2. Determine the indicated angle. a) b) c) d) e) f)

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1 U4L1 Review of Necessary Skills 1. Determine the length of x.. Determine the indicated angle. a) b) c) d) e) f) 3. From the top of a cliff 300m high, the angle of depression of a boat is o. Calculate the distance, x, from the boat to the foot of the cliff. 4. Calculate the distance between the centres of two adjacent holes on a nine-hole bolt circle with diameter 4.0 cm. 5. The shadow of a tree is 1m when the angle of elevation of the sun is 68 o. Calculate the height of the tree. Answers: 1.a) 9.4m b) 15.1m c) 13.1km d) 9.4m e) 3.m f) 13.6km.a) 45 b) c) 36 d) e) 55 f) m cm m 1

2 U4L - Special Triangles 1. Draw a sketch of the two special triangles ( and ).. Use your drawings to determine the exact value of each trig ratio. A. sin 30 B. cos 60 C. tan 45 D. sin 45 E. cos 45 F. tan 30 G. sin 60 H. cos Determine the value of θ. Refer to your triangle drawings to determine θ. A. tan θ = 1 B. cosθ = 1 E. cosθ = 1 F. sinθ = 1 C. sinθ = 1 G. tanθ = 1 3 D. sinθ = 3 H. tanθ = 3 4. If tana = 1, answer the following: 5 A. Draw the special triangle. B. Determine the value of sina. C. Determine the value of cosb. 5. If sina = 3, answer the following: 7 A. Draw the special triangle. B. Determine the value of cosa. C. Determine the value of tanb. Answers. a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 3 g) 3 h) 3 3. a) 45 o b) 60 o c) 45 o d) 60 o e) 45 o f) 30 o g) 30 o h) 60 o 4. sina = 1 6 cosb = cosa = 40 7 or tanb = 7 3 or 10 3

3 U4L3 - CAST Rule 1) a) Complete the CAST rule and label ) Label each side of a 3) Label each side of a the quadrants. 30,60,90 triangle. 45,45,90 triangle. b) What is the CAST rule used for? ) Find the exact value of each ratio. Use fractional answers. No Decimals Fully draw & label each triangle in the appropriate quadrant. a) sin 40 = b) tan 315 = c) cos( 10 ) = d) sin135= e) tan(-40) = f) cos315 = g) sin(-45)= h) tan(135) = i) cos150 = 3

4 j) sec 45 = k) csc( 135 ) = l) cot 90 = 3) Solve for angle x. 0 x 360. Use the CAST rule. Circle the final answers. Fully draw & label each triangle in the appropriate quadrant. Draw both triangles a) 1 sin x b) tan x 3 c) 1 cos x 4

5 Answers U4L3: 1 5

6 U4L4 - Applying the CAST Rule Solve each equation for. Round your answers to the nearest hundredth. 1. sin x 0.8. tan x cos x cos x sin x cos x + = 0 7. sin x 1 = tan x + 1 = 0 9. sin x 1 = a) sin 3 f) cos 1 0 b) cos 3 g) sin 4.8 c) 3 cos 1 h) 3sin 5 0 d) cos 0 i) 6 3sin 6 e) 5 4sin 3 j) 4sin 0 U4L4 Answers: , , , , , o, 46 o 7. 30, , a) 40, 300 b) 30, 330 c) 0,180,360 d) 45, 315 e) 30, 150 f) 60, 300 g) 194.5, h) 48., i) 0, 180, 360 j) 5, 315 U4L5 - Multi-Step Two Dimensional Trigonometric Problems 1. Kevin is standing at a point A, 0 m from the base of a building. From point A, the angle of elevation to point B is 8, and the angle of elevation to point C is Determine the vertical distance between points B and C.. Determine the distance AC. 6

7 3. A triangular deck is to be added to the back wall of a house. The length of the side of the deck that will be attached to the house is 0 ft, and the other two sides form angles of 3 and 65 with the side that is attached to the wall of the house. a) Draw a diagram to represent this situation. b) Determine the length of the shortest side of the deck. 4. Ella is standing on a bridge. From her location, the angle of elevation to the top of a nearby building is 8, and the angle of depression to the base of the building is 4. The least distance between Ella and the building is 40 m. Determine the height of the building. 5. From point A on the level ground, the angle of elevation of the top of a building is 5. From point B, 10 ft closer to the base of the building, the angle of elevation of the top of the building is 35. Determine the height of the building. 6. A rectangle is 16 cm long and 1 cm w wide. a) Determine the length of the diagonals of the rectangle. b) Determine the measure of the angle formed by the length of the triangle and a diagonal. Answers U4L m. 3.7m 3.b)10.7ft m ft 6.a) 0cm b) 37 o U4L6 - Multi-Step Three Dimensional Trigonometric Applications 1. Peter designed a garden pond, which is in the shape of an inverted square-based pyramid. The side lengths of the base are 3 m, and the slant height is m. a) Determine the greatest depth of the garden pond. b) Determine the maximum volume of water the pond can hold.. A drill bit is in the shape of a cone. The angle at the vertex of the cone is 15. The length of the slanted side of the drill bit is 13 mm. Determine the diameter at the top of the drill bit. 7

8 3. Louis is standing at point A on the shoreline of a river. From where he is standing, the angle of elevation to point D at the top of a cliff across the river is 5. Louis walks to point B, which is 30 m east of point A, and determines that ACB is 90 and CAB is 38. Determine the height of the cliff. 4. From point A on the level ground, the angle of elevation of the top of a building is 5. From point B, 10 ft closer to the base of the building, the angle of elevation of the top of the building is 35. Determine the height of the building. 5. Sheila is standing at point A west of a mountain in the Kootenay region of British Columbia. From point A, the angle of elevation of the top of the mountain is 38. From point B, which is ft to the east of point A, the angle of elevation of the top of the same mountain is 4. Determine the height of the mountain. 6. Michael is an architect who has designed a cottage with a roof that has two sides with different slopes. The shorter side of the roof is 1 ft in length and makes an angle of 68 with the top of the building. The longer side of the roof makes an angle of 8 with the top of the building. a) Draw a diagram to represent this situation. b) Determine the length of the other side of the roof. 7. A rocket, launched vertically from point C, is tracked by two tracking stations at A and B. Data from the launch were recorded according to the diagram below. a) Determine the height, h, of the rocket as calculated from tracking station A. b) Determine the height, h, of the rocket as calculated from tracking station B. c) Are the approximate results for the calculated heights of the rocket the same or different for parts a) and b)? Explain. 8

9 8. Faiza would like to calculate the height of a cliff. From point A where she is standing, the angle of elevation to point C at the top of the cliff is 38. If Faiza walks 80 m east to point D, ABD is formed, where B is the base of the cliff, such that DAB = 47 and ADB = 49. a) Determine the height, BC, of the cliff. b) Determine the distance from point A, where Faiza is standing, to point C at the top of the cliff. Answers U4L6: 1.a) 1.3m b) 3.9m.3.4mm m ft ft ft 7.a) 16.4m b) 16.6m c) The calculated heights of the rocket are different; they depend on the accuracy of the measurements of the angles of elevation of the rocket. 8.a) 47.3m b) 77.0m U4L7 - Ambiguous Case 1. In ΔHJK, h = 7.m, j = 8.4m, and H = 68. a) Calculate the value of sinh. b) How many solutions are possible for ΔHJK?. In ΔABC, a = 4.5cm, b = 5.cm, and A = 37. a) Calculate the value of bsina. b) How many different ΔABC are possible? c) Solve ΔABC. 3. In ΔPQR, p = 5.3km, q = 10.6km and P = 30. a) Determine the number of possible solutions for ΔPQR. b) Solve ΔPQR. 4. Given ΔMNP with m = 6.7cm, n = 1,4cm and M = 6, determine all possible values of p. 5. Sketch each triangle. Then calculate the length of the third side. Include all possible solutions. a) In ΔABC, c = 10cm, b = 1cm, and B = 44. b) In ΔRST, r = 6.m, s = 8.1m, and R = 39. c) In ΔABC, A = 30, a = 1.5 cm, and b = cm. 9

10 6. Determine the number of distinct triangles that exist in each case. a) In ΔABC, a = 0, c = 16, and A = 30. b) In ΔABC, a = 7, c = 16, and A = 30. c) In ΔABC, a = 10, b = 16, and A = 30. Answers: 1. a) 7.8 b) no solution. a) 3.1 b) two solutions c) B = 44, C = 99, c = 7.4cm or B = 136, C = 7, c = 0.9cm 3. a) one solution b) q = 90, R = 60, r = 9.km 4. p = 15.1cm or p = 7.cm 5. a) one triangle, a = 17.0cm b) two triangles, t = 9.8m or t =.8m c) c =.9 or c = a) one triangle C = 4, A = 30, and B = 16 b) no triangle c) two triangles (see below) 10