Chapter 2: Trigonometry

Transcription

1 Chapter 2: Trigonometry Section 2.1 Chapter 2: Trigonometry Section 2.1: The Tangent Ratio Sides of a Right Triangle with Respect to a Reference Angle Given a right triangle, we generally label its sides with respect to a chosen reference angle. This is shown in the diagram below. The side that touches the reference angle is referred to as the adjacent side (A). The side that is directly across from the reference angle is referred to as the opposite side (O). The longest side of the triangle, the side across from the right angle, is the hypotenuse (H). Ex. Label the adj., opp., and hyp. according to angle θ in each triangle. (a) θ (b) θ 31

2 Chapter 2: Trigonometry Section 2.1 The Tangent Ratio There is a relationship that exists between the chosen reference angle and the measure of the sides that are opposite and adjacent to it. This is known as the Tangent Ratio (Tan). Tangent Ratio If A is an acute angle in a right triangle, then C lengh of side opposite to A tan(a) = lengh of side adjacent to A tan(a) = opp adj Opposite Hypotenuse tan(a) = O A B Adjacent A Determine the Tangent Ratios for Angles Ex1. Determine Tan(D) and Tan(F) for the triangle F 3 D 4 E Ex2. Determine Tan(X) and Tan(Z) for the triangle X 12 Y 6 Z 32

3 Chapter 2: Trigonometry Section 2.1 Using the Tangent Ratio to Determine the Measure of an Angle Ex1. Determine the measures of G and J to the nearest degree. H 5 4 G I Ex2. Determine the measures of K and N to the nearest degree. M 13 9 K N 33

4 Chapter 2: Trigonometry Section 2.1 Terminology: Angle of Inclination: The acute angle that is formed between a line segment and the horizon. Angle of Inclination Using the Tangent Ratio to solve a Problem Ex1. A 10 ft. ladder leans against the side of a building with its base 4 ft. from the wall. What angle, to the nearest degree, does the ladder make with the ground? Ex2. A support cable is anchored to the ground 5 m from the base of a telephone pole. The cable is 19 m long. It is attached near the top of the pole. What angle, to the nearest degree, does the cable make with the ground? Practice Problems: 3,4,5,8,10,14,17 pg

5 Chapter 2: Trigonometry Section 2.2 Section 2.2: Using The Tangent Ratio to Determine Side Length Determining the Length of a Side Given an Angle Ex1. Determine the length of AB to the nearest centimetre. B 10 cm C 27 A Ex2. Determine the length of XY to the nearest foot. Y 5.0 ft Z 49 X 35

6 Chapter 2: Trigonometry Section 2.2 Ex3. Determine the length of EF to the nearest tenth of an inch. E 24 F 3.5 in D Ex4. Determine the length of VX to the nearest tenth of a centimetre. X 44 V 7.2 cm W 36

7 Chapter 2: Trigonometry Section 2.2 Word Problem Applications of Tangent Ex1. A searchlight beam shines vertically on a cloud. At a horizontal distance of 250 m from the line of sight to the cloud is 75. Determine the height of the cloud to the nearest metre. Ex2. At a horizontal distance of 200 m from the base of an observation tower, the angle between the ground and the line of sight to the top of the tower is 8. How high is the tower to the nearest metre? Practice Problems #3,4,5,6,7,8,9,10,13,14 pg

8 Chapter 2: Trigonometry Section 2.3 Section 2.3: Using a Clinometer Terminology: Clinometer: A device used to measure the angle between a horizontal line and the line of sight to the top of an object. NOTE 1: The reading on a clinometer gives the measurement of the angle formed between the hypotenuse and the top of the object (vertical side). NOTE 2: When calculating the height of an object using a clinometer, you must also add on the height of the person making the measurements as a clinometer measures according the user s eye level. Ex1. Determine the height of a tall tree if you are stood 10 m away from the base of the tree and the clinometer you are using gives an angle measurement of 35. Assume that you are 1.7 m tall. Ex2. Julia is 1.3 m tall. If she is stood 7.0 m away from a flag pole and uses a clinometer to determine the angle between your line of sight and the top of the pole is 15, determine the height of the flag pole. Practice Problem #2 & 3 pg 86 38

9 Chapter 2: Trigonometry Section 2.4 Section 2.4: The Sine and Cosine Ratios Sine Ratio If θ is an acute angle in a right triangle, then C lengh of side opposite to θ Sin(θ) = hypotenuse Sin(θ) = opp hyp Sin(θ) = O H Opposite B Adjacent Hypotenuse θ A Cosine Ratio If θ is an acute angle in a right triangle, then C lengh of side adjacent to θ Cos(θ) = hypotenuse Cos(θ) = adj hyp Cos(θ) = A H Opposite B Adjacent Hypotenuse θ A Sometimes it is useful to have a mnemonic to help you to remember the three trig ratios such as: 39

10 Chapter 2: Trigonometry Section 2.4 Determining the Sine and Cosine Ratio for an Angle Ex1. Determine Sin(D) and Cos(D) to the nearest hundredth E 12 cm 5cm D 13 cm F Ex2. Determine Sin(G)and Cos(G) to the nearest hundredth H 8cm J 15 cm 17 cm G 40

11 Chapter 2: Trigonometry Section 2.4 Using Sine or Cosine to Determine the Measure of an Angle Ex1. Determine the measures of G and H to the nearest tenth of a degree. K 6 G 14 H Ex2. Determine the measures of K and M to the nearest tenth of a degree. M 8 K 3 N 41

12 Chapter 2: Trigonometry Section 2.4 Using Sine or Cosine to Determine an Unknown Side Ex1. Determine the length of the unknown side in each triangle (a) x cm (b) 7 m 41 x (c) 18 ft 63 x (d) 9.5 in x 71 42

13 Chapter 2: Trigonometry Section 2.4 Terminology: Angle of Elevation The angle formed between the ground and the line of sight from an observer. Angle of Depression The angle formed between a horizon and the line of sight from an observer. Angle of Depression Angle of Elevation NOTE: The angle of elevation in a diagram will always equal the angle of depression. Using Sine or Cosine to Solve a Problem Ex1. A water bomber is flying at an altitude of 5000 ft. The plane s radar shows that it is a horizontal distance of 8000 ft. from the target site. What is the angle of elevation of the plane measured from the target site, to the nearest degree? 43

14 Chapter 2: Trigonometry Section 2.4 Ex2. An observer is sitting on a dock watching a float plane in Vancouver harbour. At a certain time, the plane is 300 m above the water and 430 m from the observer. Determine the angle of elevation of the plane measured from the observer, to the nearest degree. Ex3. A surveyor made the measurements shown in the diagram. How could the surveyor determine the distance from the transit to the survey pole to the nearest hundredth of a metre? Survey Pole 67.3 Transit m Survey Stake 44

15 Chapter 2: Trigonometry Section 2.4 Ex4. From a radar station, the angle of elevation of an approaching airplane is The horizontal distance between the plane and the radar station is 35.6 km. How far is the plane diagonally from the radar station to the nearest tenth of a kilometer? Practice Questions 5, 7, 8, 10, 12 pg , 4, 5, 6, 7, 8 pg

16 Chapter 2: Trigonometry Section 2.5 Section 2.5: Applying Trig Ratios in Multistep Questions Solving a Right Triangle Ex1. Solve XYZ. Give the measures to the nearest tenth. Y 6.0 cm 10.0 cm X Z Ex2. Solve MNK. Give measurements to the nearest tenth. K 7.0 cm M 27 N 46

17 Chapter 2: Trigonometry Section 2.5 Calculate a Side Length Using More than One Triangle Ex1. Calculate the length of CD to the nearest tenth of a centimetre. B C A 4.2 cm D Ex2. Calculate the length of XY to the nearest tenth of a centimetre. Y cm 32 X W Z 47

18 Chapter 2: Trigonometry Section 2.5 Solving a Problem with Triangles in the Same Place Ex1. From the top of a 20 m high building, a surveyor measured the angle of elevation of the top of another building to be 30 and the angle of depression to the base of that building to be 15. The surveyor sketched this plan of her measurements. Determine the height of the taller building to the nearest tenth of a metre. Taller Building Shorter Building 20m Ex2. A surveyor stands at the window on the 9 th floor of an office tower. He uses a clinometer to measure the angles of elevation to be 31 to the top and depression to be 42 to the base of a taller building. Determine the height of the taller building to the nearest tenth of a metre. 39 m Shorter Building Taller Building 48

19 Chapter 2: Trigonometry Section 2.5 Ex3. A tourist at Point Amour sees a fishing boat at an angle of depression of 23 and a sailboat at an angle of depression of 9. If the tourist is 33.5 m above the water, determine how far apart the two vessels are. Light House Fishing Boat Sail Boat Three Dimensional Problem Ex. From the top of a 90 ft observer 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? 5 2 Tower 90 ft Fire 2 Fire 1 Practice Questions #3,4,6,8,9,10,11,13,14 pg