UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet STRAND I: Geometry nd Trigonometry I1 Pythgors' Theorem nd Trigonometric Rtios Tet Contents Section I1.1 Pythgors' Theorem I1. Further Work With Pythgors' Theorem I1.3 Sine, Cosine nd Tngent I1.4 Finding Lengths in Right Angled Tringles I1.5 Finding Angles in Right Angled Tringles
UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet I1 Pythgors' Theorem nd Trigonometric Rtios I1.1 Pythgors' Theorem Pythgors' Theorem gives reltionship between the lengths of the sides of right ngled tringle. Pythgors' Theorem sttes tht: In ny right ngled tringle, the re of the squre on the hypotenuse (the side opposite the right ngle) is equl to the sum of the res of the squres on the other two sides (the two sides tht meet t the right ngle). For the tringle shown opossite, = b + c b Note The longest side of right ngled tringle is clled the hypotenuse. c Proof Drw squre of side b + c, s shown opposite. Join up the points PQ, QR, RS, SP s shown, to give qudrilterl, PQRS. In fct, PQRS is squre s ech side is equl to (s the four tringles re congruent) nd t the point P, + ngle SPQ + y = 180 But we know tht + y = 90, so b y S c c P b y y b R c c Q y b ngle SPQ = 90 Similrly for the other three ngles in PQRS. Thus PQRS is squre, nd equting res, 1 + 4 bc b c = + ( ) + bc = b + bc + c Hence = c + b 1
I1.1 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Worked Emple 1 Find the length of the hypotenuse of the tringle shown in the digrm. Give your nswer correct to deciml plces. As this is right ngled tringle, Pythgors' Theorem cn be used. If the length of the hypotenuse is, then b = 4 nd c = 6. So = b + c = 4 + 6 = 16 + 36 = 5 = 5 = 7. cm (to one deciml plce) 4 cm 6 cm Worked Emple Find the length of the side of the tringle mrked in the digrm. As this is right ngled tringle, Pythgors' Theorem cn be used. Here the length of the hypotenuse is 6 cm, so writing = 6 cm nd c = 3 cm with b =, we hve = b + c 6 = + 3 36 = + 9 36 9 36 9 7 = = = 7 = = 5. cm (to one deciml plce) 3 cm 6 cm Eercises 1. Find the length of the side mrked in ech tringle. () (b) 4 m 5 cm 3 m 1 cm
I1.1 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet (c) (d) 15 cm 10 m 6 m (e) 1 cm (f) 6 cm 6 m 8 m 10 cm (g) 0 m 5 m (h) 36 cm 15 cm. Find the length of the side mrked in ech tringle. Give your nswers correct to deciml plces. () (b) 7 cm 15 cm 11 cm 14 cm (c) 4 cm (d) 7 m 8 cm 5 m (e) (f) 8 cm 10 cm 1 cm 7 cm (g) 5 m (h) m 6 m 1 m 3
I1.1 (i) UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet (j) 18 cm 10 cm 5 m 4 m (k) (l) 3.3 m 9.4 m 4.6 m 7.8 m (m) 8.9 cm (n) 5. cm.3 m 5.4 m 3. Andre runs digonlly cross school field, while Rkeif runs round the edge. () How fr does Rkeif run? 00 m (b) How fr does Andre run? (c) How much further does Rkeif run thn Andre? Andre Rkeif 10 m 4. A guy rope is ttched to the top of tent pole, t height of 1.5 metres bove the ground, nd to tent peg metres from the bse of the pole. How long is the guy rope? Peg Guy rope m Pole 1.5 m 5. Disy is 1.4 metres tll. At certin time her shdow is metres long. Wht is the distnce from the top of her hed to the tip of her shdow? 6. A rope of length 10 metres is stretched from the top of pole 3 metres high until it reches ground level. How fr is the end of the line from the bse of the pole? 7. A rope is fied between two trees tht re 10 metres prt. When child hngs on to the centre of the rope, it sgs so tht the centre is metres below the level of the ends. Find the length of the rope. m 10 m 4
I1.1 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 8. The roof on house tht is 6 metres wide peks t height of 3 metres bove the top of the wlls. Find the length of the sloping side of the roof. 3 m 6 m 9. The picture shows shed. Find the length, AB, of the roof..5 m A B m m 10. Rohn wlks 3 km est nd then 10 km north. () How fr is he from his strting point? (b) He then wlks est until he is 0 km from his strting point. How much further est hs he wlked 11. Jodieis building shed. The bse PQRS of the shed should be rectngle mesuring.6 metres by 1.4 metres. To check tht or if the bse is rectngulr, Jodie hs to mesure the digonl PR. () Clculte the length of PR when the bse is rectngulr. You must show ll your working. (b) When building the shed Jodie finds ngle PSR > 90. She mesures PR. Which of the following sttements is true? X: PR is greter thn it should be. Y: PR is less thn it should be. Z: PR is the right length. Informtion 1.4 m.6 m The Greeks, (in their nlysis of the rcs of circles) were the first to estblish the reltionships or rtios between the sides nd the ngles of right ngled tringle. The Chinese lso recognised the rtios of sides in right ngled tringle nd some survey problems involving such rtios were quoted in Zhou Bi Sun Jing. It is interesting to note tht sound wves re relted to the sine curve. This discovery by Joseph Fourier, French mthemticin, is the essence of the electronic musicl instrument developments tody. P S Q R 5
UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet I1. Further Work with Pythgors' Theorem Worked Emple 1 Find the length of the side mrked in the digrm. D cm C y 4 cm A 4 cm B First consider tringle ABC. The unknown length of the hypotenuse hs been mrked y. y 4 cm 4 cm = ( ) + ( ) y = 16 cm + 16 cm y C 4 cm y = 3 cm A 4 cm B Tringle ACD cn now be considered, using the vlue for y. From the tringle, = y +, nd using y = 3 cm gives D cm = 3 cm + 4 cm = 36 cm = 36 cm y C = 6 cm Note A When finding the side, it is not necessry to find 3, but to simply use y = 3 cm. Worked Emple Find the vlue of s shown on the digrm. Using Pythgors' Theorem gives 13 3 = ( ) + ( ) ( ) = = ) 169 = 4 + 9 (since 4 169 = 13 13 = = 13 13 m = 361. m (to deciml plces) 3 6
I1. UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Eercises 1. Find the length of the side mrked in ech digrm. () 7 (b) 4 8 5 (c) 6 (d) 3 4 4 6 10 1 3 (e) (f) 4 10 15 3. Find the length of the side mrked in the following situtions. () (b) 0 8 4 (c) (d) 5 4 4 0 5 3. Which of the following tringles re right ngled tringles? () (b) 9 (c) (d) 10 A 6 13 B 15 6.5 C 6 10 D 130 4.5 40 7
I1. UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 4. A ldder of length 4 metres lens ginst verticl wll. The foot of the ldder is metres from the wll. A plnk tht hs length of 5 metres rests on the ldder, so tht one end is hlfwy up the ldder. () How high is the top of the ldder? (b) How high is the top of the plnk? (c) How fr is the bottom of the plnk from the wll? m 5. The digrm shows how the sign tht hngs over food store is suspended by rope nd tringulr metl brcket. Find the length of the rope. 160 cm Rope Wll 90 cm FOOD FISH AND STORE CHIPS 50 cm Metl Brcket 6. The digrm shows how cble is ttched to the mst of siling dingy. A br pushes the cble out wy from the mst. Find the totl length of the cble. 80 cm Mst Br 40 cm 40 cm Cble 80 cm Deck 7. A helicopter flies in stright line until it reches point 0 km est nd 15 km north of its strting point. It then turns through 90 nd trvels further 10 km. () (b) How fr is the helicopter from its strting point? If the helicopter turned 90 the other wy, how fr would it end up from its strting point? 8
I1. UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 8. A cone is plced on wedge. The dimensions of the wedge re shown in the digrm. The cone hs slnt height of 30 cm. Find the height of the cone. 4 cm 30 cm 0 cm 9. A simple crne is to be constructed using n isosceles tringulr metl frme. The top of the frme is to be 10 metres bove ground level nd 5 metres wy from the bse of the crne, s shown in the digrm. Find the length of ech side of the tringle. 10 m 5 m 10. A thin steel tower is supported on one side by two cbles. Find the height of the tower nd the length of the longer cble. 10 m 40 m 5 m 11. An isosceles tringle hs two sides of length 8 cm nd one of length 4 cm. Find the height of the tringle nd its re. height 1. Find the re of ech the equilterl tringles tht hve sides of lengths () 8 cm (b) 0 cm (c) cm 9
UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet I1.3 Sine, Cosine nd Tngent When working in right ngled tringle, the longest side is known s the hypotenuse. The other two sides re known s the opposite nd the djcent. The djcent is the side net to mrked ngle, nd the opposite side is opposite this ngle. Opposite Hypotenuse For right ngled tringle, the sine, cosine nd tngent of the ngle re defined s: Adjcent sin = opposite hypotenuse cos = djcent hypotenuse tn = opposite djcent Sin will lwys hve the sme vlue for ny prticulr ngle, regrdless of the size of the tringle. The sme is true for cos nd tn. Worked Emple 1 For the tringle nd ngle shown, stte which side is: () the hypotenuse (b) the djcent (c) the opposite. () (b) (c) The hypotenuse is the longest side, which for this tringle is CB. The djcent is the side tht is net to the ngle, which for this tringle is AB. The opposite side is the side tht is opposite the ngle, which for this tringle is AC. A C B Worked Emple Write down the vlues of sin, cos nd tn for the tringle shown. Then use clcultor to find the ngle in ech cse. First, opposite = 8 djcent = 6 hypotenuse = 10 8 6 10 10
I1.3 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet opposite djcent opposite sin = cos = tn = hypotenuse hypotenuse djcent 8 6 8 = = = 10 10 6 4 = 08. = 0.6 = 3 Using clcultor gives = 53. 1 (correct to 1 deciml plce) in ech cse. Note In the tringle opposite, we know tht Thus sin b sin =, cos = c + cos = b + c (hyp) c (dj) b (opp) b = + c = b + c But by Pythgors' Theorem, we lso know tht = b + c Hence This result sin + cos = = 1 sin + cos = 1 is useful result. For emple, when = 45, sin 45 = cos 45 (s the two sides c nd b re equl) nd hence, sin 45 + sin 45 = 1 sin 45 = 1 sin 45 = 1 1 or sin 45 = cos 45 = (You cn check this on your clcultor.) 11
I1.3 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Eercises 1. For ech tringle, stte which side is the hypotenuse, the djcent nd the opposite. () B (b) D (c) H G I A C E F (d) J K (e) M (f) Q P L O N R. For ech tringle, write sin, cos nd tn s frctions. () (b) (c) 3 5 5 13 8 15 4 1 17 (d) 1.5 (e) (f).5 50 48 1 1.5 3.5 14 3. Use clcultor to find the following. Give your nswers correct to 3 deciml plces. () sin 30 (b) tn 75 (c) tn 5. 6 (d) cos 66 (e) tn 33 (f) tn 45 (g) tn 37 (h) sin 88. (i) cos 45 (j) cos 48 (k) cos 46. 7 (l) sin 45 1
I1.3 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 4. Use clcultor to find in ech cse. Give your nswers correct to 1 deciml plce. () cos = 05. (b) sin = 1 (c) tn = 045. (d) sin = 0. 81 (e) sin = 075. (f) cos = 09. (g) tn = 1 (h) sin = 05. (i) tn = (j) cos = 014. (k) sin = 06. (l) tn = 55. 5. () Drw right ngled tringle with n ngle of 50 s shown in the digrm, nd mesure the length of ech side. (b) Using 50 sin = opposite hypotenuse cos = djcent hypotenuse tn = opposite djcent nd the lengths of the sides of your tringle, find sin 50, cos50 nd tn 50. (c) Use your clcultor to find sin 50, cos50 nd tn 50. (d) Compre your results to (b) nd (c). 6. For the tringle shown, write down epressions for: () cos (b) sinα (c) tn (d) cosα y α (e) sin (f) tnα z I1.4 Finding Lengths in Right Angled Tringles When one ngle nd the length of one side re known, it is possible to find the lengths of other sides in the sme tringle, by using the sine, cosine or tngent formul. For emple, sin 50 = 1 cos50 = y 1 = 1sin 50 y = 1 cos50 1 50 y 13
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Worked Emple 1 Find the length of the side mrked in the tringle shown. In this tringle, hypotenuse = 0 cm opposite = cm Choose sine becuse it involves hypotenuse nd opposite. Using sin = opposite hypotenuse gives sin 70 = 0 0 cm 70 To obtin, multiply both sides of this eqution by 0, which gives 0sin 70 = or = 0sin 70 This vlue is obtined using clcultor. = 18. 8 cm (to 1 deciml plce) Worked Emple Find the length of the side mrked in the tringle. In this tringle, opposite = djcent = 8 metres Use tngent becuse it involves the opposite nd djcent. 40 8 m Using tn = opposite djcent gives tn 40 = 8 Multiplying both sides by 8 gives 8tn 40 = or = 8tn 40 = 67. metres (to 1 deciml plce) 14
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Worked Emple 3 Find the length mrked in the tringle. 10 m In this tringle, tn 4 = opposite djcent = 10 4 Multiplying by gives tn 4 = 10 nd dividing by tn 4 gives = = 10 tn 4 10 0. 9004 = 11.1 metres (to 1 deciml plce) Worked Emple 4 Find the length of the hypotenuse, mrked, in the tringle. 10 cm 8 In this tringle, hypotenuse = opposite = 10 cm Use sine becuse it involves hypotenuse nd opposite. Using sin = opposite hypotenuse 10 gives sin 8 = where is the length of the hypotenuse. Multiplying both sides by gives sin 8 = 10, then dividing both sides by sin 8 gives = 10 sin 8 = 1. 3 cm (to 1 deciml plce) 15
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Worked Emple 5 E H 8 m F 40 I 5 m G The digrm bove, not drwn to scle, represents one fce of the roof of house in the shpe of prllelogrm EFGH. Angle EFI = 40 nd EF = 8 m. EI represents rfter plced perpendiculr to FG such tht IG = 5 m. Clculte, giving your nswers to 3 significnt figures, () the length of FI (b) the length of EI (c) the re of EFGH. () FI = 8cos 40 613. m (b) EI = 8sin 40 5. 14 m (c) Are of EFGH = FG EI ( ) = 5 + 6. 13 5. 14 = 11. 13 5. 14 57. m Eercises 1. Find the length of the side mrked in ech tringle. () (b) (c) 8 cm 1 cm 5 11 cm 50 80 16
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet (d) (e) (f) 15 cm 30 18 cm 4 cm 70 (g) (h) (i) 0 cm 45 9 m 8 6 cm 60 (j) (k) (l) 38 45 10.4 m 16.7 m 50 0 m (m) (n) (o) 48 15. m 18 1.8 m 8. m 5. A ldder lens ginst wll s shown in the digrm. 4 m 68 () (b) How fr is the top of the ldder from the ground? How fr is the bottom of the ldder from the wll? 17
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 3. A guy rope is ttched to tent peg nd the top of tent pole so tht the ngle between the peg nd the bottom of the pole is 60. () Find the height of the pole if the peg is 1 metre from the bottom of the pole. (b) If the length of the rope is 1.4 metres, find the height of the pole. (c) Find the distnce of the peg from the bse of the pole if the length of the guy rope is metres. 60 Tent 4. A child is on swing in prk. The highest position tht she reches is s shown. Find the height of the swing set bove the ground in this position. 3 m 15 3.5 m 5. A lser bem shines on the side of building. The side of the building is 500 metres from the source of the bem, which is t n ngle of 16 bove the horizontl. Find the height of the point where the bem hits the building. 6. A ship sils 400 km on bering of 075. () How fr est hs the ship siled? (b) How fr north hs the ship siled? 7. An eroplne flies 10 km on bering of 10. () How fr south hs the eroplne flown? (b) How fr west hs the eroplne flown? 8. A kite hs string of length 60 metres. On windy dy ll the string is let out nd mkes n ngle of between 0 nd 36 with the ground. Find the minimum nd mimum heights of the kite. 9. Find the length of the side mrked in ech tringle. () (b) (c) 9 cm 18 cm 48 4 10 cm 70 18
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet (d) (e) (f) 45 m 7 m 1 m 55 18 (g) (h) (i) 6 cm 17 1.8 m 58 50 4 cm 10. The digrm shows slide in plyprk. () Find the height of the top of the slide. (b) Find the length of the slide. Steps m 70 Slide 40 11. A snooker bll rests ginst the side cushion of snooker tble. It is hit so tht it moves t 40 to the side of the tble. How fr does the bll trvel before it hits the cushion on the other side of the tble? 40 80 cm 1. () Find the length of the dotted line nd the re of this tringle. 5 cm 50 1 cm (b) Find the height of this tringle nd then find formul for its re in terms of nd. 19
I1.4 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 13. A wire 18 metres long runs from the top of pole to the ground, s shown in the digrm. The wire mkes n ngle of 35 with the ground. Clculte the height of the pole. Give your nswer to resonble degree of ccurcy. 35 18 m 14. In the figure shown, clculte () the length of BD. (b) the length of BC. B A 48 4 cm D 36 C I1.5 Finding Angles in Right Angled Tringles If the lengths of ny two sides of right ngled tringle re known, then sine, cosine nd tngent cn be used to find the ngles of the tringle. 18 cm? 10 cm Worked Emple 1 Find the ngle mrked in the tringle shown. In this tringle, hypotenuse = 0 cm opposite = 14 cm Using sin = opposite hypotenuse gives sin = 14 0 = 07. So = sin 1 (0.7) 14 cm 0 cm nd using the SHIFT nd SIN buttons on clcultor gives = 44. 4 (to 1 d.p.) 0
I1.5 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet Worked Emple Find the ngle mrked in the tringle shown. In this tringle, opposite = 5 cm djcent = 4 cm 5 cm Using tn = opposite djcent gives tn = 5 4 = 65. 4 cm So = tn 1 (6.5) nd using the SHIFT nd TAN buttons on clcultor gives = 80. 9 (to 1 d.p.) Worked Emple 3 Q 0 cm P 30 S 9 cm R The digrm bove shows tringle PQR, not drwn to scle. PQ = 0 cm, QPR = 30, QS is perpendiculr to PR, SR = 9 cm, nd SQR =. Clculte () (b) the length of QS the size of ngle to the nerest degree. () QS = 0sin30 = 10 cm 1
I1.5 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet (b) tn = 9 10 = 09. = tn 1 (0.9) nd using the SHIFT nd TAN buttons on clcultor gives 41. 987 4 to the nerest degree. Eercises 1. Find the ngle of: () (b) (c) 8 m 10 m 6 cm 0 cm cm 5 cm (d) 14 cm (e) (f) 6.7 m m 15 cm 8 m (g) (h) (i) 5 m 9 m 48 mm 0.7 m 7 m 0.5 m 1 mm (j) 0.9 cm (k) (l) 1. m 16.5 m 3.6 cm 8.7 m 15.1 m
I1.5 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet. A ldder lens ginst wll. The length of the ldder is 4 metres nd the bse is metres from the wll. Find the ngle between the ldder nd the ground. 4 m m 3. As crs drive up rmp t multi-storey cr prk, they go up metres. The length of the rmp is 10 metres. Find the ngle between the rmp nd the horizontl. 10 m m 4. A flg pole is fied to wll nd supported by rope, s shown. 5 m m Find the ngle between () the rope nd the wll (b) the pole nd the wll. 5. The mst on ycht is supported by number of wire ropes. One, which hs length of 15 metres, goes from the top of the mst t height of 10 metres, to the front of the bot. () Find the ngle between the wire rope nd the mst. (b) Find the distnce between the bse of the mst nd the front of the bot. 6. A soldier runs 500 metres est nd then 600 metres north. If he hd run directly from his strting point to his finl position, wht bering should he hve run on? 7. A ship is 50 km south nd 70 km west of the port tht it is heding for. Wht bering should it sil on to rech the port? 3
I1.5 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 8. The digrm shows simple bridge, which is supported by four steel cbles. () Find the ngles t α nd β. (b) Find the length of ech cble. α β 6 m 4 m 4 m 4 m 4 m 4 m 9. In the digrm below, not drwn to scle, EFGH is rectngle. The point D on HG is such tht ED = DG =1 cm nd GDF ˆ = 43. E F 43 H D Clculte correct to one deciml plce () the length of GF (b) the length of HD (c) the size of the ngle HDE. 1 cm G 10. ABC is right ngled tringle. AB is of length 4 metres nd BC is of length 13 metres. () Clculte the length of AC. (b) Clculte the size of ngle ABC. C 13 m A 4 m B 4
I1.5 UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet 11. The digrm shows roofing frme ABCD. AB = 7 m, BC = 5 m, ngle ABD = ngle DBC = 90 D 3 m A 7 m B 5 m C () (b) Clculte the length of AD. Clculte the size of ngle DCB. 5