7. Trigonometry In this chapter you will learn aout: the relationship etween the ratio of the sides in a right-angled triangle solving prolems using the trigonometric ratios finding the lengths of unknown sides in right-angled triangles finding unknown angles in right-angled triangles Get Ready Throughout this chapter you will need: pencil ruler protractor calculator 7.1 Relationship Between Sides in Rightangled Triangles In this section you will e introduced to trigonometry. You will eplore the relationship etween the ratio of the sides in right-angled triangles, and use the terms sin, cos, and tan. Eercise 7.1 You will need a calculator throughout this eercise. 0 1 0 1 0 inch 1 2 3 4 5 6 7 8 9 10 11 12 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 inch 1 2 3 4 5 6 7 8 9 10 11 12 0 inch 1 2 3 4 5 6 7 8 9 10 11 12 4 cm Warm Up 6.5 6.5 cm cm 1 4 cm 4 cm Identifying Relationships Here is a rectangle. 13 cm 10.5 cm a Which of these rectangles are similar to the rectangle aove? 2 cm A 13 13 cm cm B 10.5 10.5 cm cm 8 cm C 2 cm 2 cm 8 cm 3.25 cm 8 cm 8 cm 8 cm 8 cm 3.25 3.25 cm cm Diagrams not drawn to scale 0 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 6.5 cm Math fact: Two shapes are similar if one is an enlargement or reduction of the other. 164 M07_ADEC_SB_09_ARW_8265_U07.indd 164 20/04/2016 11:31
Work out the scale factor for each of the similar rectangles in part a. 2 These two triangles are similar. p a Work out the length of a. Use the angle shown to help you work out which sides are corresponding. Work out the length of. 6 3 4 4 a 72 72 108 108 8 cm 10 cm 3 Eplaining Are these pairs of shapes similar? Eplain your answer. A B C 4 Which of these triangles are right-angled? Use a protractor to check. A B C D r q z y s t u 1 Diagrams not drawn to scale 42 4 cm 3 cm 2 42 5 Write the letter laelling the hypotenuse for each of these right-angled triangles. A B C 6 a Use a protractor, pencil, and ruler to draw a right-angled triangle. Measure the length of each side in cm to 1 d.p. c Measure the two other angles using a protractor. 4 Word fact: The hypotenuse in a right-angled triangle is the longest side and is opposite the right angle. First draw a horizontal line. From one end of the line, fi nd a 90 angle using your protractor. 2 165 M07_ADEC_SB_09_ARW_8265_U07.indd 165 20/04/2016 11:31
4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 96 AC AC AC AC HANDS ON ACtiVITY Main Eercise 7 Work in pairs. You will need: set of laminated triangles lank ratios tale ruler protractor dry-wipe marker 0 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 inch 1 2 3 4 5 6 7 8 9 10 11 12 a Choose one of the triangles and use your protractor to measure each of the angles of the triangle. Use the dry-wipe marker to lael the size of each of the angles on the triangle. Turn the triangle around until the right angle is on the ottom A left-hand side as shown in the diagram. Use a ruler to measure the hypotenuse (AB) in centimeters to 1 d.p. and write the measurement in the tale. C B 0 1 cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 inch 1 2 3 4 5 6 7 8 9 10 11 12 Length AB Length BC Length AC Ratio BC AB c Measure lines AC and BC to 1 d.p. and fi ll in the measurements in the tale. d Find the ratio of sides BC to AB to 2 d.p. and fi ll in the ratios in the tale. To fi nd the ratio of BC to AB, divide the length of BC y the length of AB: BC AB =?? =? e Find the ratio of the sides AC to AB and BC to AC to 2 d.p. Fill in the ratios in the tale. 7.4 Tales of Ratios Length AC Ratio AB AC Ratio BC AB Triangle Length AB Length BC Ratio BC Angle A =, Angle B =, Angle C = Length AC Ratio AB AC Ratio BC AB Triangle Length AB Length BC Ratio BC Angle A =, Angle B =, Angle C = Length AC Ratio AB AC Ratio BC AB Triangle Length AB Length BC Ratio BC Angle A =, Angle B =, Angle C = Length AC Ratio AB AC Ratio BC AB Triangle Length AB Length BC Ratio BC Pearson Education Limited Photocopiales 2016 Ratio AC AB Angle A =, Angle B =, Angle C = 0 1 Ratio BC AC?????? Word fact: Trigonometry is the ranch of mathematics concerned with the relationships etween the sides and angles of triangles. cm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 inch 1 2 3 4 5 6 7 8 9 10 11 12 166 M07_ADEC_SB_09_ARW_8265_U07.indd 166 20/04/2016 11:31
HANDS ON ACtiVITY TALK ABOUT IT TALK ABOUT IT HANDS ON ACtiVITY THINK ABOUT IT 8 a Gathering and Recording Information Repeat steps a e from Q7 with two more triangles from the set. Identifying Relationships Look at your tale. What do you notice is the same and what is different etween these three triangles? c Suggesting Conclusions What can you say aout similar right-angled triangles and the relationship etween their side lengths? d Look at the last triangle in the set. Measure the angles. i Eplaining Is this triangle similar to the other three triangles? Eplain. ii Identifying Relationships Before you measure the sides and fi nd the ratios, what do you think will e the same and what will e different to the other triangles? e Repeat steps a e from Q7 with the last triangle from the set. Do the measurements and ratios agree with your conclusions you made in part d? Discuss with your partner. 9 Suggesting Conclusions Join with two other pairs of students, each with a set of triangles with different sized angles to your set. Compare the three different tales that have een completed. What do you notice? Discuss in your group. Share with your class. What do you notice aout the ratios of the side lengths in each set of triangles? 10 You will need a set triangles with 60 angles and a lank ratios tale from your teacher. a Measure the angles in each triangle and mark on the diagrams. Measure the sides of each of the triangles in centimeters to 1 d.p. and fi ll in the lengths in the tale. c Work out the ratios for each of the pairs of sides. d Are all the triangles right-angled triangles? e Are any of the triangles similar? Eplain. f What do you notice aout the ratios of the sides in the four triangles? 11 a Are the ratios etween the sides the same for right-angled triangles that are not similar? Fatima says, How can I fi nd all the ratios for every set of triangles of different angles? What could you do to fi nd all the ratios needed for right-angled triangles of all the different angles? c What prolem could you have if you tried completing your suggestion in part? 167 M07_ADEC_SB_09_ARW_8265_U07.indd 167 20/04/2016 11:31
12 There is a function on your calculator which can e used instead of learning every ratio tale. a Find the function utton on your calculator. Press, and write down the answer. Now look at the tale of ratios for right-angled triangles with a 30 angle, that your group made in Q9. What do you notice? c sin is short for sine. What do you think sine is? d To check your idea, press and look at the tale of ratios for the 40 triangles. What do you notice? Does this agree with your answer for part c? 13 a Press, and write the answer. Look at the tale of ratios for 30 again. What do you notice? cos is short for cosine. What do you think cosine is? c Check your idea using the tale of ratios for 40. Press. Does the answer agree with your idea? d The last ratio function on your calculator is tan. This is short for tangent. What do you think tangent is? Check your idea using the tale of ratios and your calculator. A right-angled triangle has one angle of 90. The longest side of the triangle (c) is called the hypotenuse. a c The other two sides are named differently depending on y which angle is eing talked aout. For eample, if angle is eing used then the side is called the Opposite ecause it is the side opposite the angle. The side a is called the Adjacent ecause it is the side net to the angle. If angle y was eing used, then the opposite side is a and the adjacent side is. 14 Which side is opposite the angle laelled z? a c a c z z a a c c z 15 Which side is adjacent to the angle laeled y ut is not the hypotenuse? a a c c a y c The longest side net to the y angle is the hypotenuse. c y The shorter side net to the a angle is the adjacent side. Word fact: Opposite means on the other side. Adjacent means net to. 168 M07_ADEC_SB_09_ARW_8265_U07.indd 168 20/04/2016 11:31
The trigonometric ratios are: rememered as sin = side opposite sin = O H SOH hypotenuse cos = side adjacent to cos = H A CAH hypotenuse tan = side opposite tan = O A TOA side adjacent You can state the trigonometric ratios for a given right-angled triangle: sin = 12 15 cos = 15 9 tan = 12 9 15 12 9 16 Copy and lael the sides of these right-angled triangles with hypotenuse (H), opposite (O), and adjacent (A). A B C D E The opposite side and the adjacent side depend on which is the named angle: 17 State the trigonometric ratios for the given right-angled triangles: a sin =? cos =? tan =? sin =? cos =? tan =? c sin =? cos =? tan =? Use SOH, CAH, TOA. A B opposite the angle C Math fact: The symol is often used to represent the angle. is the Greek letter theta. Other Greek letters such as α (alpha) and β (eta) are sometimes used to lael angles. A opposite the angle 3 24 26 5 4 10 B C 5 13 12 169 M07_ADEC_SB_09_ARW_8265_U07.indd 169 20/04/2016 11:31
THINK ABOUT IT When the hypotenuse is 1, the trigonometric ratios of the triangle tell us the lengths of the sides: a 1 = sin 1 = cos cos a a = sin = 1 => 1 1 Scaling can e used to fi nd the length of sides of similar right-angled triangles. If the hypotenuse is douled, the other sides will also e douled: 18 Copy and state the missing trigonometric ratios of these similar triangles: 3 cos? cos 19 Use your calculator to fi nd the following ratios to 1 d.p. a cos 45 sin 50 c cos 72 d tan 42 e sin 25 sin?? 3 5 10 20 a Use your calculator to fi nd the ratio sin 65 to 1 d.p. Draw a right-angled triangle and lael one of the angles as 65. c Lael the triangle PQR where QR is the hypotenuse, and angle R is 65. Lael the length of the hypotenuse as 1. d Eplaining Use the information from a to lael the length of sin = O one of the other sides and eplain your answer. H e Eplaining Use the information from a and d to fi nd the unknown angles of this triangle ABC. Eplain how you know. A 10 This triangle is similar ut the sides are? times igger. B 9 α C f Compare triangle ABC in e with triangle DEF. What is the same and what is different aout their sides and angles? g Think aout what you know aout triangles. If angle is 65, what is angle α? Which math fact did you use to fi nd the angle? 2 cos 2 sin 2?? D E sin Look ack at Q12 to see which keys to press. 20 18 α F 170 M07_ADEC_SB_09_ARW_8265_U07.indd 170 20/04/2016 11:31
21 The right-angled triangles elow are similar. F H d A a B 1 C D 10 21 c Write in your noteook the missing sides and angles a e. What do you know aout similar triangles? What changes and what stays the same? 7 22 Eplaining Does the value of the ratio of side lengths in a right-angled triangle change with the size of the triangle or the size of the angles? Eplain your answer. E I e 46 G 171 M07_ADEC_SB_09_ARW_8265_U07.indd 171 20/04/2016 11:31
THINK ABOUT IT 7.2 Finding the Length of Unknown Sides in Right-angled Triangles In this section you will use trigonometric ratios to fi nd the length of unknown sides in right-angled triangles. You will then use these to solve prolems. Eercise 7.2 You will need a calculator throughout this eercise. Warm Up 1 Use Pythagoras Theorem to fi nd the missing side lengths of these triangles: a c 6 26 15 Math fact: Pythagoras Theorem is a rule for the side lengths of right-angled triangles. It states that a 2 + 2 = c 2, where c is the side length of the hypotenuse, and a and are the two short side lengths. 8 24 2 What information do you need to use Pythagoras Theorem to fi nd a missing side length? 3 Lael the side opposite the known angle (O), the side adjacent to the known angle (A), and the hypotenuse (H) in the following triangles: a c d 53 37 4 Copy and fi ll in the missing letters (O, A, or H) for the trigonometric ratios. a Sine = O? Cosine = H? c Tangent =?? d Copy and complete: The trigonometric ratios can e rememered as: S ine C? T? O pposite A??? H ypotenuse H??? 43 25 72 172 M07_ADEC_SB_09_ARW_8265_U07.indd 172 20/04/2016 11:31
5 State the three trigonometric ratios for each of the given right-angled triangles: a c 4.5 7.5 7.5 19.5 36 6 18 39 6 Use your calculator to fi nd the following ratios to 2 d.p.: a cos 36 sin 45 c cos 62 d tan 50 e sin 22 7 Here is a triangle with hypotenuse of 1: cos sin Copy the triangles elow and fi ll in the missing information. a 1 7 cos?? 13 sin 7 13 c? 21 Main Eercise Eample? 30 Find the type of ratio that is needed to fi nd the missing length in the triangle. adjacent Answer Step 1 Lael the sides you are using. Step 2 Use SOH, CAH, TOA to fi nd which type of ratio to use. We know the Hypotenuse, and are looking for the Adjacent d? 35? 15 Write sin =?, cos =?, tan =?. Math fact: When you only know one side length of a right-angled triangle, there is not enough information to use Pythagoras Theorem to find a missing length. If you know one of the angles other than the right angle, then trigonometry is needed to find the missing side length. SOH CAH TOA 8 Lael the sides you are using with hypotenuse, adjacent, or opposite. Using the memory aid SOH, CAH, TOA, write which type of ratio is needed to fi nd the missing length. a c 12 14 36 39 d e f 14 hypotenuse 12 42 17 37 29 9 53 50 173 M07_ADEC_SB_09_ARW_8265_U07.indd 173 20/04/2016 11:31
Eample Use trigonometric ratios to fi nd the missing length in the triangle. Answer Step 1 Lael the sides you are using. Step 2 Use SOH, CAH, TOA to fi nd which type of ratio to use. Step 3 Write the ratio to fi nd the missing length. Step 4 Rearrange the equation to fi nd the value of. Look at which sides of the triangle are eing used to see which trigonometric ratio you need. SOH CAH TOA sin 30 = O H sin 30 = 14 14 sin 30 = Step 5 Use a calculator to solve. = 7 9 Use trigonometric ratios to fi nd the missing lengths in these triangles to 1 d.p. a c d 39 17 37 10 The Municipality has put a new slide in the playground at the park. The ladder to the top of the slide is 4 meters long and is at a 55 angle from the ground. How high is the top of the slide from the ground? 11 A uilder has an etendale ladder. It etends from 2 m to 4 m to 6 m. The uilder always sets the ladder at 75 from the ground. How far away from the wall does the ladder need to e at each length? Give answers to 2 d.p. a c 2 m 75 14 4 m 75 70 opposite 30 14 30 6 m 75 22 64 hypotenuse 4m 55 174 M07_ADEC_SB_09_ARW_8265_U07.indd 174 20/04/2016 11:31
Eample hypotenuse 14 Answer Step 1 Lael the sides you are using. io n 30 Use trigonometric ratios to find the missing length in the triangle. adjacent Step 2 Use SOH, CAH, TOA to find which type of ratio to use. SOH CAH TOA Step 3 Write the ratio to find the missing length. tan 30 = ut tan 30 = i Step 4 Rearrange the equation to find the value of. The unknown is on the ottom so this will need two steps to make the suject. tan 30 = 12 tan 30 = 12 tr Multiply oth sides y. O A 12 Divide oth sides y tan 30. di s Step 5 Use a calculator to solve. = 12 tan 300 = 20.8 (1 d.p.) 12 Use trigonometric ratios to find the missing lengths in these triangles to 1 d.p. 12 c 29 36 12 42 9 53 50 Solve the following equations to find. Write your answers to 2 d.p. where necessary: 9 0.56 = 18 c 0.78 = 21 e 0.98 = 27 f 0.87 = a 0.68 = d 0.5 = sa le 13 d or a 14 Solve the following equations to find. Write your answers to 2 d.p. 15 26 8 sin 37 = 12 c cos 65 = d sin 72 = 21 e cos 54 = 38 f tan 18 = or a tan 40 = 15 A contractor has made a right angle to frame this 14 13 tf triangle window. Look at the diagram and work out how much wood he needs to make the last part of the frame, laelled. Give your answer to 1 d.p. No 120 cm 44 175 M07_ADEC_SB_09_ARW_8265_U07.indd 175 20/04/2016 11:31
16 The UAE fl agpole is 123 meters high. The fl agpole was measured at an angle of 72 from the ground to the top of the fl agpole at point A. How far away from the ase of the fl agpole is Point A? Give your answer to 1 d.p. 17 A ladder is 1.25 meters away from a wall at a 70 angle. a How long is the ladder? How high up the wall is the ladder? Give your answers to 1 d.p. Always check your answers. Are they reasonale? 1.25 m Look at where the rope makes a right angle with the pole of the tent, it has een marked. The diagram is not looking straight onto the angle. 75 18 Four lengths of rope are needed to peg out a tent. The tent is 1.3 meters high. The rope will need to e pegged at a 50 angle from the ground. What is the shortest length of rope needed for each peg? 19 A set of stairs to the alcony which is 4 meters high, make an angle of 35 with the ground. How far out from the alcony do the stairs reach? Draw a diagram of the prolem and then solve. 20 The angle from the end of the uilding s shadow to the top of the uilding was 67. The uilding is 14 meters high. How long was the shadow? Draw a diagram of the prolem and then solve. The horizontal is the ase line from where you are measuring. This may e the ground or it could e eye height. The angle of elevation is the angle the eyes look up from the horizontal to an oject. The angle of depression is the angle the eyes look down from the horizontal to an oject. If the horizontal is taken at eye height and the height of an oject is eing found, then the height of the person must e taken into account. You need to solve a prolem using a right-angled triangle, then add or sutract the height of the person s eye level. 1.3 m 123 m 50 angle of elevation angle of depression 72 oject horizontal ground level oject A 176 M07_ADEC_SB_09_ARW_8265_U07.indd 176 20/04/2016 11:31
THINK ABOUT IT 21 Look at these different angles and state if they are angles of elevation or angles of depression. a c 22 A clinometer is used to measure the height of three trees. a Work out the height of each tree. Give your answers to 1 d.p. where necessary. 54 42 i 6 m 1.5 m iii 4.5 m 1.5 m ii Eplaining Eplain why the height of eye level needs to e added to the height found. You can use a diagram to eplain. 23 The ladder from a fi re truck reached to the window of a uilding 22 m from the ground. The angle of elevation of the ladder was 70. The height of the fi re truck is 2.9 m. How far away from the uilding was the ottom of the ladder? Give your answer to 2 d.p. 24 A supporting rod measures 3.5 m and, when fi ed to the wall, makes an angle of 61 with the ground. Another, longer rod measures 5.2 m and, when fi ed to the wall, the angle increases y 14. How much further 5.2 m up the wall is the longer rod? Work out the height each rod reaches, fi rst, using the right-angled triangles. 45 10.5 m 1.5 m A clinometer measures the angle of elevation from eye level. To fi nd the total height you will need to add the distance from the ground to eye level after you have found the rest of the height of the tree. h Word fact: A clinometer is used to measure the angle of a slope. You can make a clinometer using a protractor, a piece of string, a weight such as a metal washer, and a straw or pencil. 14 61 3.5 m 177 M07_ADEC_SB_09_ARW_8265_U07.indd 177 20/04/2016 11:31
HANDS ON ACtiVITY 25 For each of these questions, draw a sketch and mark on the measurements, efore calculating. a Jasem is 1.75 m tall. He is standing 20 m from a flagpole. He looks up and measures the angle of elevation to the top of the fl agpole as 40. How high is the fl agpole? A fi shing oat at sea sees a warning light on top of a 62 m cliff at an angle of elevation of 22. How far out to sea is the oat? 26 c Badriya sat at the top of a hill. She used a clinometer to measure the angle of depression to the ottom of the hill. The clinometer measured 36. The hill is 245 meters long. How high up is Badriya sitting? You will need: clinometer measuring tape a Find the height of your school uilding y following these instructions: i Stand 25 m from the uilding. ii Use the clinometer to fi nd the angle of elevation to the top of the uilding. iii Use the clinometer to fi nd the angle of depression to the ase of the uilding. iv Draw a sketch, marking on the angles and measurements. v Use trigonometry to fi nd the height of the uilding. Find the height of a tree in the same way as a. c Find the height of your school fl agpole in the same way as a. Use dotted lines to show the horizontal and vertical lines. 178 M07_ADEC_SB_09_ARW_8265_U07.indd 178 20/04/2016 11:32
7.3 Finding the Size of Unknown Angles in Right-angled Triangles In this section you will use trigonometric ratios to fi nd the size of unknown angles in right-angled triangles. You will then use these to solve prolems. Eercise 7.3 You will need a calculator throughout this eercise. Warm Up 1 Write the trigonometric ratios for the missing length in these triangles: a c 15 47 d e f 13 67 2 For Q1a f use the trigonometric ratios to fi nd the missing lengths to 2 d.p. 3 Copy and complete the trigonometric ratios: sin =,?? =, Adjacent? =??? Adjacent 4 Draw a sketch for each of the following right-angled triangles: a = 45, hypotenuse = 15 cm, adjacent = 8 cm = 52, adjacent = 16 m, opposite = 9 m 58 49 22 14 c = 36, hypotenuse = 26 mm, opposite = 17 mm 8 31 63 d A 3.5 m ladder stands against the wall at a 65 angle of elevation. e A 30 m road has a 35 angle of depression. 5 a For Q4d fi nd how far away from the wall the ase of the ladder is. For Q4e fi nd how much higher the top of the road is than the ottom. 7 Word fact: Inverse means the opposite or reverse of something else. For eample, the inverse of addition is sutraction. The inverse of multiplication is division. 179 M07_ADEC_SB_09_ARW_8265_U07.indd 179 20/04/2016 11:32
THINK ABOUT IT READ ABOUT IT 6 The inverse operation rings us ack to the numer we started with. start 7 operation a Write the inverse operations to get ack to the numer at the start. i 12 + 9 21? 12 ii 12 9 3? 12 iii 5 9 45? 5 iv 36 9 4? 36 Copy and complete: + 4 11 i 8?? 7 8 ii 121?? 11 121 7 Match the functions to the inverse functions: Functions Inverse Functions a + 8 3 A 8 3 3 8 B 3 8 c 3 8 C 8 + 3 inverse operation d + 8 3 D + 3 8 8 An inverse operation can help us fi nd missing information, using what we already know. For eample, we know that a tale was etended y 1.5 m and is now 3.5 m. We can write this as: + 1.5 = 3.5, where is the original length of the tale. To fi nd the length of the tale efore it was etended (), we use the inverse operation ( 1.5) to solve the equation. We can write this as: 3.5 1.5 =. Write the equations and use the inverse function to solve these prolems: a A o of mangos was shared equally etween seven people. Each person has eight mangos. How many were in the o? Four equal lengths of wood make a square frame. The frame has a perimeter of 3.6 m. How long is each piece of wood? c 150 posters were shared equally etween a group to give out. Each person gave out 30 posters. How many people were in the group? 4 end 7 180 M07_ADEC_SB_09_ARW_8265_U07.indd 180 20/04/2016 11:32
THINK ABOUT IT Main Eercise When two lengths of a right-angled triangle are given, then the trigonometric ratios can e used to fi nd a missing angle. To select the appropriate ratio, lael the sides of the triangle in relation to the angle to e found. opposite Use the memory aid SOH, CAH, and TOA to fi nd the appropriate ratio. sin = y h, cos = h, tan = y To fi nd a missing angle the trigonometric ratio must e inverse. These are usually written as sin 1, cos 1, and tan 1. y h adjacent hypotenuse If you know the Opposite and the Adjacent you can inverse the tangent to fi nd the size of the angle: = tan 1 ( O H ) = tan 1 ( y ), in the diagram aove. You can read this as, The angle whose tangent is y. On your calculator the inverses can usually e found aove the sin, cos, and tan uttons: You will need to press the shift key to get the inverse ratio. 9 a If you know the Hypotenuse and the Adjacent, what can you do to fi nd the angle? Write the equation for the angle. If you know the Hypotenuse and the Opposite, what can you do to measure the angle? Write the equation for the angle. Eample Find the missing angle in this triangle: Answer Step 1 Lael the sides where the lengths are known. Step 2 Use SOH, CAH, TOA to fi nd which trigonometric ratio to use. SOH CAH TOA Step 3 Write the ratio. sin = O H Step 4 Fill in the known lengths. sin = 12 13 Step 5 Rearrange and write the inverse ratio to fi nd the missing angle. = sin Step 6 Use the inverse ratio keys on your calculator to fi nd angle to 1 d.p. = 67.4 13 hypotenuse opposite 12 1 12 13 181 M07_ADEC_SB_09_ARW_8265_U07.indd 181 20/04/2016 11:32
TALK ABOUT IT 10 Use the appropriate trigonometric ratios to fi nd the missing angles () in these triangles to 1 d.p: a c d e f 11 Look at the two triangles elow. A 9 15 21 13 9 B 12 12 a Copy the sketches of the two triangles into your noteook. Lael the given sides opposite, adjacent, or hypotenuse. c Eplaining What do you notice aout the laels you have just given? Are they the same or different? Eplain. d Use inverse trigonometric ratios to fi nd the missing angles in each triangle. e Identifying Relationships After fi nding the missing angle in triangle A, how could you fi nd the missing angle in triangle B without using trigonometric ratios? What do you know aout the angles in triangles? What do you know aout congruent triangles? 12 A 6 meter-long ramp rises 0.65 meters. What is the angle of elevation of the ramp? The angle of elevation is the angle measured from the horizontal. 13 The shadow from a tree measures 6.4 meters. The height of the tree is 4.6 meters. What is the angle of elevation to the sun? 9 7 14 12 8 0.65 m 28 9 22 7 6.4 m 6 m 4.6 m 182 M07_ADEC_SB_09_ARW_8265_U07.indd 182 20/04/2016 11:32
14 A 3.8 ladder is against a wall. The ase of the ladder is 2.2 m away from the wall. What is the angle of elevation of the ladder from the ground? 3.8 m 2.2 m 15 A ramp is uilt to rise up 1.2 meters. The ramp is 7 meters long. What is the angle of elevation of the ramp? Draw a diagram to show your answer. 16 The top of the slide is 2.6 meters from the ground. The slide is 4.2 meters long. What is the angle of depression from the top of the slide? Draw a diagram to show your answer. The question asks for the angle of depression. Use a dotted line to show the horizontal. 17 A ladder is 2 meters away from the wall and can reach 3.4 meters up the wall. The ladder etends and can now reach 7.5 meters up the wall. By how much did the angle of elevation of the ladder increase when the ladder etends? You will need to fi nd the difference etween the two angles. 18 Two groups of students are making ramps to race toy cars. The hypotenuse on each ramp is 0.75 m long. Group A made a ramp that rose to a height of 0.45 m. Group B increased the angle of elevation from Group A s ramp y 13. What height did Group B s ramp rise to? 7.5 m 3.4 m You will need to fi nd the angle of elevation of Group A s ramp fi rst. 2 m 2 1 183 M07_ADEC_SB_09_ARW_8265_U07.indd 183 20/04/2016 11:32
HANDS ON ACtiVITY 19 You will need: measuring tape chalk a Work in pairs. Choose three different times in your school day to take measurements to fi nd the lengths of shadows. Copy this tale into your noteook: Time of Day Your Height Length of Your Shadow Angle of Elevation to the Sun???????????? c Measure your height in centimeters. Fill in your height in the Your Height column on your tale. d Find a place outside on a hard surface in the sun. Use chalk to mark this place with an X. e Stand with the sun directly ehind you and mark an arrow net to the X to show which way you are facing. f Have your partner mark the end of your shadow with the chalk. g Use the measuring tape to measure the length of your shadow. Write the measurement in your tale. h Use your height measurement and your shadow length to fi nd the angle of elevation of the sun. Draw a diagram to help you fi nd which inverse trigonometric ratio to use. i Repeat at two other times of the day. 184 M07_ADEC_SB_09_ARW_8265_U07.indd 184 20/04/2016 11:32
Summary Trigonometry is the area of mathematics that relates to the fact that in similar right-angled triangles, the ratios of the side lengths are the same. A right-angled triangle has one angle of 90. The side opposite the right angle is called the hypotenuse and is the longest side of the triangle. The side opposite to the angle eing used is laeled the opposite side. The side net to the angle eing used is laeled the adjacent side. The laels for the opposite side and the adjacent side depend on which angle is eing used. The hypotenuse is always the longest side. The trigonometric ratios are named sine, cosine, and tangent and are shortened to sin, cos, and tan. The trigonometric ratios are : rememered as sin = side opposite sin = O H SOH hypotenuse cos = side adjacent to cos = A H CAH hypotenuse tan = side opposite tan = O A TOA side adjacent When the hypotenuse is 1, scaling can e used to fi nd the length of sides of similar right-angled triangles. The trigonometric ratios can e used to fi nd missing side lengths in right-angled triangles. The horizontal is the ase line from where you are measuring. This may e the ground or it could e eye height. The angle of elevation is the angle the eyes look up from the horizontal to an oject. The angle of depression is the angle the eyes look down from the horizontal to an oject. If the horizontal is taken at eye height and the height of an oject is eing found, then the height of the person must e taken into account. You need to solve a prolem using a right-angled triangle, then add or sutract the height of the person s eye level. A clinometer is used to measure the angle of a slope. A simple clinometer can e made from a protractor, a piece of string, a weight such as a metal washer, and a straw or pencil. The inverse trigonometric ratios can e used to fi nd missing angles in right-angled triangles. To fi nd a missing angle the trigonometric ratio must e inverse. The trigonometric ratios to fi nd missing angles are: sin 1 ( O H ), cos 1 ( A H ) or tan 1 ( O A ). adjacent opposite hypotenuse hypotenuse opposite adjacent 185 M07_ADEC_SB_09_ARW_8265_U07.indd 185 20/04/2016 11:32
Review 1 What do the letters SOH, CAH, TOA stand for? 2 Copy and complete the trigonometric ratios: sin =,?? =, Adjacent tan =? =???? 3 Copy and state the missing trigonometric ratios of these similar triangles: 4 Copy these triangles into your noteook. Lael the sides with hypotenuse (H), opposite (O), and adjacent (A). a c d 5 State the trigonometric ratios for the given right-angled triangle: sin =?? cos =?? tan =?? 10 6 What is the same and what is different etween these two triangles? 7 Lael the sides of these triangles, and then use the appropriate trigonometric ratio to fi nd the missing sides to 2 d.p. a 8 The height from the top step to the fl oor is 2 meters. The stairs rise at a 40 angle. What is the length of 40 the stairs? 9 A ramp connects the ground to a patio 1.5 meters from the ground. The angle of elevation is 34. How long is the ramp. Use a diagram to show your answer. 10 Look at the two triangles elow. Use trigonometric ratios to fi nd the missing angles. a 4 cos? 7.5 cos 4 13 24 12.5 14 cm 55 sin?? 34 A 6 B 45 8.5?? 9 10 12 24 8 cm C D 24 E 34 F 2 m 186 M07_ADEC_SB_09_ARW_8265_U07.indd 186 20/04/2016 11:32
Put it all together Put it all together 11 A rope is 14 meters long. It runs from the top of a 5-meter pole to the ground. What is the angle of elevation from the ground? Write your answer to 2 d.p. 12 Ali is uilding a ramp to his raised patio for his Grandmother s wheelchair to go on. The height of the patio is 30 cm. The ramp length is 120 cm. a Sketch a diagram of the ramp. What is the angle of the ramp from the ground? The recommended angle for wheelchair ramps is 4.8. c What length will the ramp need to e to have an angle of 4.8? d How much longer should Ali s ramp e to make it safe? 13 Jumaa is uilding a frame for a roof. The slope of the roof frame is 3.2 m long. The height if the roof is 2.4 m high. The height, ase, and slope make a right-angled triangle. a Sketch a diagram of the roof frame. What is the angle of elevation of the frame? c What is the length of the ase? d What would the height of the roof frame e if the angle of elevation increased y 15 ut the slope length was still 3.2 m long? e What would the new length of the ase e? I can statements What can you do? I can lael the side opposite to the angle on a right-angled triangle. I can lael the side adjacent to the angle on a right-angled triangle. I can lael the hypotenuse on a right-angled triangle. I know the trigonometric ratios and can rememer them using the memory aid SOH, CAH, TOA. I can write the trigonometric ratios using given information in a right-angled triangle. I can use trigonometric ratios to fi nd missing lengths in a right-angled triangle. I can use a calculator to solve prolems involving trigonometric ratios. I can state which is the horizontal aseline when measuring. I can identify and calculate an angle of elevation. I can identify and calculate an angle of depression. I can use a clinometer to measure the angle of a slope. I can use the inverse trigonometric ratios to fi nd missing angles in right-angled triangles. I can use the inverse function of trigonometric ratios on a calculator to fi nd missing angles in right-angled triangles. 187 M07_ADEC_SB_09_ARW_8265_U07.indd 187 20/04/2016 11:32