1 Pythgors theorem 2 Finding shorter side 3 Solving prolems using Pythgors theorem This chpter will show you how to use Pythgors theorem in right-ngled tringles how to solve prolems using Pythgors theorem how to use trigonometric rtios in right-ngled tringles how to use trigonometry to solve prolems Right-ngled tringles Pythgors theorem Trigonometry Finding shorter side Which rtio to use Solving prolems Solving prolems 4 Trigonometric rtios Using tngent Using sine Using cosine 5 Clculting ngles 6 Using the sine function 7 Using the cosine function 8 Using the tngent function 9 Which rtio to use 10 Solving prolems using trigonometry Wht you should lredy know how to find the squre nd squre root of numer how to round numers to suitle degree of ccurcy Quick check Use your clcultor to evlute the following, giving your nswers to one deciml plce. 1 2.3 2 2 15.7 2 3 0.78 2 4 8 5 260 6 0.5 111
6.1 Pythgors theorem In this section you will lern how to: clculte the length of the hypotenuse in right-ngled tringle Key words hypotenuse Pythgors theorem Pythgors, who ws philosopher s well s mthemticin, ws orn in 580 BC, on the islnd of Smos in Greece. He lter moved to Croton (Itly), where he estlished the Pythgoren Brotherhood, which ws secret society devoted to politics, mthemtics nd stronomy. It is sid tht when he discovered his fmous theorem, he ws so full of joy tht he showed his grtitude to the gods y scrificing hundred oen. Consider squres eing drwn on ech side of right-ngled tringle, with sides 3 cm, 4 cm nd 5 cm. The longest side is clled the hypotenuse nd is lwys opposite the right ngle. Pythgors theorem cn then e stted s follows: For ny right-ngled tringle, the re of the squre drwn on the hypotenuse is equl to the sum of the res of the squres drwn on the other two sides. 25 cm 2 5 cm 4 cm 16 cm 2 3 cm 9 cm 2 The form in which most of your prents would hve lernt the theorem when they were t school nd which is still in use tody is s follows: In ny right-ngled tringle, the squre of the hypotenuse is equl to the sum of the squres of the other two sides. Pythgors theorem is more usully written s formul: c 2 = 2 + 2 Rememer tht Pythgors theorem cn only e used in right-ngled tringles. c Finding the hypotenuse EXMPLE 1 Find the length of the hypotenuse, mrked on the digrm. Using Pythgors theorem gives 2 = 8 2 + 5.2 2 cm 2 = 64 + 27.04 cm 2 = 91.04 cm 2 So = 91.04 = 9.5 cm (1 deciml plce) 8 cm 5.2 cm 112
EXERCISE 6 For ech of the following tringles, clculte the length of the hypotenuse,, giving your nswers to one deciml plce. 9 cm 5 cm 5.1 cm 3 cm In these emples you re finding the hypotenuse. The squres of the two short sides re dded in every cse. 4.8 cm 7 cm 16 cm 13 cm 9 cm 11 cm 15 cm 15 cm 4 cm 3 cm 12 cm 5 cm 8 cm 6 cm The lst three emples give whole numer nswers. Sets of whole numers tht oey Pythgors theorem re clled Pythgoren triples. For emple, 3, 4, 5 5, 12, 13 nd 6, 8, 10. Note tht 6, 8, 10 re respectively multiples of 3, 4, 5. 113
6.2 Finding shorter side In this section you will lern how to: clculte the length of shorter side in right-ngled tringle Key word Pythgors theorem By rerrnging the formul for Pythgors theorem, the length of one of the shorter sides cn esily e clculted. c 2 = 2 + 2 c So, 2 = c 2 2 or 2 = c 2 2 EXMPLE 2 Find the length. is one of the shorter sides So using Pythgors theorem gives 15 cm 11 cm 2 = 15 2 11 2 cm 2 = 225 121 cm 2 = 104 cm 2 So = 104 = 10.2 cm (one deciml plce) EXERCISE 6B For ech of the following tringles, clculte the length, giving your nswers to one deciml plce. 8 cm 17 cm 24 cm 19 cm In these emples you re finding short side. The squre of the other short side is sutrcted from the squre of the hypotenuse in every cse. c d 6.4 cm 9 cm 25 cm 31 cm 114
For ech of the following tringles, clculte the length, giving your nswers to one deciml plce. 17 m 12 m 19 cm 11 cm These emples re miture. Mke sure you comine the squres of the sides correctly. c 17 m d 23 m 9 cm 8.5 cm For ech of the following tringles, find the length mrked. 8 m c d 12 m 13 m 10 m 5 m 30 cm 4 m 40 cm 6.3 Solving prolems using Pythgors theorem In this section you will lern how to: solve prolems using Pythgors theorem Key words 3-D isoceles tringle Pythgors theorem Pythgors theorem cn e used to solve certin prcticl prolems. When prolem involves two lengths only, follow these steps. Drw digrm for the prolem tht includes right-ngled tringle. Look t the digrm nd decide which side hs to e found: the hypotenuse or one of the shorter sides. Lel the unknown side. If it s the hypotenuse, then squre oth numers, dd the squres nd tke the squre root of the sum. 115
If it s one of the shorter sides, then squre oth numers, sutrct the squres nd tke the squre root of the difference. Finlly, round off the nswer to suitle degree of ccurcy. EXMPLE 3 plne leves Mnchester irport heding due est. It flies 160 km efore turning due north. It then flies further 280 km nd lnds. Wht is the distnce of the return flight if the plne flies stright ck to Mnchester irport? First, sketch the sitution. Using Pythgors theorem gives 2 = 160 2 + 280 2 km 2 = 25 600 + 78 400 km 2 280 km = 104 000 km 2 So = 104 000 = 322 km (3 significnt figures) Mnchester 160 km Rememer the following tips when solving prolems. lwys sketch the right-ngled tringle you need. Sometimes, the tringle is lredy drwn for you ut some prolems involve other lines nd tringles tht my confuse you. So identify which right-ngled tringle you need nd sketch it seprtely. Lel the tringle with necessry informtion, such s the length of its sides, tken from the question. Lel the unknown side. Set out your solution s in Emple 3. void short cuts, since they often cuse errors. You gin mrks in your emintion for clerly showing how you re pplying Pythgors theorem to the prolem. Round your nswer off to suitle degree of ccurcy. EXERCISE 6C ldder, 12 metres long, lens ginst wll. The ldder reches 10 metres up the wll. How fr wy from the foot of the wll is the foot of the ldder? 12 m 10 m model footll pitch is 2 metres long nd 0.5 metre wide. How long is the digonl? 116
How long is the digonl of squre with side of 8 metres? ship going from port to lighthouse stems 15 km est nd 12 km north. How fr is the lighthouse from the port? Some pedestrins wnt to get from point X on one rod to point Y on nother. The two rods meet t right ngles. c If they follow the rods, how fr will they wlk? Insted of wlking long the rod, they tke the shortcut, XY. Find the length of the shortcut. How much distnce do they sve? X 33 m 94 m Y mst on silot is strengthened y wire (clled sty), s shown on the digrm. The mst is 10 m tll nd the sty is 11 m long. How fr from the se of the mst does the sty rech? 10 m 11 m ldder, 4 m long, is put up ginst wll. How fr up the wll will it rech when the foot of the ldder is 1 m wy from the wll? When it reches 3.6 m up the wll, how fr is the foot of the ldder wy from the wll? pole, 8 m high, is supported y metl wires, ech 8.6 m long, ttched to the top of the pole. How fr from the foot of the pole re the wires fied to the ground? 8 m 8.6 m How long is the line tht joins the two coordintes (13, 6) nd B(1, 1)? y 6 5 4 3 2 1 0 1 B 2 3 4 5 6 7 8 9 10 11 12 13 14 15 117
The regultion for sfe use of ldders sttes tht: the foot of 5 m ldder must e plced etween 1.6 m nd 2.1 m from the foot of the wll. Wht is the mimum height the ldder cn sfely rech up the wll? Wht is the minimum height the ldder cn sfely rech up the wll? Is the tringle with sides 7 cm, 24 cm nd 25 cm, right-ngled tringle? 24 cm 7 cm 25 cm Pythgors theorem nd isosceles tringles This section shows you how to to use Pythgors theorem in isosceles tringles. Every isosceles tringle hs line of symmetry tht divides the tringle into two congruent right-ngled tringles. So when you re fced with prolem involving n isosceles tringle, e wre tht you re quite likely to hve to split tht tringle down the middle to crete right-ngled tringle which will help you to solve the prolem. EXMPLE 4 Clculte the re of this tringle. It is n isosceles tringle nd you need to clculte its height to find its re. First split the tringle into two right-ngled tringles to find its height. Let the height e. Then, using Pythgors theorem, 2 = 7.5 2 3 2 cm 2 = 56.25 9 cm 2 = 47.25 cm 2 So = 47.25 cm = 6.87 cm 7.5 cm 6 cm 7.5 cm 7.5 cm 3 cm Keep the ccurte figure in the clcultor memory. The re of the tringle is 1 2 6 6.87 cm 2 (from the clcultor memory), which is 20.6 cm 2 (1 deciml plce) 118
EXERCISE 6D Clculte the res of these isosceles tringles. c 9 cm 9 cm 3 cm 7 cm 10 cm 8 cm 2 cm Clculte the re of n isosceles tringle whose sides re 8 cm, 8 cm nd 6 cm. Clculte the re of n equilterl tringle of side 6 cm. n isosceles tringle hs sides of 5 cm nd 6 cm. Sketch the two different isosceles tringles tht fit this dt. Which of the two tringles hs the greter re? Sketch regulr hegon, showing ll its lines of symmetry. Clculte the re of the hegon if its side is 8 cm. Clculte the re of hegon of side 10 cm. Clculte the lengths mrked in these isosceles tringles. c 10 cm 12 cm 13 cm 24 cm 12 cm 20 cm Find the re first. Pythgors theorem in three dimensions This section shows you how to solve prolems in 3-D using Pythgors theorem. In your GCSE emintions, there my e questions which involve pplying Pythgors theorem in 3-D situtions. Such questions re usully ccompnied y clerly lelled digrms, which will help you to identify the lengths needed for your solutions. 119
You del with these 3-D prolems in ectly the sme wy s 2-D prolems. Identify the right-ngled tringle you need. Redrw this tringle nd lel it with the given lengths nd the length to e found, usully or y. From your digrm, decide whether it is the hypotenuse or one of the shorter sides which hs to e found. Solve the prolem, rounding off to suitle degree of ccurcy. EXMPLE 5 Wht is the longest piece of stright wire tht H cn e stored in this o mesuring 30 cm y 15 cm y 20 cm? The longest distnce cross this o is ny one 30 cm D C of the digonls G, DF, CE or HB. E Let us tke G. 15 cm First, identify right-ngled tringle contining G nd drw it. B This gives tringle FG, which contins two lengths you do not know, G nd F. Let G = nd F = y. G 20 cm G F cm y cm 15 cm F Net identify right-ngled tringle tht contins the side F nd drw it. 20 cm y cm B 30 cm F This gives tringle BF. You cn now find F. By Pythgors theorem y 2 = 30 2 + 20 2 cm 2 y 2 = 1300 cm 2 (there is no need to find y) Now find G using tringle FG. By Pythgors theorem 2 = y 2 + 15 2 cm 2 2 = 1300 + 225 = 1525 cm 2 So = 39.1 cm (1 deciml plce) So, the longest stright wire tht cn e stored in the o is 39.1 cm. 120
Note tht in ny cuoid with sides, nd c, the length of digonl is given y «««««( 2 + «««««2 + c 2 ) c EXERCISE 6E o mesures 8 cm y 12 cm y 5 cm. H G Clculte the lengths of the following. i C ii BG iii BE Clculte the digonl distnce BH. E D 8 cm F B C 12 cm 5 cm grge is 5 m long, 3 m wide nd 3 m high. Cn 7 m long pole e stored in it? Spike, spider, is t the corner S of the wedge shown in the digrm. Fred, fly, is t the corner F of the sme wedge. Clculte the two distnces Spike would hve to trvel to get to Fred if she used the edges of the wedge. S R 12 cm E P F 4 cm Q 8 cm Clculte the distnce Spike would hve to trvel cross the fce of the wedge to get directly to Fred. Fred is now t the top of ked-ens cn nd Spike is directly elow him on the se of the cn. To ctch Fred y surprise, Spike tkes digonl route round the cn. How fr does Spike trvel? Imgine the cn opened out flt. F 10 cm S 6 cm corridor is 3 m wide nd turns through right ngle, s in the digrm. Wht is the longest pole tht cn e crried long the corridor horizontlly? 3 m If the corridor is 3 m high, wht is the longest pole tht cn e crried long in ny direction? 3 m 121
The digrm shows squre-sed pyrmid with se length 8 cm nd sloping edges 9 cm. M is the mid-point of the side B, X is the mid-point of the se, nd E is directly ove X. E 9 cm c Clculte the length of the digonl C. Clculte EX, the height of the pyrmid. Using tringle BE, clculte the length EM. D M X B 8 cm C The digrm shows cuoid with sides of 40 cm, 30 cm, nd 22.5 cm. M is the mid-point of the side FG. Clculte (or write down) these lengths, giving your nswers to three significnt figures if necessry. E H F M G 40 cm H G c M d HM D C 22.5 cm B 30 cm 6.4 Trigonometric rtios In this section you will lern how to: use the three trigonometric rtios Key words djcent side cosine hypotenuse opposite side sine tngent trigonometry Trigonometry is concerned with the clcultion of sides nd ngles in tringles, nd involves the use of three importnt rtios: sine, cosine nd tngent. These rtios re defined in terms of the sides of right-ngled tringle nd n ngle. The ngle is often written s θ. In right-ngled tringle the side opposite the right ngle is clled the hypotenuse nd is the longest side the side opposite the ngle θ is clled the opposite side 122
the other side net to oth the right ngle nd the ngle θ is clled the djcent side. Hypotenuse (H) Opposite (O) djcent () The sine, cosine nd tngent rtios for θ re defined s Opposite djcent sine θ = cosine θ = tngent θ = Hypotenuse Hypotenuse These rtios re usully revited s Opposite djcent O sin θ = cos θ = tn θ = H H These revited forms re lso used on clcultor keys. Memorising these formule my e helped y mnemonic such s Silly Old Hitler Couldn t dvnce His Troops Over fric in which the first letter of ech word is tken in order to give O S = C = T = H H O O Using your clcultor You cn use your clcultor to find the sine, cosine nd tngent of ny ngle. To find the sine of n ngle, press the key lelled sin. To find the cosine of n ngle, press the key lelled cos. To find the tngent of n ngle, press the key lelled tn. Mke sure you cn find sin, cos nd tn on your clcultor. Importnt: Mke sure your clcultor is working in degrees. Depending on the type of clcultor used, you need to put it into degree mode efore you strt working on sines, cosines nd tngents. This cn e done either y using the o utton or y pressing the key ZDRG until DEG is on disply. Try this now to mke sure you cn do it. When it is in degree mode, D or DEG ppers on the clcultor disply. 123
EXMPLE 6 Use your clcultor to find the sine of 27, written s sin 27. On some scientific clcultors, the function is keyed in numers first 2 7 ß The disply should red 0.453990499. (You my hve more or fewer digits depending on your clcultor.) This is 0.454 to three significnt figures. If you hve grphics clcultor or n lgeric logic (DL) clcultor, you key in the function s it reds ß 2 7 You should get the sme vlue s ove. If you don t, then consult your clcultor mnul or your techer. EXMPLE 7 Use your clcultor to find the cosine of 56, written s cos 56. cos 56 = 0.559 192 903 = 0.559 to 3 significnt figures Check tht you gree with this, using s mny digits s your clcultor llows. EXMPLE 8 Use your clcultor to work out 3 cos 57, written s 3 cos 57. Depending on your type of clcultor, key in either 3 5 7 ç = or 3 ç 5 7 = Check tht you get n nswer of 1.633 917 105 = 1.63 to 3 significnt figures. EXERCISE 6F Find these vlues, rounding off your nswers to three significnt figures. sin 43 sin 56 c sin 67.2 d sin 90 e sin 45 f sin 20 g sin 22 h sin 0 Find these vlues, rounding off your nswers to three significnt figures. cos 43 cos 56 c cos 67.2 d cos 90 e cos 45 f cos 20 g cos 22 h cos 0 From your nswers to questions 1 nd 2, wht ngle hs the sme vlue for sine nd cosine? 124
i Wht is sin 35? ii Wht is cos 55? i Wht is sin 12? ii Wht is cos 78? c i Wht is cos 67? ii Wht is sin 23? d Wht connects the vlues in prts, nd c? e Copy nd complete these sentences. i ii sin 15 is the sme s cos cos 82 is the sme s sin iii sin is the sme s cos Use your clcultor to work out the vlues of tn 43 tn 56 c tn 67.2 d tn 90 e tn 45 f tn 20 g tn 22 h tn 0 Use your clcultor to work out the vlues of the following. sin 73 cos 26 c tn 65.2 d sin 88 e cos 35 f tn 30 g sin 28 h cos 5 Wht is so different out tn compred with oth sin nd cos? Use your clcultor to work out the vlues of the following. 5 sin 65 6 cos 42 c 6 sin 90 d 5 sin 0 Use your clcultor to work out the vlues of the following. 5 tn 65 6 tn 42 c 6 tn 90 d 5 tn 0 Use your clcultor to work out the vlues of the following. 4 sin 63 7 tn 52 c 5 tn 80 d 9 cos 8 Use your clcultor to work out the vlues of the following. 5 sin 63 6 cos 32 6 sin 90 c d 5 sin 30 Use your clcultor to work out the vlues of the following. 3 tn 64 7 tn 42 5 tn 89 c d 6 tn 40 Use your clcultor to work out the vlues of the following. 19 8 sin 75 c 7 cos 71 d sin 23 15 sin 81 125
Use your clcultor to work out the vlues of the following. 19 8 tn 75 c 7 tn 71 d tn 23 15 tn 81 Using the following tringles clculte sin, cos, nd tn. Leve your nswers s frctions. c 3 12 13 4 5 5 7 24 25 6.5 Clculting ngles In this section you will lern how to: use the trigonometric rtios to clculte n ngle Key word inverse functions The sine of 54 is 0.809 016 994 4 (to 10 deciml plces). The sine of 55 is 0.819 152 044 3 (to 10 deciml plces). Wht ngle hs sine of 0.815? Oviously, it is etween 54 nd 55, so we could proly use tril-nd-improvement method to find it. But there is n esier wy which uses the inverse functions on your clcultor. n inverse function cn e ccessed in severl different wys. For emple, the inverse function for sine my e ny of these keys: @ ß s ß I ß The inverse function printed ove the sine key is usully given in either of the following wys: sin 1 rcsin ß or ß You will need to find out how your clcultor dels with inverse functions. When you do the inverse sine of 0.815, you should get 54.587 361 89. It is norml in trigonometry to round off ngles to one deciml plce. So, the ngle with sine of 0.815 is 54.6 (1 deciml plce). This cn e written s sin 1 0.815 = 54.6. 126
EXMPLE 9 Find the ngle with cosine of 0.654. cos 1 0.654 = 49.156 131 92 = 49.2 (1 deciml plce) EXMPLE 10 Find the ngle with sine of (3 4). How you solve this will depend on your type of clcultor. So key in either 3 / 4 = I ß or I ß ( 3 / 4 ) = So sin 1 ( 3 4 ) = 48.590 377 89 = 48.6 (1 deciml plce) EXMPLE 11 Find the ngle with tngent of 0.75. tn 1 0.75 = 36.869 897 65 = 36.9 (1 deciml plce) EXERCISE 6G Use your clcultor to find the nswers to the following. Give your nswers to one deciml plce. Wht ngles hve the following sines? 0.5 0.785 c 0.64 d 0.877 e 0.999 f 0.707 Wht ngles hve the following cosines? 0.5 0.64 c 0.999 d 0.707 e 0.2 f 0.7 Wht ngles hve the following tngents? 0.6 0.38 c 0.895 d 1.05 e 2.67 f 4.38 Wht ngles hve the following sines? 4 5 2 3 c 7 10 d 5 6 e 1 24 f 5 13 Wht ngles hve the following cosines? 4 5 2 3 c 7 10 d 5 6 e 1 24 f 5 13 Wht ngles hve the following tngents? 3 5 7 9 c 2 7 d 9 5 e 11 7 f 6 5 127
Wht hppens when you try to find the ngle with sine of 1.2? Wht is the lrgest vlue of sine you cn put into your clcultor without getting n error when you sk for the inverse sine? Wht is the smllest? i Wht ngle hs sine of 0.3? (Keep the nswer in your clcultor memory.) ii Wht ngle hs cosine of 0.3? iii dd the two ccurte nswers of prts i nd ii together. Will you lwys get the sme nswer to the ove no mtter wht numer you strt with? 6.6 Using the sine function In this section you will lern how to: find lengths of sides nd ngles in right-ngled tringles using the sine function Key word sine Rememer sine θ = Opposite Hypotenuse We cn use the sine rtio to clculte the lengths of sides nd ngles in right-ngled tringles. Opposite Hypotenuse EXMPLE 12 Find the ngle θ, given tht the opposite side is 7 cm nd the hypotenuse is 10 cm. Drw digrm. (This is n essentil step.) From the informtion given, use sine. O 7 sin θ = = = 0.7 H 10 10 cm 7 cm Wht ngle hs sine of 0.7? To find out, use the inverse sine function on your clcultor. sin 1 0.7 = 44.4 (1 deciml plce) 128
EXMPLE 13 Find the length of the side mrked in this tringle. Side is the opposite side, with 12 cm s the hypotenuse, so use sine. sin θ = sin 35 = 12 So = 12 sin 35 = 6.88 cm (3 significnt figures) 12 cm O H 35 EXMPLE 14 Find the length of the hypotenuse, h, in this tringle. Note tht lthough the ngle is in the other corner, the opposite side is gin given. So use sine. O sin θ = H 8 sin 52 = h 8 So h = = 10.2 cm (3 significnt figures) sin 52 h 8 cm 52 EXERCISE 6H Find the ngle mrked in ech of these tringles. c 10 cm 3 cm 8 cm 8 cm 3 cm 15 cm Find the side mrked in ech of these tringles. c 13 cm 8 cm 46 24 32 25 cm Find the side mrked in ech of these tringles. c 3 cm 61 59 cm 41 6 cm 36 129
Find the side mrked in ech of these tringles. c d 7 cm 47 55 8 cm 64 13 cm 75 15 cm Find the vlue of in ech of these tringles. c d 15 cm 11 cm 9 cm 17 37 8 cm 13 cm 4 cm ngle θ hs sine of 3 5. Clculte the missing lengths in these tringles. 10 50 9 6.7 Using the cosine function In this section you will lern how to: find lengths of sides nd ngles in right-ngled tringles using the cosine function Key word cosine Rememer cosine θ = djcent Hypotenuse We cn use the cosine rtio to clculte the lengths of sides nd ngles in right-ngled tringles. djcent Hypotenuse 130
EXMPLE 15 Find the ngle θ, given tht the djcent side is 5 cm nd the hypotenuse is 12 cm. Drw digrm. (This is n essentil step.) From the informtion given, use cosine. 5 cos θ = = H 12 Wht ngle hs cosine of 5 12? To find out, use the inverse cosine function on your clcultor. cos 1 = 65.4 (1 deciml plce) 5 cm 12 cm EXMPLE 16 Find the length of the side mrked in this tringle. Side is the djcent side, with 9 cm s the hypotenuse, so use cosine. cos θ = cos 47 = H 9 So = 9 cos 47 = 6.14 cm (3 significnt figures) 9 cm 47 EXMPLE 17 Find the length of the hypotenuse, h, in this tringle. The djcent side is given. So use cosine. cos θ = cos 40 = So h = H 20 h 20 cos 40 = 26.1 cm (3 significnt figures) 20 cm h 40 131
EXERCISE 6I Find the ngle mrked in ech of these tringles. c 8 cm 4 cm 1 cm 5 cm 100 cm 160 cm Find the side mrked in ech of these tringles. c 9 cm 42 cm 39 44 12 140 cm Find the side mrked in ech of these tringles. c 69 14 cm 35 6 cm 125 cm 22 Find the side mrked in ech of these tringles. c d 8 cm 36 12 cm 11 cm 24 52 14 cm 48 Find the vlue of in ech of these tringles. c 6.5 cm d 49 10 cm 16 cm 13 cm 17 cm 56 11 cm ngle θ hs cosine of 5 13. Clculte the missing lengths in these tringles. 26 15 6.5 132
6.8 Using the tngent function In this section you will lern how to: find lengths of sides nd ngles in right-ngled tringles using the tngent function Key word tngent Rememer tngent θ = Opposite djcent We cn use the tngent rtio to clculte the lengths of sides nd ngles in right-ngled tringles. Opposite djcent EXMPLE 18 Find the ngle θ, given tht the opposite side is 3 cm nd the djcent side is 4 cm. Drw digrm. (This is n essentil step.) From the informtion given, use tngent. O 3 tn θ = = = 0.75 4 Wht ngle hs tngent of 0.75? To find out, use the inverse tngent function on your clcultor. tn 1 0.75 = 36.9 (1 deciml plce) 3 cm 4 cm EXMPLE 19 Find the length of the side mrked in this tringle. Side is the opposite side, with 9 cm s the djcent side, so use tngent. tn θ = tn 62 = O 9 So = 9 tn 62 = 16.9 cm (3 significnt figures) 9 cm 62 133
EXMPLE 20 Find the length of the side mrked in this tringle. Side is the djcent side nd the opposite side is given. So use tngent. tn θ = tn 35 = So = O 6 6 tn 35 = 8.57 cm (3 significnt figures) 35 6 cm EXERCISE 6J Find the ngle mrked in ech of these tringles. c 6 cm 20 cm 9 cm 15 cm 35 cm 45 cm Find the side mrked in ech of these tringles. c 34 45 cm 23 12 cm 300 cm 75 Find the side mrked in ech of these tringles. c 40 3 cm 200 cm 20 52 cm 70 Find the side mrked in ech of these tringles. c d 61 5 cm 43 7 cm 11 cm 33 6 cm 34 134
Find the vlue in ech of these tringles. c d 52 63 7 cm 8 cm 9 cm 9 cm 4 cm 3.5 cm ngle θ hs tngent of 4 3. Clculte the missing lengths in these tringles. 9 16 1.5 6.9 Which rtio to use In this section you will lern how to: decide which trigonometric rtio to use in right-ngled tringle Key words sine cosine tngent The difficulty with ny trigonometric prolem is knowing which rtio to use to solve it. The following emples show you how to determine which rtio you need in ny given sitution. EXMPLE 21 Find the length of the side mrked in this tringle. TO PGE 136 Step 1 Identify wht informtion is given nd wht needs to e found. Nmely, is opposite the ngle nd 16 cm is the hypotenuse. 16 cm Step 2 Decide which rtio to use. Only one rtio uses opposite nd hypotenuse: sine. 37 O Step 3 Rememer sin θ = H Step 4 Put in the numers nd letters: sin 37 = 16 Step 5 Rerrnge the eqution nd work out the nswer: = 16 sin 37 = 9.629 040 371 cm Step 6 Give the nswer to n pproprite degree of ccurcy: = 9.63 cm (3 significnt figures) 135
In relity, you do not write down every step s in Emple 21. Step 1 cn e done y mrking the tringle. Steps 2 nd 3 cn e done in your hed. Steps 4 to 6 re wht you write down. Rememer tht eminers will wnt to see evidence of working. ny resonle ttempt t identifying the sides nd using rtio will proly get you some method mrks ut only if the frction is the right wy round. The net emples re set out in wy tht requires the minimum mount of working ut gets mimum mrks. EXMPLE 22 Find the length of the side mrked in this tringle. 7 cm 50 Mrk on the tringle the side you know (H) nd the side you wnt to find (). Recognise it is cosine prolem ecuse you hve nd H. So cos 50 = 7 = 7 cos 50 = 4.50 cm (3 significnt figures) H 7 cm 50 EXMPLE 23 Find the ngle mrked in this tringle. 15 cm 9 cm Mrk on the tringle the sides you know. Recognise it is sine prolem ecuse you hve O nd H. 9 So sin = = 0.6 15 = sin 1 0.6 = 36.9 (1 deciml plce) H 15 cm O 9 cm 136
EXMPLE 24 Find the ngle mrked in this tringle. 12 cm 7 cm Mrk on the tringle the sides you know. Recognise it is tngent prolem ecuse you hve O nd. 12 So tn = 7 1 12 = tn 7 = 59.7 (1 deciml plce) O 12 cm 7 cm EXERCISE 6K Find the length mrked in ech of these tringles. c 20 40 39 48 50 50 d e f 20 40 52 37 76 5 Find the ngle mrked in ech of these tringles. c 20 70 14 60 50 50 d e f 20 100 4 52 13 5 137
Find the ngle or length mrked in ech of these tringles. 5 c 10 12 62 120 26 d 34 e 39 16 25 f g h 56 78 54 45 80 50 i j 59 230 23 82 In mths tetook it sys: The tngent of ny ngle is equl to the sine of the ngle divided y the cosine of the ngle. Show clerly tht this is true for n ngle of 30. Prove, y using the definitions of sin θ nd cos θ, tht the sttement is true for this right-ngled tringle. c 138
6.10 Solving prolems using trigonometry In this section you will lern how to: solve prcticl prolems using trigonometry solve prolems using n ngle of elevtion or n ngle of depression solve ering prolems using trigonometry using trigonometry to solve prolems involving isosceles tringles Key words ngle of depression ngle of elevtion ering isosceles tringle three-figure ering trigonometry Most trigonometry prolems in GCSE emintion ppers do not come s strightforwrd tringles. Usully, solving tringle is prt of solving prcticl prolem. You should follow these steps when solving prcticl prolem using trigonometry. Drw the tringle required. Put on the informtion given (ngles nd sides). Put on for the unknown ngle or side. Mrk on two of O, or H s pproprite. Choose which rtio to use. Write out the eqution with the numers in. Rerrnge the eqution if necessry, then work out the nswer. Give your nswer to sensile degree of ccurcy. nswers given to three significnt figures or to the nerest degree re cceptle in ems. EXMPLE 25 window clener hs ldder which is 7 m long. The window clener lens it ginst wll so tht the foot of the ldder is 3 m from the wll. Wht ngle does the ldder mke with the wll? Drw the sitution s right-ngled tringle. Then mrk the sides nd ngle. H 7 7 Recognise it is sine prolem ecuse you hve O nd H. 3 So sin = 7 = sin 1 3 3 3 O 7 = 25 (to the nerest degree) 139
EXERCISE 6L In these questions, give nswers involving ngles to the nerest degree. ldder, 6 m long, rests ginst wll. The foot of the ldder is 2.5 m from the se of the wll. Wht ngle does the ldder mke with the ground? The ldder in question 1 hs sfe ngle with the ground of etween 60 nd 70. Wht re the sfe limits for the distnce of the foot of the ldder from the wll? nother ldder, of length 10 m, is plced so tht it reches 7 m up the wll. Wht ngle does it mke with the ground? Yet nother ldder is plced so tht it mkes n ngle of 76 with the ground. When the foot of the ldder is 1.7 m from the foot of the wll, how high up the wll does the ldder rech? Clculte the ngle tht the digonl mkes with the long side of rectngle which mesures 10 cm y 6 cm. This digrm shows frme for ookcse. 0.9 m Wht ngle does the digonl strut mke with the long side? Use Pythgors theorem to clculte the length of the strut. 1.9 m This digrm shows roof truss. Wht ngle will the roof mke with the horizontl? 1.6 m Use Pythgors theorem to clculte the length of the sloping strut. 4.5 m lici pces out 100 m from the se of church. She then mesures the ngle to the top of the spire s 23. How high is the church spire? 23 100 m girl is flying kite on string 32 m long. The string, which is eing held t 1 m ove the ground, mkes n ngle of 39 with the horizontl. How high is the kite ove the ground? 32 m 39 140 1 m
ngle θ hs sine of 3 5. Use Pythgors theorem to clculte the missing side of this tringle. 5 3 Write down the cosine nd the tngent of θ. c Clculte the missing lengths mrked in these tringles. i ii iii 2.5 16 40 ngles of elevtion nd depression When you look up t n ircrft in the sky, the ngle through which your line of sight turns from looking stright hed (the horizontl) is clled the ngle of elevtion. When you re stnding on high point nd look down t ot, the ngle through which your line of sight turns from looking stright hed (the horizontl) is clled the ngle of depression. Line of sight Horizontl ngle of depression ngle of elevtion Line of sight Horizontl EXMPLE 26 From the top of verticl cliff, 100 m high, ndrew sees ot out t se. The ngle of depression from ndrew to the ot is 42. How fr from the se of the cliff is the ot? The digrm of the sitution is shown in figure i. From this, you get the tringle shown in figure ii. i ii 42 42 100 m O 100 m From figure ii, you see tht this is tngent prolem. 100 So tn 42 = 100 = = 111 m (3 significnt figures) tn 42 141
EXERCISE 6M In these questions, give ny nswers involving ngles to the nerest degree. Eric sees n ircrft in the sky. The ircrft is t horizontl distnce of 25 km from Eric. The ngle of elevtion is 22. How high is the ircrft? pssenger in n ircrft hers the pilot sy tht they re flying t n ltitude of 4000 m nd re 10 km from the irport. If the pssenger cn see the irport, wht is the ngle of depression? mn stnding 200 m from the se of television trnsmitter looks t the top of it nd notices tht the ngle of elevtion of the top is 65. How high is the tower? From the top of verticl cliff, 200 m high, ot hs n ngle of depression of 52. How fr from the se of the cliff is the ot? From ot, the ngle of elevtion of the foot of lighthouse on the edge of cliff is 34. If the cliff is 150 m high, how fr from the se of the cliff is the ot? If the lighthouse is 50 m high, wht would e the ngle of elevtion of the top of the lighthouse from the ot? ird flies from the top of 12 m tll tree, t n ngle of depression of 34, to ctch worm on the ground. How fr does the ird ctully fly? How fr ws the worm from the se of the tree? Sunil stnds out 50 m wy from uilding. The ngle of elevtion from Sunil to the top of the uilding is out 15. How tll is the uilding? The top of ski run is 100 m ove the finishing line. The run is 300 m long. Wht is the ngle of depression of the ski run? Trigonometry nd erings ering is the direction to one plce from nother. The usul wy of giving ering is s n ngle mesured from north in clockwise direction. This is how nvigtionl compss nd surveyor s compss mesure erings. ering is lwys written s three-digit numer, known s three-figure ering. The digrm shows how this works, using the min compss points s emples. When working with erings, follow these three rules. lwys strt from north. lwys mesure clockwise. lwys give ering in degrees nd s three-figure ering. 000 315 N 045 NW NE 270 W E 090 SW SE S 225 135 180 142
The difficulty with trigonometric prolems involving erings is deling with those ngles greter thn 90 whose trigonometric rtios hve negtive vlues. To void this, we hve to find right-ngled tringle tht we cn redily use. Emple 27 shows you how to del with such sitution. EXMPLE 27 ship sils on ering of 120 for 50 km. How fr est hs it trvelled? The digrm of the sitution is shown in figure i. From this, you cn get the cute-ngled tringle shown in figure ii. From figure ii, you see tht this is cosine prolem. i N 50 km 120 ii 30 50 H So cos 30 = 50 = 50 cos 30 = 43.301 = 43.3 km (3 significnt figures) EXERCISE 6N ship sils for 75 km on ering of 078. How fr est hs it trvelled? How fr north hs it trvelled? Lophm is 17 miles from Wth on ering of 210. How fr south of Wth is Lophm? How fr est of Lophm is Wth? plne sets off from n irport nd flies due est for 120 km, then turns to fly due south for 70 km efore lnding t Seddeth. Wht is the ering of Seddeth from the irport? helicopter leves n rmy se nd flies 60 km on ering of 278. How fr west hs the helicopter flown? How fr north hs the helicopter flown? ship sils from port on ering of 117 for 35 km efore heding due north for 40 km nd docking t ngle By. c d How fr south hd the ship siled efore turning? How fr north hd the ship siled from the port to ngle By? How fr est is ngle By from the port? Wht is the ering from the port to ngle By? Mountin is due west of wlker. Mountin B is due north of the wlker. The guideook sys tht mountin B is 4.3 km from mountin, on ering of 058. How fr is the wlker from mountin B? 143
The digrm shows the reltive distnces nd erings of three ships, B nd C. N How fr north of is B? (Distnce on digrm.) C w c d How fr north of B is C? (Distnce y on digrm.) How fr west of is C? (Distnce z on digrm.) Wht is the ering of from C? (ngle w on digrm.) y B N 062 25 km z 75 km N 332 ship sils from port for 42 km on ering of 130 to point B. It then chnges course nd sils for 24 km on ering of 040 to point C, where it reks down nd nchors. Wht distnce nd on wht ering will helicopter hve to fly from port to go directly to the ship t C? Trigonometry nd isosceles tringles Isosceles tringles often feture in trigonometry prolems ecuse such tringle cn e split into two right-ngled tringles tht re congruent. EXMPLE 28 Find the length in this isosceles tringle. Clculte the re of the tringle. 7 cm 7 cm Drw perpendiculr from the pe of the tringle to its se, splitting the tringle into two congruent, right-ngled tringles. 53 53 H 7 cm 53 y To find the length y, which is 1 2 of, use cosine. y So, cos 53 = 7 y = 7 cos 53 = 4.212 705 1 cm So the length = 2y = 8.43 cm (3 significnt figures). 144
To clculte the re of the originl tringle, you first need to find its verticl height, h. O h 53 H 7 cm You hve two choices, oth of which involve the right-ngled tringle of prt. We cn use either Pythgors theorem (h 2 + y 2 = 7 2 ) or trigonometry. It is sfer to use trigonometry gin, since we re then still using known informtion. This is sine prolem. y h So, sin 53 = 7 h = 7 sin 53 = 5.590 448 6 cm (Keep the ccurte figure in the clcultor.) The re of the tringle = 1 2 se height. (We should use the most ccurte figures we hve for this clcultion.) = 1 2 8.425 410 3 5.590 448 6 = 23.6 cm 2 (3 significnt figures) You re not epected to write down these eight-figure numers, just to use them. Note: If you use rounded-off vlues to clculte the re, the nswer would e 23.5 cm 2, which is different from the one clculted using the most ccurte dt. So never use rounded-off dt when you cn use ccurte dt unless you re just estimting. EXERCISE 6P In questions 1 4, find the side or ngle mrked. 15 cm 12 cm 32 42 72 8.6 cm 20 cm 45 cm This digrm elow shows roof truss. How wide is the roof? 2.3 m 25 Clculte the re of ech of these tringles. c d 9 cm 34 18 cm 84 58 67 14 cm 24 cm 145
footll pitch BCD is shown. The length of the pitch, B = 120 m. The width of the pitch, BC = 90 m. 120 m B BC is right ngled tringle. B = 12 cm, BC = 8 cm. Find the size of ngle CB (mrked in the digrm). Give your nswer to 1 deciml plce. C 90 m 8 cm D Clculte the length of the digonl BD. Give your nswer to 1 deciml plce. ldder is lent ginst wll. Its foot is 0.8 m from the wll nd it reches to height of 4 m up the wll. C 12 cm B PQR is right-ngled tringle. PQ = 15 cm, ngle QPR = 32. Find the length of PR (mrked y in the digrm). Give your nswer to 1 deciml plce. y R P 32 15 cm Q Clculte the length, in metres, of the ldder (mrked on the digrm). Give your nswer to suitle degree of ccurcy. In the digrm, BC is right-ngled tringle. C = 18 cm nd B = 12 cm. 12 cm 0.8 m 4 m 18 cm 6 m E F D 10 m 8 m DE = 6 m. EG = 10 m. FG = 8 m. ngle DEG = 90. ngle EFG = 90. Clculte the length of DG. Give your nswer correct to 3 significnt figures. Clculte the size of the ngle mrked. Give your nswer correct to 1 deciml plce. Edecel, Question 9, Pper 6 Higher, June 2004 lighthouse, L, is 3.2 km due West of port, P. ship, S, is 1.9 km due North of the lighthouse, L. G N B C S Digrm not ccurtely drwn Clculte the length of BC. 1.9 km N L 3.2 km P Clculte the size of the ngle mrked. Give your nswer correct to 3 significnt figures. Find the ering of the port, P, from the ship, S. Give your nswer correct to 3 significnt figures. Edecel, Question 10, Pper 6 Higher, June 2005 146
The digrm represents cuoid BCDEFGH. H G E F 3 cm C Clculte the length of G. Give your nswer correct to 3 significnt figures. Clculte the size of the ngle etween G nd the fce BCD. Give your nswer correct to 1 deciml plce. Edecel, Question 15, Pper 6 Higher, Novemer 2004 5 cm Digrm not ccurtely drwn B 7 cm B = 5 cm. BC = 7 cm. E = 3 cm. WORKED EXM QUESTION BC is right-ngled tringle. C = 19 cm nd B = 9 cm. 19 cm 9 cm B C Clculte the length of BC. PQR is right-ngled tringle. PQ = 11 cm nd QR = 24 cm. P 11 cm Solution Let BC = By Pythgors theorem 2 = 19 2 9 2 cm 2 = 280 cm 2 So = ««««280 = 16.7 cm (3 sf) Let PRQ = 11 So tn = 24 = tn 1 11 = 24.6 (1 dp) 24 c Q Clculte the size of ngle PRQ. BD nd BCD re right-ngled tringles. B = 26 cm, D = 24 cm nd ngle BCD = 35º. 26 cm 24 cm B R c In tringle BC, let BD = By Pythgors theorem 2 = 26 2 24 2 = 100 = 10 cm 24 cm D Clculte the length of BC. Give your nswer to 3 significnt figures. 35 C In tringle BCD, let BC = y 10 So sin 35º = y 10 y = sin 35º So BC = 17.4 cm (3 sf) 147
GRDE YOURSELF le to use Pythgors theorem in right-ngled tringles le to solve prolems in 2-D using Pythgors theorem le to solve prolems in 3-D using Pythgors theorem le to use trigonometry to find lengths of sides nd ngles in right-ngled tringles le to use trigonometry to solve prolems How to use Pythgors theorem Wht you should know now How to solve prolems using Pythgors theorem How to use the trigonometric rtios for sine, cosine nd tngent in right-ngled tringles How to solve prolems using trigonometry How to solve prolems using ngles of elevtion, ngles of depression nd erings 148