Trigonometry. What you will learn

Transcription

1 C H P T R 10 Trigonometry hat you will learn 10.1 Introducing trigonometry 10.2 Finding the side length of a right-angled triangle 10.3 Further problems involving side lengths 10.4 Finding the angle 10.5 Mied application problems 10.6 ngles of elevation and depression 10.7 Bearings 10.8 Problems involving two triangles

2 The ureka Tower The ureka Tower in Melbourne is 292 metres tall. Building the tower involved many logistic and mathematical problems. One of the most dangerous and challenging problems was how to dismantle and remove safely the giant 180-tonne crane in the lift well on top of the building. To do this another identical crane was placed on the top of the building to sit opposite the eisting one to remove the first crane piece by piece. The second crane was then removed using a smaller but powerful recovery crane which is easier to dismantle. This whole process required careful planning and calculation of angles and lengths of triangles to ensure that no damage was done to the ureka Tower or other surrounding buildings. Calculations involving triangles, angles and lengths require the use of trigonometry. VL Measurement, chance and data tudents estimate and measure length and angle. tudents use trigonometric ratios, sine, cosine and tangent to obtain lengths of sides, angles and the area of rightangled triangles. orking mathematically tudents choose, use and develop mathematical models and procedures to investigate and solve problems set in a wide range of practical, theoretical and historical contets. tudents select and use technology in various combinations to assist in mathematical inquiry, to manipulate and represent data, to analyse functions and carry out symbolic manipulation.

3 killsheet T CH R Do now 1 Round off each number correct to four decimal places. a b c d Round off each number correct to two decimal places. a b c d Round off each number to the nearest metre. a 4.6 m b m c m d m 4 olve each of the following equations to find. a 3 6 b 4 12 c 5 60 d 8 48 e f g h i j k l olve each of the following equations to find correct to two decimal places. a 4.7 b 7.43 c d olve each of the following equations to find correct to one decimal place a b c 15 d e 6.9 f 8.4 g 3.24 h Find the value of each pronumeral. a b c d e f nswers 1 a b c d a 4.23 b 5.68 c d a 5 b 35 c 1 d a 2 b 3 c 12 d 6 e 12 f 30 g 28 h 182 i 9 j 2 k 12 l 8 5 a b c 1.38 d a 0.6 b 0.6 c 2.1 d 0.5 e 0.6 f 2.0 g 9.1 h a 32 b 18 c 152 d 60 e 20 f ssential Mathematics VL dition Year 9

4 10.1 Introducing trigonometry Trigonometry deals with the relationship between the sides and the angles of triangles. Trigonometry enables us to calculate lengths and angles which may be difficult or impossible to measure directly. Trigonometry is used in the fields of science, engineering, surveying, astronomy, navigation and architecture. hen using trigonometry, it is important to name the sides of a right-angled triangle correctly. If the symbol is used to represent one angle, then the other sides can be named according to whether they are opposite or adjacent to that angle. a b c hypotenuse adjacent side opposite side adjacent side hypotenuse opposite side adjacent side hypotenuse opposite side For right-angled triangles, the basic trigonometric ratios are called sine, cosine and tangent and these are derived from the unit circle (a circle with radius one unit) which will be discussed in greater depth in Year 10 mathematics. Key ideas The three trigonometric ratios are defined as: length of the opposite side sine of angle (or sin ) length of the hypotenuse length of the adjacent side cosine of angle (or cos ) length of the hypotenuse length of the opposite side tangent of angle (or tan ) length of the adjacent In summary: Label each side of the triangle with O (opposite side), (adjacent side) and H (hypotenuse). Decide which two sides are involved in the problem by using: OH CH TO H O opposite sin hypotenuse adjacent cos hypotenuse tan opposite adjacent Chapter 10 Trigonometry 355

5 ample 1 Copy this triangle and label the sides as opposite to (O), adjacent to () or hypotenuse (H). olution O planation Draw the triangle and label the sides as hypotenuse (H), opposite (O) and adjacent (). H ample 2 rite trigonometric ratios (in fraction form) for each of the following triangles. a b c olution a cos H 5 7 b sin O H 4 9 planation (H) 7 (O) 5 () 9 (H) () ide length 7 is the longest side so it is (H). ide length 5 is adjacent to angle so it is (). ide length 9 is the longest side so it is (H). ide length 4 is opposite angle so it is (O). 4 (O) c tan O 3 5 () 5 (H) 3 (O) ide length 5 is the adjacent side to angle so it is (). ide length 3 is opposite angle so it is (O). 356 ssential Mathematics VL dition Year 9

6 ercise 10 ample 1 1 Copy each of these triangles and label the sides as opposite to (O), adjacent to () or hypotenuse (H). a b c d e f g ample 2 2 For the triangle shown, state which number corresponds to: a the hypotenuse α b the side opposite angle 5 4 c the side opposite angle d the side adjacent to angle e the side adjacent to angle 3 3 rite a trigonometric ratio (in fraction form) for each of the following triangles and simplify where possible. a b c 6 d 5 m n e 3 3 f 4a g 2 h 5a 4y 3 2y 4 Copy each of these triangles and mark the angle that will enable you to write a ratio for sin. a b c d H O O O H H O H Chapter 10 Trigonometry 357

7 5 For each of these triangles, write a ratio (in fraction form) for sin, cos and tan. a b c For the triangle shown on the right, write a ratio (in fraction form) for: a sin b sin c cos d tan e cos f tan α This triangle has angles 90, 60 and 30 and side lengths 1, 2 and a rite a ratio for: 2 i sin 30 ii cos 30 iii tan 30 iv sin 60 v cos 60 vi tan 60 30º b hat do you notice about the following pairs of ratios? 3 i cos 30 and sin 60 ii sin 30 and cos 60 8 a Measure all the side lengths of this triangle to the nearest millimetre. b Use your measurements from part a to find an approimate ratio for: i cos 40 ii sin 40 iii tan iv sin 50 v tan 50 vi cos 50 c Do you notice anything about the trigonometric ratios for 40 and 50? 9 For each of the following: i Use Pythagoras theorem to find the unknown side. ii Find sin, cos and tan. a b 7 c 9 d º 1 10 a Draw a right-angled triangle and mark one of the angles as. Then mark in the length of the opposite side as 15 units and the length of the hypotenuse as 17 units. b Find the length of the adjacent side using Pythagoras theorem. c Determine the ratios for sin, cos and tan. 11 Triangle BC has a right angle at B and angle C is. Distance B is 4 cm and distance C is 5 cm. a Draw the triangle. b Find distance BC. c rite the ratios for sin, cos and tan 358 ssential Mathematics VL dition Year 9

8 nrichment Th 12 a Given that is acute and cos 4 find sin and tan. Hint: Use Pythagoras 5, theorem. b For each of the following draw a right-angled triangle then use it to find the other trigonometric ratios. i sin 1 ii cos 1 iii tan For triangle BC on the right, find: a distance C b a ratio for sin, cos and tan c a ratio for sin, cos and tan 14 Investigate the Pythagorean Identity for trigonometry and test it by: C α 10 6 β B a choosing a ratio for sin b calculating cos using Pythagoras theorem and c testing the identity. Describe what an identity is and research other identities. Chapter 10 Trigonometry 359

9 TI-nspire PP DI X Using technology to determine trigonometric ratios It is difficult to determine trigonometric ratios accurately just by measuring the sides and angles of a triangle. scientific, graphics or C calculator can be used to obtain the accurate values. Before entering angles you need to make sure that the calculator is in degree mode. ample: Use a calculator to find the value of each of the following, correct to four decimal places. a cos 30 b cos 54 c tan 89 cientific calculator Graphics calculator C calculator a Press sin 30. Press I, type 30) and Press 2nd I, type 30) and press press.. 1 This gives the answer 0.5 or 2. b Press cos 54. Press CO, type 54) and Press 2nd CO, type 54) and press. press. Use to get the decimal approimation. This gives the answer which rounds up to c Press tan 89. Press T, type 89) and Press 2nd T, type 89) and press. press to get the decimal approimation. This gives the answer which rounds up to ercise 1 Use a calculator to find the value of each of the following, correct to four decimal places. a sin 10 b tan 30 c cos 40 d tan 60 e tan 80 f cos 90 g tan 10 h sin 70 i cos 60 j sin 40 k cos 80 l cos 50 2 Use a calculator to find the value of each of the following, correct to three decimal places. a sin 12 b tan 34 c cos 44 d tan 69 e tan 82 f cos 88 g tan 14 h sin 72 i cos 68 j sin 64 k cos 86 l cos ssential Mathematics VL dition Year 9

10 10.2 Finding the side length of a right-angled triangle There are many situations where we need to calculate the side length of a right-angled triangle, for eample the height of a building or the width of a river. Key ideas If the size of the angle and the length of one side of a right-angled triangle are given, the length of any other side can be found using OH CH TO. For eample, for the diagram shown: sin 30 or cos y 6 30º y ample 3 Find in the equation cos 20 correct to two decimal places. 3, olution planation cos cos Multiply both sides of the equation by 3. valuate. ample 4 For each triangle find the value of correct to two decimal places. a b c Chapter 10 Trigonometry 361

11 olution planation a sin 38 O H ince (O) and the (H) are given, the sin ratio must be used. (O) 7 (H) sin sin Multiply both sides by 7. valuate. 38 () b tan 42 O tan 42 4 ince (O) and the () are given, the tan ratio must be used. (H) (O) 4 tan Multiply both sides by 4. valuate () c cos 24 H cos cos ince () and the (H) are given, the cos ratio must be used. Multiply both sides by 10. valuate. (O) () (H) ercise 10B ample 3 1 In each of the following find the value of correct to two decimal places. a sin 20 b cos 43 c 3 7 tan 87 5 ample 4 d tan 39 e sin 16 f cos g cos 8.7 h i sin tan For the triangles given below, find the value of correct to two decimal places. a 1 b c 4 17 d e f 12 g 20 h ssential Mathematics VL dition Year 9

12 i j k 34 l m n o p Determine the height of each of these triangles correct to two decimal places. a b c 15 m m m 4 ndrew walks 3.2 km up a hill which is inclined at 12 to the horizontal. How high (correct to two decimal places) is he above his starting point? 3.2 km 12 5 Leonie wanted to measure the width of a river. he placed two markers, and B, 72 m apart along the bank. C is a point directly opposite marker B. Leonie measured angle CB to be 32. Find the width of the river correct to two decimal places. 32 C width B 72 m 6 One end of a 12.2-m rope is tied to a boat. The other end is tied to an anchor, which is holding the boat steady in the water. If the anchor is making an angle of 34 with the vertical, how deep is the water correct to two decimal places? m 7 n escalator is 11 m long and slopes at an angle of 42 to the horizontal. How high up (correct to two decimal places) is a shopper who is at the top of the escalator? 11 m 42 Chapter 10 Trigonometry 363

13 nrichment Th 8 n isosceles triangle has a base length of 24 cm and base angles of cm a Find the height correct to two decimal places. b Use Pythagoras theorem to find the value of correct to one decimal place. c Find the perimeter and area of the large triangle correct to the nearest cm. 9 In this diagram you should see three right-angled triangles. D 2 B 60 C a b c d Find the length BC correct to two decimal places. Find the length D correct to two decimal places. Find length C correct to one decimal place. Investigate how changing the length DC changes the answers to parts a to c above. 364 ssential Mathematics VL dition Year 9

14 10.3 Further problems involving side lengths hen finding the hypotenuse length or other side length of a triangle, the unknown value sometimes appears in the denominator of the equation. Key ideas If the unknown value of a trigonometric ratio is in the denominator you need to rearrange the equation to make the pronumeral the subject. For eample, for the triangle shown: cos 30 5 which gives 5 cos ample 5 Find in the equation cos 35 2 correct to two decimal places., olution planation cos 35 2 cos cos Multiply both sides of the equation by. Divide both sides of the equation by cos 35. valuate and round off to two decimal places. ample 6 Find the values of the pronumerals correct to two decimal places. a b 28 5 y Chapter 10 Trigonometry 365

15 olution planation a b sin 35 O H sin 35 5 sin sin tan 28 O tan tan tan y y y (O) 5 35 () ince (O) and the () are given, use sin. Multiply both sides of the equation by. Divide both sides of the equation by sin 35. valuate and round off to two decimal places. () ince (O) and () are 28 given use tan. 19 (O) (H) y Multiply both sides of the equation by. Divide both sides of the equation by tan 28. valuate and round off to two decimal places. Find y by using Pythagoras theorem and substitute the eact value of stored in your calculator. () 28 (H) y (H) 19 (O) lternatively, y can be found by using sin. ercise 10C ample 5 1 For each of the following equations find the value of correct to two decimal places. a cos 35 2 b sin 27 3 c tan 11 5 ample 6a d sin e tan 67 f cos g tan 49 h cos i sin 54 2 Find the value of correct to two decimal places. a b c d ssential Mathematics VL dition Year 9

16 e f g 25 h i 14 j k l ample 6b 3 Find the value of each pronumeral correct to one decimal place. a b c d b b a y a y e y f 9.6 g n h y m y 4 ladder is inclined at an angle of 32 to the horizontal. If the ladder reaches 9.2 m up a wall, what is the length of the ladder, correct to one decimal place? 5 kite is flying at a height of 27 m. If the string is inclined at 42 to the horizontal, find the length of the string correct to the nearest metre. 27 m 42 6 glider flying at a height of 800 m descends at an angle of 12 to the horizontal. How far (to the nearest metre) has it travelled in descending to the ground? 12 distance in descent 800 m m mine shaft is dug at an angle of 15 to the horizontal. How far (to the nearest metre): a below ground level is the end of the shaft? b is the end of the shaft horizontally from the opening? ground m Chapter 10 Trigonometry 367

17 8 In the shed shown on the right, how long will one of the sloping timber beams be if they are each inclined at an angle of 24 to the horizontal? Give your answer correct to two decimal places m nrichment Th 9 Lena wants to find the height of a tree. From a distance of 25 metres from the base of the tree she measures the angle shown as a Given that Lena is 1.64 metres tall find the height of the tree correct to two decimal places. b hat effect would it have on the calculated height of the tree if the horizontal distance (25 metres) was overestimated or underestimated by 50 cm? c hat effect would it have on the calculated height of the tree if the angle was overestimated or underestimated by 0.5? d hat do you notice? 10 a For the diagram shown find the value of. 2 b hat is the value of if the given side length is 30 i doubled? ii halved? Describe any pattern you observe. c hat is the value of if the side length is left unchanged but the angle is: i doubled? ii halved? Describe any pattern you observe. 368 ssential Mathematics VL dition Year 9

18 10.4 Finding the angle ometimes it is useful to know the angles in a right-angled triangle. For eample, you may need to know the angle a wire makes with a vertical pole or the bearing a plane might be travelling on. Key ideas Given two side lengths of a right-angled triangle you can find an angle within the triangle. 3 2 If you know which trigonometric ratio is relevant you can use one of the keys below to work out the the angle. cos 1 sin 1 tan 1 To find the angle if sin : on a scientific on a graphics on a C calculator, calculator, press: calculator, press: press: sin nd sin 1 (0.5446) TR sin 1 (0.5446) TR ample 7 Find the value of to the level of accuracy indicated. a sin (nearest degree) b tan 1 (one decimal place) 2 a olution sin sin planation Use the sin 1 key on your calculator. Round off to the nearest whole number. Chapter 10 Trigonometry 369

19 b tan 1 2 tan 1 a 1 2 b Use the tan 1 key on your calculator. Don t forget to close the brackets Round off to one decimal place ample 8 Find the value of to the nearest degree. olution 10 planation 6 sin O H 6 10 sin 1 a 6 10 b 37 (H) 10 6 (O) () ince (O) and the (H) are given, use sin. Use the sin 1 key on your calculator. Round off to the nearest degree. ercise 10D ample 7a 1 Find the value of to the nearest degree. a sin 0.5 b cos 0.5 c tan 1 d tan e sin f sin g cos 1 h tan i cos 0 j tan k sin 1 l cos m cos n tan o cos valuate each of the following to the nearest degree. a sin 1 (0.6884) b cos 1 (0.9763) c tan 1 (0.8541) d sin 1 (0.4305) e tan 1 (1.126) f cos 1 (0.997) g cos 1 (0.1971) h sin 1 (0.1817) i sin 1 (0.7051) 370 ssential Mathematics VL dition Year 9

20 ample 7b 3 Find the angle correct to two decimal places. a sin 3 5 b sin 1 7 c sin 7 8 d cos 1 2 e cos 4 5 f cos 7 9 g tan 2 3 h tan 5 3 i tan 10 4 a The sine of angle is hat is the value of angle to the nearest degree? b The cosine of angle is hat is the value of angle to the nearest degree? 5 hich trigonometric ratio should be used to solve for? a b c d 7 m 9 m 5 m ample 8 6 Find the value of to the nearest degree. a b c d e f 7 g 32 h road rises at a grade of 3 in 10. Find the angle (to the nearest degree) the road makes with the horizontal ramp is 6 m long and 2 m high. Find the angle (correct to two decimal places) the ramp makes with the ground. 6 m 2 m 9 hen a 2.8-m long seesaw is at its maimum height it is 1.1 m off the ground. hat angle (correct to two decimal places) does the seesaw make with the ground? Chapter 10 Trigonometry 371

21 10 dam, who is 1.8 m tall, holds up a plank of wood 4.2 m long. Find the angle that the plank makes with the ground correct one decimal place. 11 children s slide has a length of 5.8 m. The vertical ladder is 2.6 m above the ground. Find the angle the slide makes with the ground correct to one decimal place. 12 The leaning tower of Pisa was 54.6 m tall when it was built in the 12th century. Today the tower is leaning over and it is about 440 cm out of line at the top. Find its inclination to the vertical correct to two decimal places. 1.8 m plank (4.2 m) nrichment Th 13 a If sin cos and is acute what is the value of angle? b For each of the following equations find two different values for if i sin 1 ii cos 1 iii tan c Investigate angles larger than 180 which satisfy the equations in part b. 14 olve each triangle, that is, find the length of all sides and the value of all angles correct to one decimal place. a b ssential Mathematics VL dition Year 9

22 10.5 Mied application problems ome problems may involve finding more than one length or angle. You may, for eample, want to know the length and height of an escalator as well as the angle it makes with the ground. Key ideas To solve application problems involving trigonometry: Draw a diagram and label the key information. Identify and draw the appropriate right-angled triangles separately. olve using trigonometry to find the missing measurements. press your answer in words. ample 9 flagpole is supported by a wire running from the top of the pole to a point on the ground 6.2 m from the base of the pole. If the wire makes an angle of 36 with the ground, find the height of the pole correct to one decimal place. olution Let the height of the flagpole be h metres. planation Define the unknown length. (O) h m (H) Draw a diagram m () tan 36 O tan 36 h 6.2 h 6.2 tan o the height of the flagpole is 4.5 m. ince the opposite side (O) and the adjacent side () are given, use tan. Multiply both sides by 6.2. valuate. press the answer in words. Chapter 10 Trigonometry 373

23 ample 10 plane flying at 1500 m starts to climb at an angle of 15 to the horizontal when the pilot sees a mountain peak 2120 m high, 2400 m away from him in a horizontal direction. ill the pilot clear the mountain? 1500 m O 2400 m M 2120 m P olution planation M P 1500 m 15 H 2400 m O Draw a diagram. The plane will clear the mountain if M ( ) m That is: M 620 m From triangle MP: tan 15 M 2400 M 2400 tan ince M 620 m the plane will clear the mountain peak. ince (O) and () are given use tan. Multiply by Round off to the nearest metre. rite the answer in words. It will be ( ) 23 metres above the mountain peak when it reaches it. ercise 10 ample 9 1 flagpole is supported by a wire running from the top of the pole to a point on the ground 4.6 m from the base of the pole. If the wire makes an angle of 28 with the ground, find the height of the pole correct to two decimal places. 2 large advertising balloon is tied to the roof of a 20-m high building by a 50-m rope which makes an angle of 42 with the horizontal. Find the height of the balloon above the ground correct to two decimal places. 3 ramp for wheelchairs and prams runs from street level to the entrance of a building, which is 0.8 m above street level. How long (correct to two decimal places) is the ramp if it makes an angle of 10 with the horizontal? 374 ssential Mathematics VL dition Year 9

24 4 train travels up a slope, making an angle of 7 with the horizontal. hen the train is at a height of 3 m above its starting point, find the distance it has travelled up the slope, to the nearest metre. 5 Toro the human cannon ball is catapulted into the air at an angle of 70 to the horizontal. hat distance (to the nearest metre) will he have travelled when he has reached a height of 30 m? 6 pendulum on a grandfather clock is 70 cm long and swings through a total angle of 12. hat is the straight-line distance (to the nearest cm) between the etreme positions of the bob on the end of the pendulum as it swings? 7 ski lift travelling up a mountain is inclined at 15 to the horizontal. If the ski lift is 560 m long, how high (to the nearest metre) is the top of the ski lift vertically from the foot of the mountain? 8 Madeline tries to swim across a 40-m-wide river. The current pushes her off course at an angle of 26 to her direct route across the river. How far (to the nearest metre) does she actually swim to reach the other side? 9 ship out at sea observes a lighthouse on top of an 82-m cliff. If the ship is 180 m from the base of the cliff find the value of the observation angle from horizontal to the nearest degree. 10 removalist van has a ramp, which is used to move furniture from ground level to inside the van. If the floor of the van is 1.2 m off the ground and the ramp is 2.4 m in length what angle (to the nearest degree) does the ramp make with the ground? ample n escalator rises 3 metres for every 7 metres horizontally. Give your answer for each of the following correct to one decimal place. a hat angle does the escalator make with the horizontal ground? b If the total height of the escalator is 6 m, how long is the escalator? 12 road has a steady gradient of 1 in 10. a hat angle does the road make with the horizontal? Give your answer to the nearest degree. b car starts from the bottom of the inclined road and drives 2 km along the road. How high vertically is the car? Give your answer correct to the nearest metre and use your answer from part a. 13 plane flying at 1850 m starts to climb at an angle of 18 to the horizontal when the pilot sees a mountain peak 2450 m high, 2600 m away from him in a horizontal direction. ill the pilot clear the mountain? 1850 m 2450 m 2600 m P Chapter 10 Trigonometry 375

25 14 house is to be built using the design shown on the right. The eaves are 600 mm and the house is 7200 mm wide, ecluding the eaves. Calculate the length (to the nearest mm) of a sloping edge of the roof, which is pitched at 25 to the horizontal. 15 garage is to be built with measurements as shown in the diagram on the right. Calculate the sloping length and pitch of the roof if the eaves etend 500 mm on each side. Give your answers correct to the nearest unit. 16 roof has a horizontal span of 10.2 m and a pitch of 24. Find the height of the roof, correct to the nearest millimetre. 17 kite is 160 cm long and 75 cm wide. The shorter edges each make an angle of 60 with the horizontal, as shown in the diagram on the right. Find the value of and y correct to the nearest cm. 18 The chains on a swing are 3.2 m long and the seat is 0.5 m off the ground when it is in the vertical position. hen the swing is pulled as far back as possible, the chains make an angle of 40 with the vertical. How high off the ground, to the nearest cm, is the seat when it is at this etreme position? nrichment 600 mm 2700 mm 75 cm cm mm 3200 mm 160 cm y cm 600 mm 1820 mm Th 19 n aeroplane takes off and climbs at an angle of 20 to the horizontal, at 190 km/h along its flight path for 15 minutes. a Find correct to two decimal places: i the distance the aeroplane travels ii the height the aeroplane reaches b If the angle at which the plane climbs is twice the given angle but its speed is halved, will it reach a greater height after 15 minutes? c If the plane speed is doubled and its climbing angle is halved, will the plane reach a greater height after 15 minutes? 20 The residents of keville live 12 km from an airport. They maintain that any plane flying lower than 4 km disturbs their peace. ach unday they have an outdoor concert from noon till 2.00 pm. a ill a plane taking off from the airport at an angle of 15 over keville disturb the residents? b hen the plane in part a is directly above keville, how far (to the nearest m) has it flown? c If the plane leaves the airport at am on unday and travels at an average speed of 180 km/h, will it disturb the start of the concert? d Investigate what average speed the plane can travel at in order not to disturb the concert. 376 ssential Mathematics VL dition Year 9

26 10.6 ngles of elevation and depression ngles of elevation and depression are measured from the horizontal, for eample the angle to the top of a building from ground level or the angle from the top of a building down to the ground. Key ideas The angle of elevation or depression of a point, Q, from another point, P, is given by the angle the line PQ makes with the horizontal. P line of sight angle of elevation horizontal ngles of elevation or depression are always measured from the horizontal. It is important to note that the angle of elevation of Q from P is equal to the angle of depression of P from Q, because they are alternate angles. Q P P horizontal angle of depression line of sight Q Q ample 11 The angle of elevation of the top of a tower from a point on the ground 30 m away from the base of the tower is 28. Find the height of the tower to the nearest metre. olution Let the height of the tower be h m. h m m angle of elevation planation ince (O) and () are given, use tan. line of sight (H) h m (O) tan 28 O m () h 30 h 30 tan The height is 16 m. Multiply both sides by 30. implify to the nearest metre. rite the answer in words. Chapter 10 Trigonometry 377

27 ample 12 From the top of a vertical cliff ndrea spots a boat out at sea. If the top of the cliff is 42 m above sea level and the boat is 90 m away from the base of the cliff, find ndrea s angle of depression to the nearest degree. olution 42 m 90 m tan O The angle of depression is 25. planation Draw a diagram and label all the given measurements. ince (O) and () are given, use tan. Use the tan 1 key on your calculator and round off to the nearest degree. press the answer in words. ercise 10F ample 11 1 The angle of elevation of the top of a tower from a point on the ground 40 m away from the base of the tower is 36. Find the height of the tower to the nearest metre. 2 The angle of elevation from the point on the ground to the top of a building is 65. If the horizontal distance to the building is 84 m find the height of the building to the nearest metre. h m m angle of elevation m line of sight 3 Tran is 34 m away from a tree and the angle of elevation of the top of the tree from the ground is 53. hat is the height of the tree to the nearest metre? 4 The angle of elevation of the top of a castle wall from a point on the ground 25 m away from the base of the castle wall is 32. Find the height of the castle wall to the nearest metre. 5 From a point on the ground mma measures the angle of elevation of an 80-m tower to be 17. Find how far from the base of the tower mma is, to the nearest metre m 378 ssential Mathematics VL dition Year 9

28 6 From a pedestrian overpass Terry spots a landmark at an angle of depression of 32. How far away (to the nearest metre) is the landmark from the base of the 24-m-high overpass? 7 From a lookout tower David spots a bushfire at an angle of depression of 25. If the lookout tower is 42 m high, how far away (to the nearest metre) is the bushfire from the base of the tower? 8 ngela is 1.5 m tall. How long (correct to one decimal place) is her shadow when the angle of elevation of the un is 58? ample 12 Th 9 From the top of a vertical cliff Bruce spots a swimmer out at sea. If the top of the cliff is 38 m above sea level and the swimmer is 50 m away from the base of the cliff, find Bruce s angle of depression to the nearest degree. 10 From the top of a viewing platform 20 m high a wombat is spotted in the bush below at a horizontal distance of 15 m. Find the angle of depression from the viewing platform to the wombat to the nearest degree. 11 From a ship a person is spotted floating in the sea 200 m away. If the viewing position on the ship is 20 m above sea level find the angle of depression from the ship to person in the sea to the nearest degree. 12 power line is stretched from a pole to the top of a house. The house is 4.1 m high and the power pole is 6.2 m high. The horizontal distance between the house and the power pole is 12 m. Find the angle of elevation of the top of the power pole from the top of the house to the nearest degree. nrichment 13 Chau observes a plane flying directly overhead at a height of 820 m. Twenty seconds later, the angle of elevation of the plane from Chau is 32. a How far (to the nearest metre) did the plane fly in 20 seconds? b hat is the plane s speed in km/h correct to two decimal places? 14 a Bertha observes a stationary hot air balloon hovering at a height of 120 m at an angle of elevation of 24 measured from her eye level. Her eyes are 1.5 m above the ground. i How far away (to the nearest metre) is the balloon along her line of vision? ii How far does Bertha need to walk to be directly underneath the balloon? b Bertha starts walking towards the balloon. fter 20 seconds she stops and looks up at the balloon. The balloon s angle of elevation is now 32. i How far has Bertha walked during this time? ii How much further does Bertha need to walk to be directly under the balloon? Give all answers correct to the nearest unit. Chapter 10 Trigonometry 379

29 10.7 Bearings compass can be used to show the direction in which someone may wish to travel or the bearing of one object from another. Bearings can be epressed as: surveyors bearings true bearings Key ideas urveyors bearings are based on the compass directions north, south, east and west. e usually start at south or north, and move east or west. ach bearing is described as a number of degrees east or west from north or south. e.g. 25 or 30 True bearings describe the angle in a clockwise direction from north. They are written using three digits, for eample 008, 032 or true true 000 true 120 T 090 true To describe the true bearing of an object positioned at from an object positioned at O, we need to start at O, face north then turn clockwise through the required angle to face the object at. O 180 true bearing of from O bearing of O from ample 13 For the following diagram write: a the surveyors bearing b the true bearing 42 olution planation a 42 tart from south and turn 42 towards east ssential Mathematics VL dition Year 9

30 b ngle is 138 T 138 T = 138 ample 14 For the diagram shown write a the true bearing of from O b the true bearing of O from O 120 olution a The bearing of from O is 120 T. b 120 O planation tart at O, face north and turn clockwise until you are facing. tart at, face north and turn clockwise until you are facing O. Therefore the bearing of O from is: (360 60) T 300 T ample 15 bushwalker walks 3 km on a true bearing of 060 T from point to point B. Find how far (correct to one decimal place) east point B is from point. B 60 3 km olution Let the distance travelled towards the east be d km. 3 km 30 d km planation Define the distance required. Draw a triangle. cos 30 d 3 d 3 cos The distance east is 2.6 km. ince the adjacent side () and the hypotenuse (H) are given use cos. Multiply both sides of the equation by 3. valuate and round off to one decimal place. press the answer in words. Chapter 10 Trigonometry 381

31 ample 16 fishing boat starts from point O and sails 75 km on a bearing of 160 to point B. a How far east (to the nearest metre) of its starting point is the boat? b hat is the bearing of O from B? O km B a olution Let the distance travelled towards the east be d km. O km sin 20 d 75 d 75 sin planation Define the distance required. Draw a diagram and label all the given measurements. ince (O) and (H) are given use sin. Multiply both sides of the equation by 75. valuate and round off to the nearest metre. d km B 340 b The boat has travelled 26 km to the east of its starting point. The bearing of O from B is ( )T 340 T rite the answer in words. tart at B, face north then turn clockwise to face O. ercise 10G ample 13 ample 14 1 For each of the following diagrams write: i the surveyor s bearing ii the true bearing a b c d For each diagram shown write: i the true bearing of from O ii the true bearing of O from. a b c d 40 O O O O ssential Mathematics VL dition Year 9

32 ample 15 3 bushwalker walks 4 km on a true bearing of 55 from point to point B. Find how far east point B is from point correct to two decimal places. 55 B 4 km 4 yacht sails 80 km on a true bearing of 048. Find how far east of its starting point the yacht is correct to two decimal places km 5 fter walking due east, then turning and walking due south, a hiker is 4 km 32 from her starting point. Find how far she walked in a southerly direction correct to one decimal place. ample km 6 four-wheel-drive vehicle travels for 32 km on a true bearing of 200. How far west (to the nearest km) of its starting point is it? 7 fishing boat starts from point O and sails 60 km on a true bearing of 140 to point B. a How far east of its starting point is the boat, to the nearest kilometre? b hat is the bearing of O from B to the nearest degree? 8 Two towns, and B, are 12 km apart. The true bearing of B from is 250. a How far is B west of correct to the nearest km? b Find the bearing of from B to the nearest degree. B 12 km 140 O 60 km 250 B 9 cyclist starts from O and rides 7.6 km on a bearing of 20 to point. a How far (to the nearest metre) west has she travelled from her starting point? b hat is the bearing of O from correct to one decimal place? Chapter 10 Trigonometry 383

33 10 submarine travels 720 km on a bearing of 130, then travels 40 km due east. a b How far east (to the nearest km) of its starting point is the submarine? Find how far south (to the nearest km) of its starting point is the submarine. 11 helicopter flies on a bearing of 40 for 210 km, then flies due east for 175 km. How far east (to the nearest km) has the helicopter travelled from its starting point? 12 Christopher walks 5 km south, then walks 36 until he is due east of his starting point. How far is he from his starting point to the nearest kilometre? 13 Two cyclists leave from the same starting point. One cyclist travels due west while the other travels on a bearing of 22. fter travelling for 18 km, the second cyclist is due south of the first cyclist. How far (to the nearest metre) has the first cyclist travelled? nrichment Th 14 plane flies on a bearing of 168 for two hours at an average speed of 310 km/h. How far (to the nearest kilometre): a has the plane travelled? b to the south of its starting point has the plane travelled? c to the east of its starting point has the plane travelled? 15 pilot intends to fly directly to nderly which is 240 km due north of his starting point. The trip usually takes 50 minutes. Due to a storm, the pilot changes course and flies on a true bearing of 320 for 150 km, at an average speed of 180 km/h, to Boleigh. a Find to the nearest kilometre how far i north the plane has travelled from its starting point ii west the plane has travelled from its starting point b How many kilometres is the plane off course? c From Boleigh the pilot flies directly to nderly at 240 km/h. i Compared to the usual route, how many etra kilometres has the pilot travelled in reaching nderly? ii Compared to the usual trip, how many etra minutes did the trip to nderly take? 384 ssential Mathematics VL dition Year 9

34 10.8 Problems involving two triangles Problems involving the solution of two triangles arise in various situations such as viewing a tower from two different positions. Key ideas In some situations it may be necessary to find missing values on two triangles to obtain the answer we are looking for. For eample, to find the value of y in triangle BD shown on the right,you need to: 1 find in triangle BCD 2 find y in triangle BD D y B 10 C ample 17 D Find the values of the pronumerals in the diagram shown, correct to two decimal places. olution D tan tan planation B C 6 Draw triangle BCD; find. ides (O) and () are given so use tan. Multiply both sides by 6. valuate and round to two decimal places. y B 32 6 C y 40 D B sin 40 y y sin 40 y sin Draw triangle BD (use the eact value of stored in your calculator). ides (O) and (H) are given so use sin. Multiply both sides by y. Divide both sides by sin 40. valuate using the stored value of and round to two decimal places. Chapter 10 Trigonometry 385

35 ample 18 M ship (at P) is 9 km due east of a lighthouse (L). The captain takes bearings from two landmarks, M and Q, which are due north and due south of the lighthouse respectively. The bearings of the M and Q from the ship are 44 and 73 respectively. How far apart (correct to two decimal places) are the two landmarks? L Q 44 P km M L olution 46 9 km P tan 46 ML 9 ML 9 tan km L 9 km P 17 tan 17 LQ 9 Q LQ 9 tan km ML LQ km The distance between the two landmarks is km. planation Distance between M and Q ML LQ Draw triangle LMP. ides (O) and () are given so use tan. Multiply both sides by 9. Calculate and round to three decimal places or store the eact answer in your calculator. Draw triangle LPQ. ides (O) and () are given so use tan. Multiply both sides by 9. Calculate and round to three decimal places. Calculate the distance between M and Q and round off to two decimal places. nswer in words. ercise 10H ample 17 1 Find the values of the pronumerals in each diagram, correct to two decimal places. a b c D y y C D B B D C y B C Find the length of C and of BC in each diagram, correct to two decimal places. a B b c C B cm C D C D D 386 ssential Mathematics VL dition Year 9

36 ample 18 3 flagpole is secured by ropes as shown in the diagram on the right. Find correct to one decimal place the a height of the flagpole b distance (BC) from the base of the flagpole to the base of the rope on the right 4 n engineer made the measurements shown when she was designing a P bridge. Find PO and PB correct to two decimal O Q places n observer is 50 m horizontally from a hot air balloon. The angle of elevation to the top of the balloon is 60 and to the bottom of the balloon s basket is 40. Find the height (to the nearest metre) a of B and T b of the balloon and its basket 6 PQ and R are the walls of two buildings which are 100 m apart. Regina is standing at point T, midway between the two buildings. From her eye level the angle of elevation of Q is 20 and the angle of elevation of is 32. Her eyes are 1.5 m above the ground. Calculate correct to one decimal place: Q a the height of the wall QP b the height of the wall R 7 ship (at P) is 24 km due east of a lighthouse (L). The captain takes bearings from two landmarks, M and Q, which are due north and due south of the lighthouse respectively. The bearings of M and Q from the ship are 38 and 64 respectively. How far apart (correct to two decimal places) are the two landmarks? P B D O m M L B 5 m m T 100 m P R T C B Q km Chapter 10 Trigonometry 387

37 8 From the top of a 90-m cliff the angles of depression of two boats in the water, in the same direction, are 25 and 38 respectively. hat is the distance between the two boats to the nearest metre? m 9 shark is observed on the seabed at an angle of depression of 26 from a submarine which is 32 m above the seabed. ome 32 m time later, the same shark is seen from the same position, in the same direction, but at an angle of depression of 34. How far (to the nearest metre) has the shark travelled since the first sighting? 10 n aeroplane starts at point O and flies due east for B 90 km to point. B is a town 60 km from O on a 60 km true bearing of 042. Find the distance (to the 42 nearest km) of the: O C a aeroplane from B when it is at point C as 90 km shown on the diagram b aeroplane from town B when it has reached its destination 11 Vicky cycles 22 km from O to on a bearing of 20, then turns and cycles 45 km due east to B. X is the point on B that is due north of O. Find to one decimal place: a X b OX c OB 12 Town and Town B are 7 km apart on a coastline that runs east west. yacht, C, is at sea on a true bearing of 055 from town, and on a bearing of 325 from town B. Calculate: C a angle CB b the distance of the yacht from town c the distance of the yacht from town B B d the distance of the yacht from the 7 km nearest point on the coast Give all answers correct to one decimal place. 388 ssential Mathematics VL dition Year 9

38 13 innie is standing on the edge of the beach (at ) directly opposite a windsurfer in the water (at ). innie walks 40 m along the edge of the beach, at which point she thinks the windsurfer is at an angle of 60 from where she is standing. 40 m a b How far is innie from the windsurfer? innie walks further along the edge of the beach until the windsurfer is now at an angle of about 30. How much further did she walk? 14 television antenna is on top of a building. From a point on the ground 30 m from the building, the angle of elevation of the top of the antenna is 54 and of the bottom of the antenna is 50. a Find the height of the antenna to the nearest metre. b If antennas over 8 m tall are not allowed, is this antenna too tall? nrichment Th 15 yacht race starts and finishes at point O. The yachts must pass around the outside of the buoys at points O, and B. Buoy is 12 km from O at a true bearing of 042, and buoy B is due north of O and at a true bearing of 325 from. hat is the total length of the race correct to two decimal places? 16 a Two fishing boats, the nchor and the Barrier, leave port at the same time. The nchor travels on a true bearing of 120 for 75 km while the Barrier travels on a true bearing of 170 for 60 km. b i Find how many kilometres east each of the boats has travelled correct to two decimal places. ii Find how many kilometres south each of the boats has travelled correct to two decimal places. The Barrier is in distress and its radio is not working, so the crew decides to release a flare which is visible from a distance of 30 km. ill the distress signal be visible from the nchor? Investigate what the minimum visible flare distance is in this situation. Chapter 10 Trigonometry 389

39 O R K I G Trigonometry Calculating heights or widths Mathematically It is not always possible or practical to measure the height of an object directly, for eample the height of a skyscraper or a tall tree. In this investigation you will use trigonometry and an inclinometer to find such heights. You will need to work in groups of three or four as directed by the teacher. Com ID PL Th ssential Mathematics VL Projects Constructing an inclinometer n inclinometer is an instrument used to measure angles of elevation and depression. Use the diagram below to help you to construct an inclinometer. You will need: a drinking straw a weight a piece of cardboard cut into a semicircle and marked off accurately every 10 string tape The height of a landmark elect a landmark, such as a tall building, a tree or a bridge which you cannot measure the height of directly. The base of the landmark must be accessible. a b c d e f g string drinking straw 90 read off and subtract this angle from 90 to give the angle to be measured weight Clearly describe the situation and draw a diagram to illustrate it. stimate the height of your chosen landmark. Measure the angle of elevation and the horizontal distance from the point where you are standing. Calculate the height of your chosen landmark. (Remember to take into account the height of line of sight your eye level.) Repeat parts a to d for different viewing postions, one closer and two further away from angle of elevation the landmark. Present your results in a table. horizontal distance Find the average of the heights to obtain a more accurate value for the height. Compare this value with the height you estimated earlier. hy didn t you get the same answer for the height each time? hich measurement is the most important, the distance or angle? hat happens if you are out by just a small amount on one or both measurements? Investigate and discuss. eye angle to be measured 390 ssential Mathematics VL dition Year 9

40 Height of an inaccessible landmark Use your inclinometer and your knowledge of trigonometry to determine the height of an object which is inaccessible for eample, a tree, a house or a tower with an inaccessible base. a i Choose two points which are perpendicular to each other. For each position measure the angle of elevation, the distance between the two points and the angles a and b as shown on the b c diagram (along the ground). ii Draw a diagram. a b distance between the two points iii Calculate the height of the landmark, using your collected measurements and the three right-angled triangles in your diagram. Repeat your measurements and calculations from two other directions and at different distances from the landmark. Discuss your results and any possible sources of error. The width of a river a b c d e f g Identify a landmark, P, on a riverbank on the side opposite where you are standing. Mark a point, Q, directly opposite P, on the riverbank on your side of the river. Mark another point, R, a fied distance on the riverbank from Q on your side of the river such that PQR 90. Measure distance QR with a tape measure or other measuring device. Measure angle PRQ. Calculate the approimate width of the river. Repeat your measurements and calculations for two different distances of QR. rite a paragraph discussing your findings and any possible sources of error. ssential Mathematics VL Projects Chapter 10 Trigonometry 391

41 Chapter summary Review Right-angled triangles For right-angled triangles, the basic trigonometric ratios are called sine, cosine and tangent. OH CH TO sin O H cos H tan O H O Using a calculator Before entering angles in degree mode, you need to make sure that the calculator is set for degrees. Finding the angle Given two side lengths of a right-angled triangle you can find an angle within the triangle by using the inverse trigonometric keys on your calculator. cos 1 sin 1 tan 1 ngles of elevation or depression line of sight Q P horizontal angle of depression line of sight P angle of elevation horizontal Q Bearings urveyors bearings use the compass directions north, south, east and west. tart at south or north, and turn east or west. True bearings describe the angle in a clockwise direction from north. 360 true 000 true 270 true 090 true 180 true To describe the bearing of an object positioned at from an object positioned at O, we need to start at O, face north then turn clockwise through the required angle to face the object at. 392 ssential Mathematics VL dition Year 9

42 Multiple-choice questions 1 For the triangle shown sin a B sin c C sin a b a c D sin b sin c c b 2 The value of tan 32 correct to four decimal places is B C D In the diagram the value of correct to two decimal places is 8 40 B C 4.70 D hich of the following could be used to find the value of in the triangle shown? 9 9cos 23 B 23cos 9 C cos 23 9 D The value of correct to four decimal places is B 0.83 C 7.9 D The length of in the triangle is given by 8 sin 46 B 8 cos 46 C 8 D cos 23 8 sin 46 cos 23 sin cos 46 8 cos 46 7 The value of a in the diagram correct to one decimal place is 5.15 B 23.5 C 24.7 D ladder is inclined at an angle of 28 to the horizontal. If the ladder reaches 8.9 m up the wall, the length of the ladder correct to the nearest metre is 19 m B 4 m C 2 m ladder 8.9 m D 33 m 24 m 9 The value of in the diagram correct to two decimal places is 0.73 B C D To calculate the value of you need to evaluate B C sin 1 a 3 tan 1 a 2 tan 1 a 3 2 b 3 b 2 b 8 8 a 3 c b a Review D cos 1 a 3 2 b sin 1 a 2 3 b Chapter 10 Trigonometry 393

43 hort-answer questions Review 1 Find the value of each of the following, correct to two decimal places. a sin 40 b tan 66 c cos 44 2 Find the value of each pronumeral, correct to two decimal places. a b c 14 y ramp runs from street level to the entrance of a building which is 0.7 m above street level. How long is the ramp if it makes an angle of 8 with the horizontal, to one decimal place? 4 The angle of elevation of the top of a lighthouse from a point on the ground 40 m away from its base is 35. Find the height of the lighthouse to two decimal places m 5 train travels up a slope, making an angle of 7 with the horizontal. hen the train is at a height of 3 m above its starting point, find the distance it has travelled up the slope, to the nearest metre. 6 yacht sails 80 km on a true bearing of 048. a How far east of its starting point is the yacht correct to two decimal places? b How far north of its starting point is the yacht correct to two decimal places? 7 From a point on the ground, Geoff measures the angle of elevation of a 120-m tower to be 34. How far from the base of the tower is Geoff, correct to two decimal places? 8 ship leaves Coffs Harbour and sails 320 km east. It then changes direction and sails 240 km due north to its destination. hat will the ship s bearing be from Coffs Harbour when it reaches its destination, correct to two decimal places? 9 From the roof of a skyscraper, isha spots a car at an angle of depression of 51 from the roof of the skyscraper. If the skyscraper is 78 m high how far away is the car from the base of the skyscraper, correct to one decimal place? 10 Penny wants to measure the width of a river. he places two markers, and B, 10 m apart along one bank. C is a point directly opposite marker B. Penny measures angle BC to be 28. Find the width of the river to one decimal place. 11 n aeroplane takes off and climbs at an angle of 15 to the horizontal, at 210 km/h along its flight path for 15 minutes. Find correct to two decimal places: a the distance the aeroplane travels b the height the aeroplane reaches m km 3 m 394 ssential Mathematics VL dition Year 9

44 tended-response questions 1 From the top of a 100-m cliff kevi sees a boat out at sea at an angle of depression of 12. a Draw a diagram for this situation. b Find how far out to sea the boat is to the nearest metre. c swimmer is 2 km away from the base of the cliff and in line with the boat. hat is the angle of depression to the swimmer to the nearest degree? d How far away is the boat from the swimmer to the nearest metre? 2 pilot takes off from mber Island and flies for 150 km at 040 true to Barter Island where she unloads her first cargo. he intends to fly to Dream Island but a bad thunderstorm between Barter and Dream Islands forces her to fly off course for 60 km to Crater toll on a bearing of 060 true before turning on a bearing of 140 true and flying for 100 km until she reaches Dream Island where she unloads her second cargo. he then takes off and flies 180 km on a bearing of 55 true to merald Island. a How many etra kilometres did she fly trying to avoid the storm? Round to the nearest km. b From merald Island she flies directly back to mber Island. How many kilometres did she travel on her return trip? Round to the nearest km. Review MC TT D&D Chapter 10 Trigonometry 395 TT