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MBF3C Unit 3 - Foundations for College Mathematics Lesson Eleven: Trigonometry Ratios Label parts of a right triangle i.e. opposite, adjacent and hypotenuse Recognize parts of a right triangle i.e. opposite, adjacent and hypotenuse Using the trigonometry ratios to find unknown sides of a right triangle Using the trigonometry ratios to find unknown interior angles of a right triangle Trigonometry: The Sine Ratio The sine ratio can be used calculate an unknown angle or and unknown side in a right triangle. To find the either of these unknowns the opposite side and hypotenuse are used from an angle of reference. Example Suppose we use A as the angle of reference then: Sine always used the hypotenuse and opposite sides of a right triangle. Sine Opposite Hypotenuse: SOH When A is an acute angle in a right triangle, then Sin A = Length of side Opposite angle A Length of Hypotenuse side Sin A = Opposite Hypotenuse MBF3C Page 1 MBF3C Unit 3 - Foundations for College Mathematics How to find the missing side of a right triangle using the sine ratio. Example 1 Find the value of x. Solution Sin A = MBF3C O H Page 2 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 1. In each triangle, name the side: a. opposite E 2. b. the hypotenuse Calculate the Sin A and Sin B in each triangle. a. 3. Calculate. a. Sin 35 4. b. b. Sin 71 c. Sin 55 d. Sin 90 Calculate the value of x. a. MBF3C b. Page 3 MBF3C Unit 3 - Foundations for College Mathematics Support Questions (con’t) c. d. How to find the missing angle of a right triangle using the sine ratio. Example 2 Find A. Solution O Sin A = H MBF3C Page 4 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 5. Calculate. a. Sin 1 0.725 6. c. Sin 1 3 7 d. Sin 1 5 12 Calculate E to the nearest degree. a. Sin E = 0.625 7. b. Sin 1 0.325 b. Sin E = 0. 812 c. Sin E = 3 5 d. Sin E = 7 11 Calculate E. MBF3C Page 5 MBF3C Unit 3 - Foundations for College Mathematics Trigonometry: The Cosine Ratio The cosine ratio can be used calculate an unknown angle or and unknown side in a right triangle. To find the either of these unknowns the adjacent side and hypotenuse are used from an angle of reference. Example Suppose we use A as the angle of reference then: Cosine always used the hypotenuse and adjacent sides of a right triangle. Cosine Adjacent Hypotenuse: CAH When A is an acute angle in a right triangle, then Cos A = Length of side Adjacent angle A Length of Hypotenuse side Cos A = Adjacent Hypotenuse MBF3C Page 6 MBF3C Unit 3 - Foundations for College Mathematics How to find the missing side of a right triangle using the Cosine ratio. Example 3 Find the value of x. Solution Cos A = MBF3C A H Page 7 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 8. In each triangle, name the side: a. adjacent E 9. b. adjacent Y Calculate the Cos A and Cos B in each triangle. a. 10. Calculate. a. Cos 35 11. b. b. Cos 71 c. Cos 55 d. Cos 90 Calculate the value of x. a. MBF3C b. Page 8 MBF3C Unit 3 - Foundations for College Mathematics Support Questions (con’t) c. d. How to find the missing angle of a right triangle using the Cosine ratio. Example 4 Find C. Solution MBF3C Page 9 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 12. Calculate. a. Cos 1 0.725 13. c. Cos 1 3 7 d. Cos 1 5 12 Calculate E to the nearest degree. a. Cos E = 0.625 14. b. Cos 1 0.325 b. Cos E = 0. 812 c. Cos E = 3 5 d. Cos E = 7 11 Calculate E. MBF3C Page 10 MBF3C Unit 3 - Foundations for College Mathematics Trigonometry: The Tangent Ratio The Tangent ratio can also be used calculate an unknown angle or and unknown side in a right triangle. To find the either of these unknowns the adjacent side and opposite sides are used from an angle of reference. Example Suppose we use A as the angle of reference then: Tangent always used the opposite and adjacent sides of a right triangle. Tangent Opposite Adjacent: TOA When A is an acute angle in a right triangle, then Tan A = Length of side opposite angle A Length of Adjacent side Tan A = Opposite Adjacent MBF3C Page 11 MBF3C Unit 3 - Foundations for College Mathematics How to find the missing side of a right triangle using the Tangent ratio. Example 5 Find the value of x. Solution Tan A = MBF3C A H Page 12 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 15. In each triangle, name the side: a. adjacent F 16. b. opposite Y Calculate the Tan A and Tan B in each triangle. a. 17. Calculate. a. Tan 35 18. b. b. Tan 71 c. Tan 55 d. Tan 90 Calculate the value of x. a. MBF3C b. Page 13 MBF3C Unit 3 - Foundations for College Mathematics Support Questions c. d. How to find the missing angle of a right triangle using the Cosine ratio. Example 6 Find A. Solution MBF3C Page 14 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 19. Calculate. a. Tan 1 0.725 20. c. Tan 1 3 7 d. Tan 1 5 12 Calculate E to the nearest degree. a. Tan E = 0.625 21. b. Tan 1 0.325 b. Tan E = 0. 812 c. Tan E = 3 5 d. Tan E = 7 11 Calculate E. MBF3C Page 15 MBF3C Unit 3 - Foundations for College Mathematics Key Question #11 1. In each triangle, name the side: 2. Calculate the Sin A and Sin B in each triangle. 3. Calculate. a. Sin 42 4. b. Cos 68 c. Tan 12 Calculate the value of x. a. MBF3C b. Page 16 MBF3C Unit 3 - Foundations for College Mathematics Key Question #11 (con’t) 5. Calculate. a. Sin 1 0.612 6. b. Cos 1 0.825 c. Cos 1 2 5 d. Tan 1 3 13 Calculate E to the nearest degree. a. Sin E = 0.387 b. Sin E = 0. 900 c. Cos E = 12 29 d. Tan E = 13 5 7. Calculate E. 8. A guy wire is 13.5 m long. It supports a vertical power pole that is 8.7 m tall. Calculate the distance between where the guy wire is anchored into the ground from the base of the power pole. ? MBF3C Page 17 MBF3C Unit 3 - Foundations for College Mathematics Key Question #11 (con’t) 9. A 5.0 m ladder is leaning 3.7 m up a wall. What is the angle the ladder makes with the ground? 10. A kite has a string 100 m long anchored to the ground. The string makes and angle with the ground of 68. What is the horizontal distance of the kite from the anchor? 11. A ladder is leaned 10 m up a wall with its base 6 m from the wall. What angle does the ladder make with the ground? 12. The acronym SOHCAHTOA is often used in trigonometry. What do you think each letter stands for and give an example finding either an unknown side or an unknown angle using a portion of this acronym. MBF3C Page 18 MBF3C Unit 3 - Foundations for College Mathematics Lesson Twelve: Law of Sines drawing a labelling triangle angles and sides interpreting information and drawing diagrams based on that information using the sine law to find an unknown side in a triangle using the sine law to find an unknown side in a triangle drawing diagrams and solving questions, using the Sine Law, involving the angle of elevation and depression Trigonometry: The Sine Law The sine law is the relationship between the ratios of the sines of the angles of a triangle and the lengths of the opposite sides. For the triangle given below: The Sine Law can be written as: Sin A Sin B Sin C a b c or a b c Sin A Sin B Sin C Recognizing when to use the Sine Law. You need: MBF3C an angle and the value of its side opposite a second angle and the need to find its unknown opposite side Page 19 MBF3C Unit 3 - Foundations for College Mathematics OR an angle and the value of its side opposite a second side and the need to find its unknown opposite angle Example 1 x Sin24 8 Sin39 x Sin24 8 Sin39 Sin24 Sin24 8 Sin39 Sin24 x 12.4 x MBF3C Page 20 MBF3C MBF3C Unit 3 - Foundations for College Mathematics Page 21 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 1. Given each of the ratios below, state whether the unknown angle will be larger or smaller than the one given. a. 2. Sin 67 Sin X 13 7.2 b. Sin Y Sin 81 7.6 8.4 c. Sin 37 Sin A 4 8 Given each of the ratios below, state whether the unknown angle will be larger or smaller than the one given. a. Sin 55 Sin 71 13 x b. Sin 16 Sin 77 12.5 y c. Sin 83 Sin 41 a 7.2 3. Find the length of the indicated side. 4. Find the measure of angle A. 5. In PQR, find the value of q, if R = 73, Q = 32, and r = 23 cm. 6. In ABC, find the value of c, if A = 52, C = 47, and a = 12 m. 7. In TUV, find the value of U, if T = 37, u = 16 cm, and t = 22 cm. 8. In XYZ, find the value of Z, if X = 72, x = 41 m, and z = 11 m. 9. A radio tower is supported by two wires on opposite sides. The wires form an angle of 60 at the top of the post. On the ground, the ends of the wire are 15 m apart, and one wire is at a 45 angle to the ground. How long will the wires be? MBF3C Page 22 MBF3C Unit 3 - Foundations for College Mathematics Key Question #12 1. Given each of the ratios below, state whether the unknown angle will be larger or smaller than the one given. a. 2. Sin B Sin 35 10 5.1 b. Sin Y Sin 64 9.1 8.4 c. Sin 37 Sin A 6.1 3.8 Given each of the ratios below, state whether the unknown angle will be larger or smaller than the one given. a. Sin 13 Sin 52 8 x b. Sin 54 Sin 23 12.5 y c. Sin 17 Sin 76 a 7.2 3. Find the length of the indicated side. 4. Find the measure of angle C. 5. In TUV, find the value of t, if T = 61, U = 24, and u = 17 cm. 6. In EFG, find the value of e, if F = 52, G = 47, and f = 40 m. 7. In ABC, find the value of A, if B = 37, a = 10 cm, and b = 17 cm. 8. In QRS, find the value of Q, if S = 72, q = 26 m, and s = 35 m. 9. An architect designs a house that is 10 m wide. The rafters holding up the roof are equal length and meet at an angle of 65. The rafters extend 0.4 m beyond the supporting wall. How long are the rafters? 10. Does the sine law apply to right triangles? Explain your answer. MBF3C Page 23 MBF3C Unit 3 - Foundations for College Mathematics Lesson Thirteen: Law of Cosine drawing a labelling triangle angles and sides interpreting information and drawing diagrams based on that information using the cosine law to find an unknown side in a triangle using the cosine law to find an unknown side in a triangle Trigonometry: The Cosine Law The cosine law is used to find the third side of a triangle when two sides and a contained angle are known or to find an angle measure when the length of three sides are known. The contained angle in a triangle is the angle between the two given sides of the triangle. In the example given below C is the contained angle between the sides CA and CB. The Cosine Law states that for a ΔABC : c 2 a2 b2 2ab CosC b2 a2 c 2 2ac CosB a2 b2 c 2 2bc CosA MBF3C Page 24 MBF3C Unit 3 - Foundations for College Mathematics Example 1 Find the missing side shown in the diagram below using the cosine law. Solution a 2 b 2 c 2 2bc CosA a 2 (16) 2 (11) 2 2(16)(11) CosA Example 1 cont. So the missing length is 23.3 units long. Example 2 Find the angle A using the cosine law. MBF3C Page 25 MBF3C Unit 3 - Foundations for College Mathematics Solution First using algebra manipulate the formula to isolate Cos A. a 2 b 2 c 2 2bc CosA a 2 - b 2 c 2 2bc CosA a 2 - b 2 c 2 2bc CosA 2bc 2bc a2 - b2 c 2 CosA 2bc c 2 b2 - a2 CosA 2bc Example 2 cont. MBF3C Page 26 MBF3C Unit 3 - Foundations for College Mathematics Support Questions 1. Find the length of the indicated side. 2. Find the measure of angle X. 3. In PQR, find the value of r, if p = 13 cm, q = 8 cm and R = 63. 4. In ABC, find the value of a, if b = 17 cm, c = 28 cm and A = 105. 5. In TUV, find the value of U, if v = 10 cm, u = 16 cm, and t = 22 cm. 6. In XYZ, find the value of Z, if x = 41 m, y = 32 m, and z = 28 m. 7. The bases on a softball diamond are 15 m apart. A player picks up a fair ground ball 3 m from third base, along the line from second to third base. How far must he throw it to first base? 8. Find the measure, to the nearest degree, of the middle angle in a triangle that has side lengths of 8 cm, 19 cm, and 21 cm. 9. A plane leaves Hamilton and flies due east for 75 km. At the same time, a second plane flies in a direction 40 southeast for 120 km. How far apart are the planes when they reach their destinations? MBF3C Page 27 MBF3C Unit 3 - Foundations for College Mathematics Key Question #13 1. Find the length of the indicated side. 2. Find the measure of angle C. 3. In EFG, find the value of f, if e = 7.1 cm, g = 8.9 cm and F = 71. 4. In LMO, find the value of m, if o = 34 cm, l = 62 cm and M = 98. 5. In BCD, find the value of B, if b = 46 cm, c = 70 cm, and d = 58 cm. 6. In ABC, find the value of C, if a = 5.2 m, b = 3.8 m, and c = 6.7 m. 7. Two paths diverge at a 48 angle. Two mountain bike riders take separate routes at 8 km/hr and 12 km/hr. How far apart are they after 2 hours? Include a diagram. 8. Draw a diagram to solve this problem: Oshawa is 10 km due east of Ajax. Uxbridge is 18 km NW of Oshawa. How far is it from Ajax to Uxbridge? Explain whether you have enough information to solve this problem. 9. A golfer hits a tee shot on a 325 m long straight golf hole. The ball is hooked (hit at an angle) 18 to the left. The ball lands 185 m from the tee. How far is the ball from the hole? 10. Given the points A(0, 0), B(3, 1) and C(1, 4), what is the measure of ABC? MBF3C Page 28 MBF3C Unit 3 - Foundations for College Mathematics Lesson Fourteen: Trigonometry and Problem Solving Identifying the trigonometry strategy needed to solve various real life situations Trigonometry Strategies One of the most important strategies to solve the question is to draw a diagram and fill as much information as possible. Often more that one strategy can be used to solve such questions. The strategy shown or suggested is considered the most direct or logical route but not necessarily the only. Example A telephone pole and an electrical pole are 200 metres away from each other. From a point halfway between the two poles the angle of elevation of their tops is 8 and 12 . How tall are the poles? Solution First draw the diagram to help organize the information given. Since the diagram contains two right angle triangles then SOHCAHTOA can be used to solve the question. Pole 1 x Tan 8 100 100 Tan8 x 14.5 x Pole 2 x 100 100 Tan12 x 21.3 x Tan 12 the two poles are approximately 14.5 m and 21.3 m tall. MBF3C Page 29 MBF3C Unit 3 - Foundations for College Mathematics Example 2 The edges of a tooth are 1.1 cm and 0.8 cm long. The base of the tooth is 0.6 cm long. Determine the angle at which the edges meet. Solution Draw the diagram and label appropriately. There are no angles given for the triangle but all three sides are present. To find an angle when no other angles are present the Law of Cosine will need to be used. 2 2 2 1.1 0.8 0.6 Cos 2(1.1)(0.8) Cos 0.847 Cos 1 (0.847) 32 Support Questions 1. Two buildings are 38 m apart. From the top of the shorter building, the angle of elevation of the top of the taller building is 31 and the angle of depression of the base is 40 . Determine the height of each building. MBF3C Page 30 MBF3C Unit 3 - Foundations for College Mathematics Support Questions (con’t) 2. The area of a triangle is given by the formula A base height . 2 Determine the area of the triangle given below. 3. Calculate the height of the building in the diagram given below. 4. Two chains support a weight at the same point. The chains are attached to the ceiling and are 6.0 m and 5.5 m in length. The angle between the chains is 105 . How far apart are the chains at the ceiling? Key Question #14 1. Noah and Brianna want to calculate the distance between their houses which are opposite sides of a water park. They mark a point, A, 150 m along the edge of the water park from Brianna’s house. The measure NBA as 80 and BAN as 75 . Determine the distance between their houses. MBF3C Page 31 MBF3C Unit 3 - Foundations for College Mathematics Key Question #14 (con’t) 2. From an observation tower that overlooks a small lake, the angles of depression of point A, on one side of the lake, and point B, on the opposite side of the lake are, 7 and 13 , respectively. The points and the tower are in the same vertical plane and the distance from A to B is 1 km. Determine the height of the tower. 3. A children’s play area is triangular. The sides of the play area measure 100m, 250m and 275m respectively. Calculate the area of the play area. 4. A baseball diamond is a square with sides 27.4 m. The pitcher’s mound is 18.4 m from home plate on the line joining home plate and second base. How far is the pitcher’s mound from first base? MBF3C Page 32 MBF3C Unit 3 - Foundations for College Mathematics Lesson Fifteen: Creating Nets Use of nets to transform 2 dimensional drawings into 3 dimensional figures Nets A lot of 3 dimensional objects are created from flat material. Cardboard boxes and sheet metal in duct work are two such examples. Material is cut, and bent or folded to form objects. Example Draw one example of a net that would produce a cube. Solution Support Questions 1. Below are examples of nets. Which will produce closed objects? Explain why the others will not. MBF3C Page 33 MBF3C Unit 3 - Foundations for College Mathematics Key Question #15 1. For each of the nets given below draw another net that will produce the same object. 2. Create a net for each of the object given below 3. Identify the patterns that will produce a closed 3-dimensional object when folded. Explain your reasoning. MBF3C Page 34 MBF3C Unit 3 - Foundations for College Mathematics Unit 3 – Support Question Answers Lesson 11 1. a. FG b. YZ 2. a. SinA 3. a. 0.574 4. a. 3 .7 1 .5 ; SinB 4 4 b. SinA b. 0.946 c. 0.819 x 18 x 18 Sin 23 x 7.03 cm Sin 23 c. d. 4 x 4 x Sin 48 13.3 x 13.4 x Sin 62 Sin 48 Sin 62 4 x Sin 48 Sin 48 Sin 48 4 x Sin 48 5.38 x 13.4 x Sin 62 Sin 62 Sin 62 13.4 x Sin 62 15.2 x 5. a. 46 b. 19 6. Sin1 0.625 a. 39 Sin1 0.812 b. 54 c. a. b. c. SinE 2 .1 3 .7 E Sin1 E 35 8. a. GE MBF3C d. 1 b. x Sin 37 3 .7 x 3.7 Sin 37 x 2.23 cm 7. 12 13.4 ; SinB 18 18 SinE 2 .1 3 .7 c. 25 10 18 E Sin1 Sin1 3 5 d. 4 9 E Sin1 E 34 7 d. 11 40 Sin 1 37 SinE 10 18 d. 25 E 26 13.4 16.1 13.4 E Sin1 16.1 E 56 SinE 4 9 b. XY Page 35 MBF3C Unit 3 - Foundations for College Mathematics 1.5 3.7 ; CosB 4 4 a. CosA 10. a. 0.819 b. 0.326 c. 0.574 d. 0 11. a. b. c. d. x 3.7 x 3.7 Cos 37 x 2.95 cm a. 44 13. a. 14. a. 2 .1 CosE 3.7 E Cos 1 15. a. FG 16. a. TanA 17. a. 0.700 MBF3C Cos 62 4 x Cos 48 Cos 48 Cos 48 4 x Cos 48 5.98 x 13.4 xCos 62 Cos 62 Cos 62 13.4 x Cos 62 28.5 x b. 71 2 .1 3.7 b. c. 65 Cos1 0.812 36 c. Cos 1 d. 65 3 5 c. 10 CosE 18 4 CosE 9 10 18 E 56 d. Cos 1 7 11 50 53 b. E Cos 1 13.3 x 13.4 x Cos 62 Cos 48 Cos 23 Cos 1 0.625 51 E 55 4 x 4 x Cos 48 x 18 x 18 Cos 23 x 16.57 cm Cos 37 12. b. CosA 13.4 12 ; CosB 18 18 9. d. E Cos 1 E 64 13.4 16.1 13.4 E Cos 1 16.1 E 34 CosE 4 9 b. XZ 3 .7 1 .5 ; TanB 1.5 3.7 b. 2.90 b. TanA 12 13.4 ; TanB 13.4 12 c. 1.43 d. undefined (error) Page 36 MBF3C 18. Unit 3 - Foundations for College Mathematics a. b. c. 18 x 18 x Tan 23 d. Tan 23 x 3.7 x 3.7 Tan 37 x 2.79 cm Tan 37 19, a. 36 20. a. 21. 18 x Tan 23 Tan 23 Tan 23 18 x Tan 23 42.4 x b. 18 Tan 1 0.625 32 a. Tan 1 0.812 39 E 60 E Tan 1 c. Tan 1 d. 23 3 5 d. E 61 E Tan 1 E 66 7 11 d. 9 TanE 4 18 10 Tan 1 32 31 c. 18 TanE 10 3 .7 2 .1 Tan 62 c. 23 b. 3 .7 TanE 2 .1 E Tan 1 b. x 13.4 x 13.4 Tan 62 x 25.2 cm x 4 x 4 Tan 48 x 4.44 m Tan 48 16.1 13.4 16.1 E Tan 1 13.4 E 50 TanE 9 4 Lesson 12 1. a. 7.2 < 13 so X is smaller than 67 c. 8 > 4 so A is larger than 37 b. 7.6 < 8.4 so Y is smaller than 81 2. a. 71 > 55 so x is larger than 13 c. 83 > 41 so a is larger than 7.2 b. 77 > 16 so y is larger than 12.5 3. Sin54 Sin27 x 8 8 Sin54 x Sin27 8 Sin54 x Sin27 Sin27 Sin27 8 Sin54 x Sin27 14.3 x MBF3C Page 37 MBF3C Unit 3 - Foundations for College Mathematics 4. SinA Sin26 6.7 3.2 6.7 Sin26 3.2 SinA 6.7 Sin26 3.2 SinA 3.2 3.2 6.7 Sin26 Sin A 3.2 0.9178 Sin A sin-1 0.9178 A 66.6 A 5. Sin32 Sin73 q 23 23 Sin32 q Sin73 23 Sin32 q Sin73 Sin73 Sin73 23 Sin32 q Sin73 12.7 q 6. Sin47 Sin52 c 12 12 Sin47 c Sin52 12 Sin47 c Sin52 Sin52 Sin52 12 Sin47 c Sin52 11.1 c MBF3C Page 38 MBF3C Unit 3 - Foundations for College Mathematics 7. SinU Sin37 16 22 16 Sin37 22 SinU 16 Sin37 22 SinU 22 22 16 Sin37 Sin U 22 0.4377 Sin U sin-1 0.4377 U 26 U 8. SinZ Sin72 11 41 11 Sin72 41 SinZ 11 Sin72 41 SinZ 41 41 11 Sin72 Sin Z 41 0.255 Sin Z sin -1 0.255 Z 15.0 Z 9. Sin60 Sin45 15 x 15 Sin45 x Sin60 15 Sin45 x Sin60 Sin60 Sin60 15 Sin45 x Sin60 12.2 x Sin60 Sin75 15 y 15 Sin75 y Sin60 15 Sin75 y Sin60 Sin60 Sin60 15 Sin75 y Sin60 16.7 y the lengths of the wires are 12.2 m and 16.7 m MBF3C Page 39 MBF3C Unit 3 - Foundations for College Mathematics Lesson 13 1. x 2 30 2 50 2 2(30)(50)C os100 x 2 3920.9 x 62.6 2. 35 2 17 2 28 2 2(35)(17) CosX 0.6134 CosX X Cos 1 0.6134 X 52.2 3. r 2 13 2 8 2 2(13)(8)Co s63 r 2 138.57 r 11.8 cm 4. a 2 17 2 28 2 2(17)(28)C os105 a 2 1319.4 a 36.3 cm 5. 10 2 22 2 16 2 2(10)(22) CosU 0.7455 CosU U Cos 1 0.7455 U 41.8 6. 412 32 2 28 2 2(41)(32) CosZ 0.732 CosZ Z Cos 1 0.732 Z 42.9 MBF3C Page 40 MBF3C Unit 3 - Foundations for College Mathematics 7. x 2 12 2 15 2 2(12)(15)C os90 x 2 369 x 19.2 8. 8 2 212 19 2 2(8)(21) CosX 0.4286 CosX X Cos 1 0.4286 X 64.6 9. x 2 75 2 120 2 2(75)(120) Cos40 x 2 6237 x 78.97 km Lesson 14 1. A 38 A 38 Tan 40 A 31.88m Tan 40 B2 38 B2 38 Tan 31 B2 22.83m Tan 31 Therefore Building A is 31.88 m tall and Building two is 54.71 m tall (22.83+31.88) MBF3C Page 41 MBF3C Unit 3 - Foundations for College Mathematics 2. h 7.8 h 7.8 Sin 39 h 4.9 cm Sin 39 Area bh (13.8)( 4.9) 67.62 33.81 cm 2 2 2 2 3. s 75 s 75 Tan 45 s 75 m Tan 45 H 75 H 75 Tan 61 H 135.3 m Tan 61 MBF3C Page 42 MBF3C Unit 3 - Foundations for College Mathematics 4. x 2 6 2 5.5 2 2(6)(5.5)C os105 x 2 83.33 x 9.1 m Lesson 15 1. Net A is closed and will produce a square based pyramid. Net B is closed and will produce a rectangular prism. Net C is not closed the edges of partial circle (arc) needs to be straight to go around the edges of the complete circle given in the net. Net D is closed and will produce a cylinder. Net E is closed and will produce a triangular prism. MBF3C Page 43