Unit 1 Trigonometry. Topics and Assignments. General Outcome: Develop spatial sense and proportional reasoning. Specific Outcomes:

Transcription

1 1 Unit 1 Trigonometry General Outcome: Develop spatial sense and proportional reasoning. Specific Outcomes: 1.1 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles. Topics and Assignments The Tangent Ratio (Outcome 1.1) Page 2 16, 17, 19, 20 Finding Side Lengths Using Tangent (Outcome 1.1) Page 13 The Sine & Cosine Ratios (Outcome 1.1) Page 26 Solving Triangles & Applying (Outcome 1.1) Page 34 Trigonometric Ratios Problems Involving Two Triangles (Outcome 1.1) Page 42

2 2 Unit 1 Trigonometry Naming the Sides of a Right Triangle: A right angle is an angle that measures 90. A right triangle is a triangle that has a 90 angle. Hypotenuse: is the side opposite the 90 (right) angle Opposite: is the side opposite the angle being referenced Adjacent: is the side next to the angle being referenced (not the hypotenuse) Ex) Label the sides of the following triangles in reference to the angle indicated. Use angle A as the reference A Use angle C as the reference A B C B C Use angle D as the reference Use angle Y as the reference D E X Z F Y

3 3 The Tangent Ratio: tan θ = lenth of side opposite angle θ lenth of side adjacent to angle θ Ex) Determine the tangent ratio for angle A. A Conclusion:

4 4 Ex) Determine the following tangent ratios. A tan A = 38 tan C = B 58 C 8 cm Z X tan X = Y 3 cm tan Y = Ex) Determine the actual measures of angles X and Y from the previous example.

5 5 Ex) Determine the measure of each angle identified. a) J K 7 3 K L L b) A A B 4.4 m C 6.1 m B Ex) Determine the tangent ratio for each of the following angles. Round your answers to 4 decimal places. a) tan 10 b) tan 30 c) tan 45 d) tan 75 e) tan 85 f) tan 90

6 6 Angle of Inclination: This is the angle that a line or line segment makes with the horizontal. This is always an acute angle (an angle between 0 and 90 ). Angle of Inclination Ex) Solar panels should have an angle of inclination approximately equal to the latitude of the building. The latitude of Spruce Grove is approximately Is the design for the solar panel shown below best for Spruce Grove. Justify your answer. Panel 6.2 ft 8.4 ft

7 7 Ex) A 12 ft. ladder leans against the side of a building with its base 5 ft. from the base of the wall. What angle does the ladder make with the ground to the nearest degree? Ex) A support cable is anchored to the ground 30 m from the base of a communications tower. The cable is 50 m long and is attached near the top of the tower. What angle does the cable make with the ground? Round your answer to the nearest degree.

8 8 The Tangent Ration Assignment: 1) In each triangle, write the tangent ratio for each acute angle. a) A b) D B 6 C 7 3 E 5 F c) d) C P 5 Q A B R 2) To the nearest tenth of a degree, determine the measure of for each value of tan. a) tan = 0.25 b) tan = 1.25 c) tan = 2.50

9 9 3) In each case given below, determine the measure of to the nearest tenth of a degree. a) b) c) d) e) 5.9 cm f) 2.4 cm 3.5 cm 6.3 cm

10 10 4) Is tan 60 greater than or less than 1? How do you know without using a calculator? 5) The grade or inclination of a road is often expressed as a percent. When a road has a grade of 20%, it increases 20 ft. in altitude for every 100 ft. of horizontal distance. 20% Grade 100 ft. 20 ft. Calculate the angle of inclination, to the nearest degree, of a road with each grade. a) 20% b) 25% c) 10% d) 15%

11 11 6) Determine the measures of all the acute angles in the diagram below. Round your answers to the nearest tenth of a degree. P S 5 cm Q 12 cm R 7) In a right isosceles triangle, why is the tangent of an acute angle equal to 1? 8) A playground slide starts 107 cm above the ground and is 250 cm long. What angle does the slide make with the ground? Round your answer to the nearest degree.

12 12 9) From a rectangular board, a carpenter cuts a stinger to support some stairs. Each stair rises 7.5 in. and has a tread of 11.0 in. To the nearest degree, at what angle should the carpenter cut the board? 10) For safety reasons, a ladder is positioned so that the distance between its base and the wall is no greater than 1 4 the length of the ladder. To the nearest degree, what is the least angle of inclination allowed for a ladder?

13 13 Finding Side Lengths Using Tangent: By using our calculators, we can look up the tangent ratio of any angle we encounter and then use this to determine the length of an unknown side in a right triangle. Ex) Determine the indicated side length for each case given. Round your answers to the nearest tenth. a) E side EF F D b) X 70 side XY Y 12.4 cm Z c) A C side AC B

14 14 d) V W X side VW e) A side AB 45 B 19 m C f) J 65 side JL L 17.8 cm K

15 15 Naming a Triangle: When naming a triangle we typically use capital letters to represent the vertices or angles and the corresponding lower case letter to represent the side opposite each angle. For example side a will be opposite angle A Ex) In ABC, C = 32, B = 90, and c = 5.4 m. Determine the length of side a to the nearest tenth. Ex) In DEF, e = 12.7 cm, D = 90 and F = 17. Find the length of side f.

16 16 Ex) The line of sight to the top of flagpole is 67 when taken 20 ft. from the base of the flagpole. How tall is the pole to the nearest tenth of a foot? Ex) You are in a boat and you must look up at an angle of elevation of 5 to see the top of a cliff that is 125 m high. How far are you from the base of the cliff? Assume your eyes are 3 m above the water due to your height and the height of the boat.

17 17 Finding Side Lengths Using Tangent Assignment: 1) Determine the length of each indicated side to the nearest tenth. a) A b) D b cm B 5.0 cm C E d F c) d) B X y Z 20.6 m 53 d 4.1 m C 31 D Y

18 18 e) f) P 6.2 ft Q A B p b 9.6 ft R C g) F h) X y Z 5.4 m m Y 48 D f E i) C j) P r b 1.8 km cm A B 66 Q R

19 19 2) In ABC, A = 56, C = 90, and b = 15.4 m. Determine the measure of side a to the nearest tenth. 3) In DEF, E = 72, F = 90, and e = 2.7 cm. Determine the measure of side d to the nearest tenth. 4) In PQR, Q = 90, R = 18, and r = 44.5 ft. Determine the measure of side p to the nearest tenth. 5) In XYZ, X = 90, Z = 26, and y = 13.1km. Determine the measure of side z to the nearest tenth.

20 20 6) The angle between one longer side of a rectangle and a diagonal is 34. One shorter side of the rectangle is 2.3 cm. Determine the length of the rectangle to the nearest tenth of a cm. 7) In PQR, R = 90, P = 58, and q = 7.1 cm. Determine the area of PQR to the nearest tenth of a cm 2.

21 21 The Sine and Cosine Ratios: Like tangent the sine and cosine ratios are based on the sides of a right angle triangle. sin θ = length of side opposite angle θ length of hypotenuse cos θ = length of side adjacent to angle θ length of hypotenuse These ratios work the same way a tangent ratio work in terms of how we use them to solve problems. SOH CAH TOA sin θ = opposite hypotenuse cos θ = adjacent hypotenuse tan θ = opposite adjacent **Remember that the sides of a triangle are labeled in reference to the angle being used.

22 22 Ex) Determine the trigonometric ratios for the identified angles given below. a) A 30 sin A = 50 cos A = B 40 C tan A = b) 12 Z X sin Y = 5 Y cos Y = tan Y =

23 23 Ex) In each case below, determine the measure of the missing angles. Round answers to the nearest tenth. a) J L K b) D 75 cm E 83 cm F c) A 7 m 6 m B C

24 24 Ex) In each case below determine the missing side lengths. Round your answers to the nearest tenth. a) P 9.4 Q 65 R b) A B 10.2 C 40 c) D 6.7 cm E 30 F

25 25 Ex) In DEF, d = 15.4 cm, E= 90 and F = 28. Find the length of side e. Ex) In PQR, q = 8.3 m, Q = 90 and P = 28. Find the length of side p.

26 26 The Sine and Cosine Ratios Assignment: 1) Determine the sine and cosine value for each of the following angles. Round your answers to the nearest thousandth. a) 57 b) 5 c) 19 d) 81 2) To the nearest tenth of a degree, determine the measure of angle in each case below. a) sin = 0.25 b) cos = 0.64 c) 6 sin = d) 11 7 cos = 9 3) In each case, determine the measure of angle to the nearest tenth of a degree. a) D E b) F 24.6

27 27 c) d) e) f) g) h)

28 28 i) j) ) Determine the value of the indicated value for each case given. Round your answers to the nearest tenth. a) b) A r Q P 8 14 cm 58 c 43.4 m R B C c) d) F Z y 3.0 cm e X 61 E 17 9 ft D Y

29 29 e) f) A B Z cm X a 6.8 m C 33 z Y g) F h) Q d P 34 p in D 94.5 km E R i) j) B c M n O D 48.0 m C 8.7 cm N

30 30 5) Sketch a triangle and label its sides for each ratio given. a) 3 sin B = b) 5 5 cos = 8 c) 1 sin = d) 4 4 cos A = 9 6) In ABC, B = 12, C = 90, and c = 10.5 cm. Determine the measure of side b correct to the nearest tenth of a cm. 7) In XYZ, Z = 90, y = 6.1cm, and z = 19.3 cm. Determine the measure of X correct to the nearest tenth of a degree.

31 31 8) In DEF, D = 90, d = 4.8 ft, and e = 2.3ft. Determine the measure of E correct to the nearest tenth of a degree. 9) In PQR, P = 90, Q = 58, and r = 32.4 m. Determine the measure of side p correct to the nearest tenth of a m. 10) In MNO, M = 90, m = 70.8 cm, and o = 25.6 cm. Determine the measure of N correct to the nearest tenth of a degree. 11) In ABC, A = 24, B = 90, and b = 11.9 km. Determine the measure of side a correct to the nearest tenth of a km.

32 32 12) A ladder is 6.5 m long. It leans against a wall. The base of the ladder is 1.2 m from the wall. Determine the angle of inclination of the ladder to the nearest tenth of a degree. 13) An airplane approaches an airport. At a certain time, it is 939 m high. Its angle of elevation measured from the airport is Determine how far the plane is from the airport correct to the nearest tenth metre. 14) A fire truck has an aerial ladder that extends 30.5 m measured from the ground. The angle of inclination of the ladder is 77. Determine how far up the wall of an apartment building the ladder will reach correct to the nearest tenth of a metre. 15) A rope that supports a tent is 2.4 m long. The rope is attached to the tent at a point that is 2.1 m above the ground. Determine the angle of the inclination of the rope correct to the nearest degree.

33 33 16) Determine the perimeter, correct to the nearest tenth of a cm, of each shape given below. a) C D 8.8 cm 34 E b) A B cm C D

34 34 Solving Triangles & Applying Trigonometric Ratios: When solving a triangle, we are required to find all missing sides and all missing angles. Tools: sin θ = opposite hypotenuse cos θ = adjacent hypotenuse tan θ = opposite adjacent a 2 + b 2 = c 2 Sum of all Angle in a triangle = 180 Ex) Solve the following angles. Round all answers to the nearest tenth. a) A 68 C 52 B

35 35 b) J L 47 feet 73 K c) In PQR, p = 17.4 m, R = 90 and Q = 38.

36 Ex) A table is in the shape of a regular octagon. The distance to one vertex to the opposite vertex, measured through the centre of the table, is approximately 4 feet. There is a strip of wood veneer around the edge of the table. What is the length of the veneer to the nearest foot? 36

37 37 Solving Triangles & Applying Trigonometric Ratios Assignment: 1) Determine the length of each indicated side. Round your answers to the nearest tenth of a unit. a) C 8.9 cm B b) R b 60.0 in 54 t A 76 T S c) P d) D 48 E r ft d cm F R Q

38 38 2) Determine the measure of each indicated angle correct to the nearest tenth of a degree. a) b) c) d)

39 39 3) Solve the following triangles (determine the measure of all missing sides and all missing angles). Round all values to the nearest tenth. a) A 17.6 cm B b) M 41 C 5.1 m 4.4 m N O c) Q d) D 2.4 ft R 3.8 ft E P 73.5 cm 10 F

40 40 4) In DEF, D = 90, d = 16.8 cm, and f = 9.2cm. Solve the triangle (determine the measure of all missing sides and angles). Round all answers to the nearest tenth of unit. 5) In XYZ, Y = 72, Z = 90, and y = 3.3ft. Solve the triangle (determine the measure of all missing sides and angles). Round all answers to the nearest tenth of a unit. 6) In ABC, C = 90, a = 1.7 cm, and b = 4.3cm. Solve the triangle (determine the measure of all missing sides and angles). Round all answers to the nearest tenth of a unit.

41 41 7) The world s tallest totem pole is in Alert Bay, B.C., home of the Nimpkish First Nation. Twenty feet from the base of the totem pole, the angle of elevation to the top of the pole is Determine how tall the totem pole is to the nearest foot. 8) A helicopter leaves its base, and flies 35 km due west to pick up a sick person. It then flies 58 km due north to a hospital. When the helicopter is at the hospital, determine how far it is from its base to the nearest km. 9) Determine the area of each shape given below. Round your answers to the nearest cm 2. a) b) cm 5.6 cm

42 42 Problems Involving Two Triangles: These problems involve two triangles and will require multiple steps to arrive at the solution. Ex) Calculate the length of CD to the nearest tenth. Ex) Find the measure of angle XYZ.

43 Ex) Determine the height of the taller building to the nearest tenth. 43

44 Ex) From the top of a 90 foot observation tower, a fire ranger observes one fire due west of the tower at an angle of depression of 5, and another fire due south of the tower at an angle of depression of 2. How far apart are the fires to the nearest foot? 44

45 45 Problems Involving Two Triangles Assignment: 1) In each triangle, determine the length of JK to the nearest tenth of a cm. a) b) c) d)

46 46 2) In each quadrilateral, calculate the length of GH to the nearest tenth of a cm. a) b) c)

47 47 3) In each case given below, calculate the measure of XYZ to the nearest tenth of a degree. a) b) c) d)

48 48 4) From a window on the second floor of her house, a student measured the angles of elevation and depression to the top and base of a nearby tree. The student knows that she made the measurements 16 ft. above the ground. Determine the height of the tree to the nearest tenth of a foot. 5) At the Muttart Conservatory, the arid pyramid has 4 congruent triangular faces. The base of each face has length 19.5 m and the slant height of the pyramid is 20.5 m. Determine the measure of each of the three angles in the face to the nearest tenth of a degree.

49 49 6) Two office towers are 50 m apart. From the top of the shorter tower, the angle of depression to the base of the taller tower is 35. The angle of elevation to the top of this tower is 25. Determine the height of each tower to the nearest metre. 7) A communications tower is supported by guy wires. One guy wire is anchored at a point that is 8.9 m from the base of the tower and has an angle of inclination of 36. From this point, the angle of elevation to the top of the tower is 59. Determine how far, to the nearest tenth of a metre, from the top of the tower the guy wire is attached to the tower.

50 50 8) Determine the length of AF, the body diagonal and the measure of AFH, the angle between the body diagonal and a diagonal of the base of the prism. Round your answers to the nearest tenth.

51 51 Answers The Tangent Ratio Assignment: 1. a) c) 6 tan A =, 7 8 tanq =, 5 7 tan B = 6 b) 5 tan R = 8 d) 5 tan D =, 3 6 tan A =, a) = 14.0 b) = 51.3 c) = tan F = 5 11 tan B = 6 3. a) = 26.6 b) = 45 c) = 60.9 d) = 69.4 e) = 36.4 f) = As the size of an angle increases from 0 to 90 the tangent value of the angle also increases. Since tan 45 = 1, tan 60 will be greater than 1 (as 60 is greater than 45 ). 5. a) 11 b) 14 c) 6 d) 9 6. P = 67.4, R = 22.6, PQS = 22.6, SQR = In a right isosceles triangle the lengths of the two legs are the same. The tangent of an acute angle will equal a fraction whose numerator and denominator are equal, thus the tangent of an acute angle will equal Finding Side Lengths Using Tangent Assignment: 1. a) 2.5 cm b) 1.4 cm c) 3.1 m d) 34.3 m e) 5.2 ft f) 2.6 ft g) 4.9 m h) m i) 3.5 km j) 4.5 cm 2. a = 22.8 m 3. d = 0.9 cm 4. p = ft z = km cm cm 2

52 52 The Sine and Cosine Ratios Assignment: 1. a) sin 57 = cos57 = b) sin 5 = cos5 = c) sin19 = cos19 = d) sin81 = cos81 = a) = 14.5 b) = 50.2 c) = 33.1 d) = a) = 65.1 b) = 50.2 c) = 33.1 d) = 75.7 e) = 68.7 f) = 63.7 g) = 53.1 h) = 28.1 i) = 45.0 j) = a) r = 11.9 cm b) c = 43.0 m c) y = 18.6 ft d) e = 10.3 cm f) a = 3.9 cm g) d = km h) p = 4.0 in i) n = 51.1m j) c = 7.8 cm 5. a) 3 5 B c) 6. b = 2.2 cm 7. X = E = p = 61.1m 10. N = a = 4.8km m m a) 12.0 cm b) 15.1 cm b) d) A 9 Solving Triangles & Applying Trigonometric Ratios Assignment: 1. a) b = 6.5 cm b) t = 58.2 in c) r = cm d) d = 9.8 ft 2. a) = 34.8 b) = 18.2 c) = 60.3 d) = 50.8

53 53 3. a) A = 49, a = 13.3cm, b = 11.5 cm b) M = 30.4, N = 59.6, m = 2.6 m c) P = 32.3, R = 57.7, q = 4.5ft d) D = 80, e = 74.6 cm, f = 13.0cm 4. E = 56.8, F = 33.2, e = 14.1cm 5. X = 18, x = 1.1ft, z = 3.5 ft 6. A = 21.6, B = 68.4, c = 4.6 cm ft km cm cm 2 Problems Involving Two Triangles Assignment: 1. a) 6.0 cm b) 6.0 cm c) 4.3 cm d) 3.6 cm 2. a) 5.7 cm b) 4.9 cm c) 5.7 cm 3. a) 93.2 b) c) 11.1 d) ft , 64.6, and m m 8. AF = 5.4cm AFH = 33.9