2. Solve ABC given C = 90, a = 88 cm, and c = 117 cm. 1.2 Solve Problems Using Trigonometric Ratios, pages 16-23 3. A flagpole casts a shadow 17.7 m long when the angle of elevation of the sun is 66.4. How tall is the flagpole? 4. A boat is off course by 11 after travelling 27.8 km. How far off course is the boat? 1.3 The Sine Law, pages 24-33 5. What information do you need about a triangle to solve it using the sine law? 1.5 Make Decisions Using Trigonometry, pages 42-51 10. The Bermuda Triangle is an area off the coast of Miami, extending to the islands of Bermuda and Puerto Rico. The distance from Miami to Bermuda is 1680 km, from Bermuda to Puerto Rico is 1760 km, and from Puerto Rico to Miami is 1600 km. Find the measures of the angles of this triangle. 11. To create a dramatic lighting effect during a play, the lighting crew has installed three lights in the arrangement shown. How far apart are the Lights A and B? 6. Solve ABC given b = 17 ft, B = 62, and A = 34. 1.4 The Cosine Law, pages 34-41 7. In XYZ, X = 66, XY = 111 m, and XZ = 222 m. Find the length of YZ. BLM 1 14 Chapter 1 Review
Chapter 1 Practice Test BLM 1 15... (page 1) 1. Is each statement true (T) or false (F)? You can solve a triangle using the sine law if you are given the length of all three sides. The sine, cosine, and tangent ratios can only be used for right triangles. You can use the sine law or cosine law to solve a right triangle. d) An angle of elevation is always measured from the horizontal. 2. Which statement is true about right triangles? A The opposite side is always the shortest side in the triangle. B The adjacent side is always the shortest side. C The hypotenuse is always the longest side. D All angles are equal. 3. A 10-m long ladder is resting against a wall. The top of the ladder is 9 m above the ground. What angle does the ladder make with the ground? A 5.8 B 64.2 C 90 D 115.8 4. If a golfer uses a 64 wedge, he will send the ball into the air at an angle of 64. He is standing 5 yd in front of a 35 ft tree. Will he be able to hit the ball over the tree? 5. Solve the right triangle. 6. Solve the triangle. 7. Solve for the unknown in each triangle. BLM 1 15 Chapter 1 Practice Test
8. To temporarily support a wall during construction, the workers allow the wall to tilt towards them at an angle of 85 to the floor. If they support the wall with a piece of wood 8 ft up the wall and nailed 6 ft from the base of the wall, how long does this support need to be? 9. A new park is to be constructed with three climbing areas A, B, and C as shown. Determine the lengths of the other two sides in this triangle. 10. Two sailboats plan to meet at a mid-point. The boats are 10 km apart and are moving towards this point as shown in the diagram. How far will each boat need to travel to reach the meeting point? BLM 1 15... (page 2) 11. To measure the distance across a river, a surveyor took measurements and drew the diagram shown. Determine the distance from X to Y. 12. A surveyor uses a transit instrument that is 1.4 m off of the ground to measure the height of a hill that is along a planned path of a road. She measures the angle of elevation to the top of the hill to be 5.7. She knows that the horizontal distance to the top of the hill is 775 m. How high does the current road surface need to climb to reach the top of the hill? 13. From the top of her apartment building, Simone estimates that the angle of elevation to the top of the building across from her is 56. The angle of depression to the bottom of the same building is 32. If the two buildings are 25 m apart, what is the height of each building? BLM 1 15 Chapter 1 Practice Test
Chapter 1 Test BLM 1 16... 1. Is each statement true (T) or false (F)? You can solve a triangle using the cosine law if you are given the length of all three sides. The sine and tangent ratios can be used in all triangles. A right triangle cannot be solved using only the sine law. d) If you are given the measure of all three angles of a triangle, you can solve the triangle using the sine law. 2. Solve the right triangle. 5. Will a golfer using a wedge that sends the ball into the air at an angle of 64, be able to hit her ball over a 30 ft tree if she is 7 yd in front of the tree? 3. Solve the triangle. 4. Solve for the unknown in each triangle. 6. Two ski poles are resting against each other. The tips of ski poles are stuck into the ground 17 cm apart and the handles meet 110 cm above the ground. What angle do the ski poles form where they touch? 7. A coast guard station has received a distress call from a fishing boat 12 km away. There is a rescue boat 5 km away from the station. From the coast guard station, the angle between the two boats is 85. How far is the rescue boat from the fishing boat? BLM 1 16 Chapter 1 Test
BLM Answers BLM 1 18... (page 1) BLM 1-1 Prerequisite Skills 1. x = 3 x = 5 x = 15 d) x = ±40 e) x = ±11 f) x = ±5 2. 39 12 56 3. 8 ft 4. 1:3 2:7 2:5 d) 3:1 5. oil 1:8; gas 7:8 6. 2:1 7. x = 2 x = 10 x = 55 d) y = 12 e) b = 4 f) x = 9, y = 81 8. 9 g walnuts 9. 7.5 14.1 3.5 d) 20.4 10. a = 60º, b = 60º d = 35, e = 110 BLM 1-3 Section 1.1 Revisit the Primary Trigonometric Ratios 1. AB and AC DE and EF GI and GH 2. 0.707 0.5 0.577 3. A = 56.68 B = 75.78 C = 61.81 4. BC = 17 cm, AB = 20 cm, A = 60 5. 9.4 m 32 58 6. A = 73, BC = 13.1 m, AC = 13.7 m 7. 2.2 m 8. x = 11.9 m, y = 8.3 m BLM 1- Section 1.2 Solving Problems Using Trigonometric Ratios 1. The angle of elevation is the angle between the horizontal and the sight line from the observer's eye to some object above eye level. The angle of depression is the angle between the horizontal and the sight line from the observer s eye to a point below eye level. 2. 66.4 3. 81.5 m 4. 53 m 5. 187 m 6. 104.3 m 7. 23 m 8. 1150 m 9. 46.5 m BLM 1-8 Section 1.3 The Sine Law a b c sin A sin B sin C 1. = = or = = sin A sin B sin C a b c 2. 36.6 52.4 66.1 3. 20.8 ft 50.2 m 27.6 cm 4. At least one angle in the triangle is needed to solve using the sine law. 5. A = 51, a = 17 m, c = 20 m E = 75, d = 9 ft, f = 7 ft I = 78, H = 30, i = 15 cm 6. 47 in. 7. 52.6 m 8. 27.4 m 14.3 m 9. b = 28 ft, c = 20 ft 10., x = 10.4 Since sin 90 = 1, the two methods reduce to the 8 same equation: x = sin50. BLM 1-10 Section 1.4 The Cosine Law 1. a 2 = b 2 + c 2 2bc cos A, b 2 = a 2 + c 2 2ac cos B, c 2 = a 2 + b 2 2ab cos C 2 2 2 2 2 2 b + c a a + c b 2. cos A =, cos B =, 2bc 2ac 2 2 2 a + b c cos C =, 2ab 3. C = 39 D = 61 K = 77 4. b = 34.9 ft d = 13.9 ft y = 6.5 in 5. B = 83, C = 45, a = 20.8 cm 6. A = 73.4, B = 48.2, C = 58.4 7. 6.9 km 8. AC = 7.4 cm, BD = 11.8 cm BLM 1-12 Section 1.5 Make Decisions Using Trigonometry 1. cosine law sine law primary trigonometric ratios 2. first hotel: 472 m; second hotel: 430 m 3. Jayveer: 693 m; Seema: 191 m 151 m 4. 75 m 5. 21.8 cm 6. 3902 m 7. 24.6 m Chapter 1 Practice Masters Answers
BLM 1-15 Chapter Review 1. R = 25, QR = 15.4 ft, PQ = 6.5 ft XY = 14.0 m, Z = 36, X = 54 2. b = 77.1 cm, B = 41.2, A = 48.8 3. 40.5 m 4. 5.3 km 5. Two angle measures and one side measure, or an angle measure and two side measures, provided one of the sides is opposite the given angle. 6. C = 84, c = 19.1 ft, a = 10.8 ft 7. 204 m 8. 66.9, 53.6, 59.5 9. 118.4 m 10. Miami 64.8, Bermuda 55.3, Puerto Rico 59.9 11. 60.5 m BLM 1-16 Practice Test BLM 1 18... (page 2) 1. F T T d) T 2. C 3. B 4. no 5. A = 62.7, B = 27.3, AC = 103 yd 6. M = 57.8, L = 80.2, KM = 21.6 cm 7. a = 15.6 e = 7.3 cm I = 28.6 8. 9.6 ft 9. AC = 95 m, BC = 105 m 10. Boat 1: 7.7 km; Boat 2: 9.3 km 11. 285.5 m 12. 78.8 m 13. Simone's building: 13.2 m; other building: 33.9 m BLM 1-17 Chapter Test 1. T F F d) F 2. AC = 120 yd, B = 37, C = 53 3. XZ = 12.8 cm, Y = 53, Z = 51 4. a = 17 e = 7 cm Q = 37 5. yes 6. 8.9 7. 12.6 km Chapter 1 Practice Masters Answers