ONLINE PAGE PROOFS. Trigonometry. 6.1 Overview. topic 6. Why learn this? What do you know? Learning sequence. measurement and geometry

Transcription

1 mesurement nd geometry topic 6 Trigonometry 6.1 Overview Why lern this? Pythgors ws gret mthemticin nd philosopher who lived in the 6th century BCE. He is est known for the theorem tht ers his nme. It concerns the reltionship etween the lengths of the sides in right-ngled tringle. Geometry nd trigonometry re rnches of mthemtics where Pythgors theorem is still widely pplied. Trigonometry is rnch of mthemtics tht llows us to relte the side lengths of tringles to ngles. Comining trigonometry with Pythgors theorem llows us to solve mny prolems involving tringles. Wht do you know? 1 think List wht you know out trigonometry. Use thinking tool such s concept mp to show your list. 2 PIr Shre wht you know with prtner nd then with smll group. 3 shre As clss, crete thinking tool such s lrge concept mp to show your clss s knowledge of trigonometry. Lerning sequence 6.1 Overview 6.2 Wht is trigonometry? 6.3 Clculting unknown side lengths 6.4 Clculting unknown ngles 6.5 Angles of elevtion nd depression 6.6 Review ONLINE ONLY 174 Mths Quest 9

2 WtCH this video The story of mthemtics: Secret society serchlight ID: eles-1693

3 mesurement nd geometry 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship etween the sides nd the ngles of tringle. Modern dy uses of trigonometry include surveying lnd, rchitecture, mesuring distnces nd determining heights of inccessile ojects. In this chpter reltionships etween the sides nd ngles of right-ngled tringle will e explored. nming the sides of right-ngled tringle The longest side of right-ngled tringle (the side opposite the right ngle) is clled the hypotenuse. In order to nme the remining two sides nother ngle, clled the reference ngle, must e dded to the digrm. The side tht is cross from the reference ngle,, is clled the opposite side, nd the remining side (the side next to the reference ngle) is clled the djcent side. Note: If there is no reference ngle mrked, only the hypotenuse cn e nmed. WorKeD exmple 1 Lel the sides of the right-ngled tringle shown using the words hypotenuse, djcent nd opposite. think 1 The hypotenuse is opposite the right ngle. Hypotenuse Opposite WrIte/DrW Hypotenuse Adjcent 2 Lel the side next to ngle s djcent nd the side opposite ngle s opposite. Hypotenuse Opposite Adjcent 176 Mths Quest 9

4 mesurement AND geometry Similr right-ngled tringles Consider the two right-ngled tringles shown elow. The second tringle (ΔDEF) is n enlrgement of the first, using scle fctor of 2. Therefore, the tringles re similr (ΔABC ~ ΔDEF), nd BCA = EFD = x. D opposite side For ΔABC, djcent side = 3 4, nd opposite side for ΔDEF, djcent side = 6 8 = 3 4. Now complete the tle elow. 3 A B Opposite side hypotenuse Adjcent side hypotenuse 5 4 x C 6 E 10 8 ΔABC In similr right-ngled tringles, the rtios of corresponding sides re equl. Experiment A. Using protrctor nd ruler, crefully mesure nd drw right-ngled tringle of se 10 cm nd ngle of 60 s shown in the digrm. Mesure the length of the other two sides to the nerest mm, nd mrk these lengths on the digrm s well. Use your mesurements to clculte these rtios correct to 2 deciml plces: x F ΔDEF opposite djcent =, opposite hypotenuse =, djcent hypotenuse = cm B. Drw nother tringle, similr to the one in prt A (ll ngles the sme), mking the se length nything tht you choose, nd mesuring the length of ll the sides. Once gin, clculte the three rtios correct to 2 deciml plces: opposite djcent =, opposite hypotenuse =, djcent hypotenuse = C. Compre your results with the rest of the clss. Wht conclusions cn you drw? Topic 6 Trigonometry 177

5 mesurement nd geometry Trigonometric rtios Trigonometry is sed upon the rtios etween pirs of side lengths, nd ech one is given specil nme s follows. In ny right-ngled tringle: int-0744 sine () = opposite hypotenuse cosine () = djcent hypotenuse tngent () = opposite djcent These rules re revited to: sin() = O H, cos() = A H nd tn() = O A. The following mnemonic cn e used to help rememer the trigonometric rtios. SOH CAH TOA WorKeD exmple 2 For this tringle, write the equtions for the sine, cosine nd tngent rtios of the given ngle. think 1 Lel the sides of the tringle WrIte/DrW 5 Opposite SOH Hypotenuse Adjcent O Hypotenuse 13 H CAH Opposite H A TOA O A 12 Adjcent 2 Write the trigonometric rtios. sin() = O H, cos() = A H, tn() = O A 3 Sustitute the vlues of A, O nd H into ech formul. sin() = 5 12, cos() = 13 13, tn() = Mths Quest 9

6 mesurement nd geometry WorKeD exmple 3 Write the trigonometric rtio tht reltes the two given sides nd the reference ngle in ech of the following tringles x think 1 Lel the given sides. 2 We re given O nd H. These re used in SOH. Write the rtio. 3 Sustitute the vlues of the pronumerls into the rtio. WrIte/DrW Opposite 6 sin() = O H sin() = Simplify the frction. sin() = Lel the given sides. 2 We re given A nd O. These re used in TOA. Write the rtio. 3 Sustitute the vlues of the ngle nd the pronumerls into the rtio. Hypotenuse 15 Adjcent 18 Opposite x 50 tn() = O A tn(50 ) = x 18 Exercise 6.2 Wht is trigonometry? InDIvIDul PtHWys PrCtIse Questions: 1 8 ConsolIDte Questions: 1 11 mster Questions: 1 12 reflection Why does sin(30 ) = cos(60 )? Individul pthwy interctivity int-4498 Topic 6 Trigonometry 179

7 mesurement nd geometry FluenCy 1 WE1 Lel the sides of the following right-ngled tringles using the words hypotenuse, djcent nd opposite. c doc doc d e 2 Lel the hypotenuse, djcent nd opposite sides, nd reference ngle, where pproprite, in ech of the following right-ngled tringles. c D F Adjcent E Opposite G I H f J Adjcent 3 For ech tringle elow, crefully mesure correct to the nerest degree, then crefully mesure ech side correct to the nerest mm. Use this informtion to copy nd complete the tle elow. O A H sin() cos() tn() 4 MC Which lterntive correctly nmes the sides nd ngle of A the tringle t right? C =, AB = djcent side, AC = hypotenuse, AC = opposite side B C =, AB = opposite side, AC = hypotenuse, B AC = djcent side C A =, AB = opposite side, AC = hypotenuse, BC = djcent side D C =, AB = opposite side, AC = hypotenuse, BC = djcent side K L C 180 Mths Quest 9

8 mesurement AND geometry 5 WE2 For ech of the following tringles, write the expressions for rtios of ech of the given ngles: i sine ii cosine iii tngent. d 7 6 γ e i β g 6 WE3 Write the trigonometric rtio tht reltes the two given sides nd the reference ngle in ech of the following tringles. d g x 15 p e h c t h α 17 i c f 0.9 v β t u γ c 5 4 f 14.3 α α UNDERSTANDING 7 MC Wht is the correct trigonometric rtio for the tringle shown t right? γ A tn(γ) = c B sin(γ) = c C cos(γ) = c D sin(γ) = c c Topic 6 Trigonometry 181

9 mesurement AND geometry Which trigonometric rtio for the tringle shown t right is incorrect? A sin(α) = c C cos(α) = c B sin(α) = c D tn(α) = α c REASONING 8 Consider the right-ngled tringle shown t right. α Lel ech of the sides using the letters O, A, H with respect to the 41 ngle. Mesure the side lengths (to the nerest millimetre). c Determine the vlue of ech trigonometric rtio. (Where pplicle, nswers should e given correct to 2 deciml plces.) 41 i sin(41 ) ii cos(41 ) iii tn(41 ) d Wht is the vlue of the unknown ngle, α? e Determine the vlue of ech of these trigonometric rtios, correct to 2 deciml plces. i sin(α) ii cos(α) iii tn(α) (Hint: First re-lel the sides of the tringle with respect to ngle α.) f Wht do you notice out the reltionship etween sin(41 ) nd cos(α)? g Wht do you notice out the reltionship etween sin(α) nd cos(41 )? h Mke generl sttement out the two ngles. 9 Given the tringle shown: why does =? wht would the vlue of tn(45 ) e? Prolem solving 10 If right-ngled tringle hs side lengths m, (m + n) nd (m n), which one of the lengths is the hypotenuse nd 45 why? 11 A ldder lens on wll s shown. Use the informtion from the digrm to nswer the following questions. In reltion to the ngle given, wht prt of the imge represents: the djcent side the hypotenuse c the opposite side? 12 Use sketches of right-ngled tringles to investigte the following. As the cute ngle increses in size, wht hppens to the rtio of the length of the opposite side to the length of the hypotenuse in ny right-ngled tringle? As the cute ngle increses in size, wht hppens to the other two rtios (i.e. the rtio of the length of the djcent side to the length of the hypotenuse nd tht of the opposite side to the djcent)? c Wht is the lrgest possile vlue for: i sin() ii cos() iii tn()? 182 Mths Quest 9

10 mesurement nd geometry 6.3 Clculting unknown side lengths Vlues of trigonometric rtios The vlues of trigonometric rtios cn e found using clcultor. Ech clcultor hs severl modes. For the following clcultions, your clcultor must e in degree mode. WorKeD exmple 4 Evlute ech of the following, giving nswers correct to 4 deciml plces. sin(53 ) cos(31 ) c tn(79 ) think 1 Set the clcultor to degree mode. Write the first 5 deciml plces. WrIte sin(53 ) = Round correct to 4 deciml plces Write the first 5 deciml plces. cos(31 ) = Round correct to 4 deciml plces c 1 Write the first 5 deciml plces. c tn(79 ) = Round correct to 4 deciml plces Finding side lengths If reference ngle nd ny side length of right-ngled tringle re known, it is possile to find the other sides using trigonometry. WorKeD exmple 5 Use the pproprite trigonometric rtio to find the length of the unknown side in the tringle shown. Give your nswer correct to 2 deciml plces. think 1 Lel the given sides. WrIte/DrW Adjcent 16.2 m m 58 Opposite x x 2 These sides re used in TOA. Write the rtio. tn() = O A 3 x Sustitute the vlues of, O nd A into the tn(58 ) = tngent rtio Solve the eqution for x tn(58 ) = x x = 16.2 tn(58 ) 5 Clculte the vlue of x to 3 deciml plces, then round the nswer to 2 deciml plces. x = x m Topic 6 Trigonometry 183

11 mesurement nd geometry WorKeD exmple 6 Find the length of the side mrked m in the tringle t right. Give your nswer correct to 2 deciml plces. think 1 Lel the given sides. WrIte/DrW Adjcent 17.4 cm m 17.4 cm 22 3 Sustitute the vlues of, A nd H into the cosine rtio. 4 Solve for m: Multiply oth sides y m. Divide oth sides y cos(22 ). 5 Clculte the vlue of m to 3 deciml plces, then round the nswer to 2 deciml plces. 22 m Hypotenuse 2 These sides re used in CAH. Write the rtio. cos() = A H cos(22 ) = 17.4 m WorKeD exmple 7 Benjmin set out on ushwlking expedition. Using compss, he set off on course N 70 E (or 070 T) nd trvelled distnce of 5 km from his se cmp. 70 Bse cmp 5 km N How fr est hs he trvelled? How fr north hs he trvelled from the se cmp? Give nswers correct to 2 deciml plces. think 1 Lel the esterly distnce x. Lel the northerly distnce y. Lel the sides of the tringle: Hypotenuse, Opposite, Adjcent. E m cos(22 ) = 17.4 m = 17.4 cos (22 ) m = m cm WrIte/DrW Adjcent y 70 Opposite x 5 km Hypotenuse 184 Mths Quest 9

12 mesurement nd geometry 2 To clculte the vlue of x, use the sides sin() = O of the tringle: x = O, 5 = H. H These re used in SOH. Write the rtio. 3 Sustitute the vlues of the ngle nd the sin(70 ) = x pronumerls into the sine rtio. 5 4 Mke x the suject of the eqution. x = 5 sin(70 ) 5 Evlute x to 3 deciml plces, using = clcultor. 6 Round to 2 deciml plces km 7 Answer the question in sentence form. Benjmin hs trvelled 4.70 km est of the se cmp. 1 To clculte the vlue of y, use the sides: cos() = A y = A, 5 = H. H These re used in CAH. Write the rtio. 2 Sustitute the vlues of the ngle nd the cos(70 ) = y pronumerls into the cosine rtio. 5 3 Mke y the suject of the eqution. y = 5 cos(70 ) 4 Evlute y using clcultor. = Round the nswer to 2 deciml plces km 6 Answer the question in sentence form. Benjmin hs trvelled 1.71 km north of the se cmp. Exercise 6.3 Clculting unknown side lengths InDIvIDul PtHWys PrCtIse Questions: 1 3, 4 c, 5 c, 6 f, 7 9, 11, 12 ConsolIDte Questions: 1 3, 4 d, 5 d, 6d i, 7 10, Individul pthwy interctivity int-4499 mster Questions: 1 3, 4d f, 5d f, 6g i, 7 10, 11, FluenCy 1 WE4 Evlute the following correct to 4 deciml plces. i sin(55 ) ii sin(11.6 ) Copy nd complete the tle elow. (Use your clcultor to find ech vlue of sin() correct to 2 deciml plces.) sin() c Summrise the trend in these vlues. 2 Evlute the following correct to 4 deciml plces. i cos(38 ) ii cos(53.71 ) reflection Wht does sin(60 ) ctully men? doc doc doc Topic 6 Trigonometry 185

13 mesurement nd geometry Copy nd complete the tle elow. (Use your clcultor to find ech vlue of cos() correct to 2 deciml plces.) cos() eles-0116 c Summrise the trend in these vlues. 3 Evlute the following correct to 4 deciml plces. i tn(18 ) ii tn(51.9 ) Copy nd complete the tle elow. (Use your clcultor to find ech vlue of tn() correct to 2 deciml plces.) tn() c Find the vlue of tn(89 ) nd tn(89.9 ). d Summrise the trend in these vlues. 4 WE5 Use the pproprite trigonometric rtios to find the length of the unknown side in ech of the tringles shown. Give the nswers correct to 2 deciml plces. c d 17 m 50 p cm x e 7.9 m 29.5 m y z f m s 46 mm 5 WE6 Use the pproprite trigonometric rtio to find the length of the unknown side in ech of the tringles shown. Give the nswers correct to 2 deciml plces. k cm 16 cm 52 s c 16.1 cm 22 q 5 z d 5.72 km 66 e e f t p 7.7 km m 186 Mths Quest 9

14 mesurement AND geometry 6 Find the length of the unknown side in ech of the following tringles, correct to 2 deciml plces. (Note: In some cses the unknown will e in the numertor nd in other cses it will e in the denomintor.) c d g 13 y 63.2 m l mm km m e h n m cm 46.7 cm d f i 75 x m z y 0.95 km mm 7 Find the lengths of the unknown sides in the tringles shown, correct to 2 deciml plces. c UNDERSTANDING c 8 MC The vlue of x correct to 2 deciml plces is: A B C D The vlue of x correct to 2 deciml plces is: A mm B mm C mm D mm c The vlue of y correct to 2 deciml plces is: A B 7.94 C 1.37 D 0.23 d The vlue of y correct to 2 deciml plces is: A 0.76 km B 1.79 km C 3.83 km D 3.47 km 25 y c y x 1.62 km mm 47 x Topic 6 Trigonometry 187

15 mesurement AND geometry 9 WE7 A ship tht ws to trvel due north veered off course nd trvelled N 80 E (or 080 T) for distnce of 280 km, s shown in the digrm. How fr est hd the ship trvelled? How fr north hd the ship trvelled? N km E 10 A rescue helicopter spots missing surfer drifting out to se on his dmged ord. The helicopter descends verticlly to height of 19 m ove se level nd drops down n emergency rope, which the surfer grips. Due to the wind the rope swings t n ngle of 27 to the verticl, s shown in the digrm. Wht is the length of the rope? 11 Wlking long the costline, Michelle (M) looks up through n ngle of 55 nd sees her friend Helen (H) on top of the cliff t the lookout point. How high is the cliff if Michelle is 200 m from its se? (Assume oth girls re the sme height.) H m m M 188 Mths Quest 9

16 mesurement AND geometry REASONING 12 One method for determining the distnce cross ody of wter is shown in the digrm elow. B A 50 m C The required distnce is AB. A surveyor moves t right ngles 50 m to point C nd uses tool clled trnsit to mesure the ngle ( ACB). If = 12.3, show tht the length AB is m. Show tht vlue of = gives length of AB = 100 m. c Find rule tht cn e used to clculte the length AC. 13 Using digrm, explin why sin(70 ) = cos(20 ) nd cos(70 ) = sin(20 ). In generl, sin() will e equl to which cosine? Prolem solving 14 Clculte the vlue of the pronumerl in ech of the following tringles. c x 10 m 12.5º 6.2 m 29º h x 1.6 m 38 Topic 6 Trigonometry 189

17 mesurement nd geometry 15 A tile is in the shpe of prllelogrm with mesurements s shown. Clculte the width of the tile, w, to the nerest mm mm 16 w A pole is supported y two wires s shown. If the length of the lower wire is 4.3 m, clculte to 1 deciml plce: the length of the top wire the height of the pole The frme of kite is uilt from 6 wooden rods s shown. Clculte the totl length of wood used to mke the frme of the kite to the nerest metre. doc cm CHllenge Clculting unknown ngles inverse trigonometric rtios We hve seen tht sin(30 ) = 0.5; therefore, 30 is the inverse sine of 0.5. This is written s sin 1(0.5) = 30. The expression sin 1(x) is red s the inverse sine of x. The expression cos 1(x) is red s the inverse cosine of x. The expression tn 1(x) is red s the inverse tngent of x. Experiment 1. Use your clcultor to find sin(30 ), then find the inverse sine of the nswer. Choose nother ngle nd do the sme thing. 2. Now find cos(30 ) nd then find the inverse cosine (cos 1) of the nswer. Choose nother ngle nd do the sme thing. 190 Mths Quest 9 c06trigonometry.indd /07/14 2:22 PM

18 MEASUREMENT AND GEOMETRY 3. Lstly, find tn(45 ) nd then find the inverse tngent (tn 1 ) of the nswer. Try this with other ngles. The fct tht sin nd sin 1 cncel ech other out is useful in solving equtions such s: sin() = 0.3 (Tke the inverse sine of oth sides.) sin 1 (sin()) = sin 1 (0.3) = sin 1 (0.3) sin 1 (x) = 15 (Tke the sine of oth sides.) sin(sin 1 (x)) = sin(15 ) x = sin(15 ) Similrly, cos() = mens tht = cos 1 (0.522) nd tn() = 1.25 mens tht WORKED EXAMPLE 8 = tn 1 (1.25). Evlute cos 1 (0.3678), correct to the nerest degree. THINK WRITE 1 Set your clcultor to degree mode. cos 1 (0.3678) = Round the nswer to the nerest whole numer nd include the degree symol. WORKED EXAMPLE 9 68 Determine the size of ngle in ech of the following. Give nswers correct to the nerest degree. sin() = tn() = THINK WRITE 1 is the inverse sine of sin() = = sin 1 (0.6543) 2 Clculte nd record the nswer. = Round the nswer to the nerest degree is the inverse tngent of tn() = = tn 1 (1.745) 2 Use the inverse tngent function on clcultor. Record the numer shown. = Round the nswer to the nerest degree. 60 Topic 6 Trigonometry 191

19 mesurement nd geometry Finding the ngle when 2 sides re known Knowing ny 2 sides of right-ngled tringle, it is possile to find n ngle using inverse sine, inverse cosine or inverse tngent. WorKeD exmple 10 Determine the vlue of in the tringle t right. Give your nswer correct to the nerest degree. 63 think 1 Lel the given sides. These re used in CAH. Write the rtio. WrIte/DrW Hypotenuse Adjcent cos() = A H 2 Sustitute the given vlues into the cosine rtio. cos() = is the inverse cosine of = cos Evlute. = Round the nswer to the nerest degree. 79 WorKeD exmple 11 Roert enjoys wter skiing nd is out to try new rmp on the Hwkesury River. The inclined rmp rises 1.5 m ove the wter level nd spns horizontl distnce of 6.4 m. Wht is the mgnitude (size) of the ngle tht the rmp mkes with the wter? Give the nswer correct to the nerest degree. think 1 Drw simple digrm, showing the known lengths nd the ngle to e found. 2 Lel the given sides. These re used in TOA. Write the rtio. 3 Sustitute the vlues of the pronumerls into the tngent rtio. WrIte/DrW 6.4 Adjcent tn() = O A tn() = Opposite is the inverse inverse tngent of = tn m m 192 Mths Quest 9

20 mesurement nd geometry 5 Evlute. = Round the nswer to the nerest degree Write the nswer in words. The rmp mkes n ngle of 13 with the wter. Exercise 6.4 Clculting unknown ngles InDIvIDul PtHWys PrCtIse Questions: 1 c, 2, 3 f, 4 10 ConsolIDte Questions: 1d f, 2, 3d h, 4 11 mster Questions: 1g i, 2, 3e i, 4 13 FluenCy 1 WE8 Evlute ech of the following, correct to the nerest degree. sin 1 (0.6294) cos 1 (0.3110) c tn 1 (0.7409) d tn 1 (1.3061) e sin 1 (0.9357) f cos 1 (0.3275) g cos 1 (0.1928) h tn 1 (4.1966) i sin 1 (0.2554) 2 WE9 Determine the size of the ngle in ech of the following. Give nswers correct to the nerest degree. sin() = sin() = c sin(β) = d cos(β) = e cos(α) = f cos(α) = g tn() = h tn() = 1 i tn() = j sin(c) = k cos() = l tn(α) = WE10 Determine the vlue of in ech of the following tringles. Give nswers correct to the nerest degree. c d Individul pthwy interctivity int-4500 e f reflection Why does cos(0 ) = 1? doc g 26 h i Topic 6 Trigonometry 193

21 mesurement AND geometry 4 MC If cos() = , the vlue of correct to 2 deciml plces is: A B C D If sin() = , the vlue of correct to 2 deciml plces is: A B C D c The vlue of in the tringle shown, correct to 2 deciml plces, is: A B C D d The vlue of in the tringle shown, correct to 2 deciml plces, is: A B C D Copy nd fill in the tle elow. x y = cos 1 (x) Plot the ove tle on grph pper or with spredsheet or suitle clcultor. UNDERSTANDING 6 A piece of fric mesuring 2.54 m y 1.5 m hs design consisting of prllel digonl stripes. Wht ngle does ech digonl mke with the length of the fric? Give your nswer correct to 2 deciml plces WE11 Dnny Dingo is perched on top of cliff 20 m high wtching n emu feeding 8 m from the se of the cliff. Dnny hs purchsed flying contrption, which he hopes will help him cpture the emu. At wht ngle to the cliff must he swoop to ctch his prey? Give your nswer correct to 2 deciml plces m m 20 m 8 m 194 Mths Quest 9

22 MEASUREMENT AND GEOMETRY REASONING 8 Jenny nd Les re cmping with friends Mrk nd Susie. Both couples hve 2-m-high tent. The top of 2-m tent pole is to e tied with piece of rope tht will e used to keep the pole upright. So tht the rope doesn t trip pssery, Jenny nd Les decide tht the ngle etween the rope nd the ground should e 80 o. Answer the following questions, correct to 2 deciml plces. Find the length of the rope needed from the top of the tent pole to the ground to support their tent pole. Further down the cmping ground, Mrk nd Susie lso set up their tent. However, they wnt to use piece of rope tht they know is in the rnge of 2 to 3 metres in length. i Explin why the rope will hve to e greter thn 2 metres in length. ii Show tht the minimum ngle the rope will mke with the ground will e Sfety guidelines for wheelchir ccess rmps used to stte tht the grdient hd to e in the rtio 1 : 20. Using this rtio, show tht the ngle tht the rmp hd to mke with the horizontl is closest to 3. New regultions hve chnged the rtio of the grdient, so the ngle the rmp must mke with the horizontl is now closest to 6. Explin why, using this ngle size, the new rtio could e 1 to 9.5. PROBLEM SOLVING 10 Clculte the vlue of the pronumerl in ech of the following to 2 deciml plces. c 12 cm 5.4 cm 0.75 m 1.2 m x 8 m 0.9 m Topic 6 Trigonometry 195

23 mesurement nd geometry 11 A fmily is uilding ptio t the ck of the house. One section of the ptio will hve gle roof. A similr structure is pictured with the plnned post heights nd spn shown. To llow more light in, the fmily wnts the pek (highest point) of the gle roof to e t lest 5 m ove deck level. According to uilding regultions, the slope of the roof (i.e. the ngle tht the sloping edge mkes with the horizontl) must e 22. Use trigonometry to clculte whether the roof would e high enough if the ngle ws 22. Use trigonometry to clculte the size of the otuse ngle formed t the pek of the roof. 6m 3.2 m 12 Use the formuls sin() = sin() o nd cos() = to prove tht tn() =. cos() h h CHllenge Mths Quest 9 c06trigonometry.indd /07/14 2:22 PM

24 mesurement nd geometry 6.5 Angles of elevtion nd depression When looking up towrds n oject, n ngle of elevtion is the ngle etween the horizontl line nd the line of vision. Line of vision Angle of elevtion Horizontl When looking down t n oject, n ngle of depression is the ngle etween the horizontl line nd the line of vision. Angles of elevtion nd depression re mesured from horizontl lines. WorKeD exmple 12 At point 10 m from the se of tree, the ngle of elevtion of the treetop is 38. How tll is the tree to the nerest centimetre? think 1 Drw simple digrm. The ngle of elevtion is 38 from the horizontl. 2 Lel the given sides of the tringle. These sides re used in TOA. Write the rtio. WrIte/DrW Adjcent tn(38 ) = h 10 3 Multiply oth sides y tn(38 ) = h 4 Clculte correct to 3 deciml plces. h = Round to 2 deciml plces Horizontl 6 Write the nswer in words. The tree is 7.81 m tll. Angle of depression Line of vision h Opposite WorKeD exmple 13 A lighthouse, 30 m tll, is uilt on top of cliff tht is 180 m high. Find the ngle of depression () of ship from the top of the lighthouse if the ship is 3700 m from the ottom of the cliff. Angle of depression 30 m 180 m 3700 m Topic 6 Trigonometry 197

25 mesurement nd geometry think 1 Drw simple digrm to represent the sitution. The height of the tringle is = 210 m. Drw horizontl line from the top of the tringle nd mrk the ngle of depression,. Also mrk the lternte ngle. WrIte/DrW S 3700 Adjcent 2 Lel the tringle. These sides re used in TOA. tn() = O A Write the rtio. 3 Sustitute the given vlues into the rtio. tn() = is the inverse tngent of = tn Evlute. = Round the nswer to the nerest degree. 3 T Opposite Write the nswer in words. The ngle of depression of the ship from the top of the lighthouse is 3. Note: In Worked exmple 13, the ngle of depression from the top of the lighthouse to the ship is equl to the ngle of elevtion from the ship to the top of the lighthouse. This is ecuse the ngle of depression nd the ngle of elevtion re lternte (or Z ) ngles. This cn e generlised s follows: For ny two ojects, A nd B, the ngle of elevtion of B, s seen from A, is equl to the ngle of depression of A, s seen from B. A Angle of depression of A from B Angle of elevtion of B from A B C Angle of depression Angle of elevtion Exercise 6.5 Angles of elevtion nd depression InDIvIDul PtHWys reflection Why does the ngle of elevtion hve the sme vlue s the ngle of depression? PrCtIse Questions: 1 6, 8, 10, 12 ConsolIDte Questions: 1 6, 8, Individul pthwy interctivity int-4501 mster Questions: 1 3, 6, 7, Mths Quest 9

26 mesurement nd geometry FluenCy 1 WE12 Building specifictions require the ngle of elevtion of ny rmp constructed for pulic use to e less thn 3. 1 m 7 m doc Rmps eing constructed t new shopping centre re ech mde in the rtio 7 m horizontl length to 1 m verticl height. Find the ngle of elevtion of these rmps nd, hence, decide whether they meet uilding specifictions. 2 A lifesver stnding on his tower 3 m ove the ground spots swimmer experiencing 12 difficulty. The ngle of depression of the swimmer from the lifesver is 12. How fr is the swimmer 3 m from the lifesver s tower? (Give your nswer correct to 2 deciml plces.) 3 From the top of lookout 50 m ove the ground, the ngle of depression of cmp site tht is level with the se of the lookout is 37. How fr is the cmp site from the se of the lookout? understnding 4 From rescue helicopter 80 m ove the ocen, the ngles of depression of two shipwreck survivors re 40 nd 60 respectively. If the two silors nd the helicopter re in line with ech other: drw lelled digrm to represent the sitution clculte the distnce etween the two silors, to the nerest metre. 5 The ngle of elevtion of the top of tree from point on the ground, 60 m from the tree, is 35. Drw lelled digrm to represent the sitution. Find the height of the tree to the nerest metre. 50 m Topic 6 Trigonometry 199

27 mesurement AND geometry 6 Mirim, n vid cmerwomn from Perth, wnts to record her dughter Alexndr s first ttempts t crwling. As Alexndr lies on the floor nd looks up t her mother, the ngle of elevtion is 17. If Alexndr is 5.2 m wy from her mother, how tll is Mirim? m 7 WE13 Stn, who is 1.95 m tll, mesures the length of the shdow he csts long the ground s 0.98 m. Find the ngle of depression of the sun s rys to the nerest degree m = ngle of depression 8 Wht ngle does 3.8-m ldder mke with the ground if it reches 2.1 m up the wll? How fr is the foot of the ldder from the wll? (Give your nswers to the nerest degree nd the nerest metre.) 3.8 m 2.1 m 9 Con nd John re prctising shots on gol. Con is 3.6 m wy from the gol nd John is 4.2 m wy, s shown in the digrm. If the height of the gol post is 2.44 m, wht is the mximum ngle of elevtion, to the nerest degree, tht ech cn kick the ll in order to score gol? Con 2.44 m 3.6 m 4.2 m John 200 Mths Quest 9

28 mesurement AND geometry 10 MC The ngle of elevtion of the top of lighthouse tower 78 m tll, from point B on the sme level s the se of the tower, is 60. The correct digrm for this informtion is: A B 78 m B m C B m REASONING 60 B 11 Lifesver Smi spots some dolphins plying ner mrker t se directly in front of him. He is sitting in tower tht is situted 10 m from the wter s edge nd is 4 m tll. The mrker is 20 m from the wter s edge. Drw digrm to represent this informtion. Show tht the ngle of depression of Smi s view of the dolphins, correct to 1 deciml plce, is 7.6. c As the dolphins swim towrds Smi, would the ngle of depression increse or decrese? Justify your nswer in terms of the tngent rtios. 12 Two uildings re 100 m nd 75 m high. From the top of the north side of the tller uilding, the ngle of depression to the top of the south side of the smller uilding is 20, s shown elow. Show tht the horizontl distnce etween the north side of the tller uilding nd the south side of the smller uilding is closest to 69 metres. 75 m South side D 20 B 60 North side 78 m 100 m Prolem solving 13 Rouk ws hiking in the mountins when she spotted n egle sitting up in tree. The ngle of elevtion of her view of the egle ws 35. She then wlked 20 metres towrds the tree nd her ngle of elevtion ws 50. The height of the egle from the ground ws 35.5 metres. Drw lelled digrm to represent this informtion. Determine how tll Rouk is, if her eyes re 9 cm from the top of her hed. Write your nswer in metres, correct to the nerest centimetre. Topic 6 Trigonometry 201

29 mesurement nd geometry 14 A lookout in lighthouse tower cn see two ships pproching the cost. Their ngles of depression re 25 nd 30. If the ships re 100 m prt, show tht the height of the lighthouse, to the nerest metre, is 242 metres. 15 At certin distnce wy, the ngle of elevtion to the top of uilding is 60. From 12 m further ck, the ngle of elevtion is 45 s shown in the digrm elow. C doc Show tht the height of the uilding is 28.4 metres. 16 A tll gum tree stnds in courtyrd in the middle of some office uildings. Three Yer 9 students, Jckie, Pho nd Theo mesure the ngle of elevtion from three different positions. They re unle to mesure the distnce to the se of the tree ecuse of the steel tree gurd round the se. The digrm elow shows the ngles of elevtion nd the distnces mesured. Theo Not to scle A 12 m D d B 41 β α 12 m x Pho 15 m h Building Jckie Height from ground to eye level 15 tn α Show tht x =, where x is the distnce, in metres, from the se of tn β tn α the tree to Pho s position. The girls estimte the tree to e 15 m tller thn them. Pho mesured the ngle of elevtion to e 72. Wht should Jckie hve mesured her ngle of elevtion to e, if these mesurements re ssumed to e correct? Write your nswer to the nerest degree. c Theo did some clcultions nd determined tht the tree ws only out 10.4 m tller thn them. Jckie clims tht Theo s clcultion of 10.4 m is incorrect. i Is Jckie s clim correct? Show how Theo clculted height of 10.4 m. ii If the height of the tree ws ctully 15 metres ove the height of the students, determine the horizontl distnce Theo should hve used in his clcultions. Write your nswer to the nerest centimetre. 202 Mths Quest 9

30 MEASUREMENT AND GEOMETRY ONLINE ONLY 6.6 Review The Mths Quest Review is ville in customisle formt for students to demonstrte their knowledge of this topic. The Review contins: Fluency questions llowing students to demonstrte the skills they hve developed to efficiently nswer questions using the most pproprite methods Prolem Solving questions llowing students to demonstrte their ility to mke smrt choices, to model nd investigte prolems, nd to communicte solutions effectively. A summry of the key points covered nd concept mp summry of this topic re ville s digitl documents. Review questions Downlod the Review questions document from the links found in your ebookplus. Lnguge int-0889 int-0703 ngle of depression ngle of elevtion djcent cosine rtio hypotenuse inverse opposite right-ngled tringle sine rtio tngent rtio trigonometric inverses trigonometric rtios int-3206 Link to ssesson for questions to test your rediness FOR lerning, your progress AS you lern nd your levels OF chievement. ssesson provides sets of questions for every topic in your course, s well s giving instnt feedck nd worked solutions to help improve your mthemticl skills. The story of mthemtics is n exclusive Jcrnd video series tht explores the history of mthemtics nd how it helped shpe the world we live in tody. Secret society (eles-1693) delves into the world of Pythgors nd his followers, known s the Pythgorens. It highlights the structure of the society in which they lived nd how the Pythgorens cme to influence our world tody. Topic 6 Trigonometry 203 c06trigonometry.indd /07/14 11:13 AM

31 <InvestIgtIon> For rich tsk or <mesurement nd geometry> For PuZZle InvestIgtIon rich tsk The gret Pyrmid of giz yers go. it ws r nd hlf thousnd fou er ov ilt u s w z gi s nd took over The gret Pyrmid of tngulr grnite lock rec ly te xim se nd its constructed using ppro sured 230 m t the me ns sio en dim its ilt, When u 20 yers to complete. m verticl height ws Mths Quest 9 c06trigonometry.indd /07/14 2:23 PM

32 mesurement nd geometry Wll rces In the uilding industry, wll frmes re strengthened with the use of rces. These rces run etween the top nd ottom horizontl sections of the frme. Industry stndrds stipulte tht the cute ngle the rce mkes with the horizontl sections lies in the rnge 37 to 53. Sometimes, more thn one rce my e required if the frme is long one. Brce 37 to 53 Brce Brce Brce Brce 1 Cut thin strips of crdord nd rrnge them in the shpe of rectngle to represent rectngulr frme. Pin the corners to hold them together. Notice tht the frme moves out of shpe esily. Attch rce ccording to the ngle stipultion of the uilding industry. Write rief comment to descrie wht effect the rce hd on the frme. 2 Investigte wht hppens to the length of the rce required s the cute ngle with the se increses from 37 to Use your fi nding from question 2 to stte which ngle requires the shortest rce nd which ngle requires the longest rce. Most contemporry houses re constructed with ceiling height of 2.4 metres; tht is, the height of the wlls from the floor to the ceiling. Use this fct to ssist in your clcultions for the following questions. 4 Assume you hve section of wll tht is 3.5 metres long. Wht would e the length of the longest rce possile? Drw digrm nd show your working to support your nswer in the spce elow. 5 Wht would e the minimum wll length in which two rces were required? Show your working, long with digrm, in the spce provided. 6 Some older houses hve ceilings over 2.4 metres. Repet questions 4 nd 5 for frme with height of 3 metres. Drw digrms nd show your workings to support your nswers in the spce elow. 7 Tke the mesurements of wll without windows in your school or t home. Drw scle drwing of the frme on seprte sheet of pper nd show the positions in which rce or rces might lie. Clculte the length nd ngle of ech rce. Topic 6 Trigonometry 205

33 <InvestIgtIon> mesurement nd For geometry rich tsk or <mesurement nd geometry> For PuZZle CoDe PuZZle Wht does it men? The vlues of lettered ngles to the nerest degree give the puzzle s nswer code. 4 m q c 7 m 17 m 6 m 50 m n 9 m Vcuum clener: Dust: Egg: Fodder: 5 m 71 m m j 8 m t 9.1 m e r 8 m 0.3 m 0.8 m 9.3 cm 15.2 m 4 m 11.4 m o d 7.4 cm s 21 m m 19 m h 4 m 3 m 9 m m 8.2 m u w 7 m m 5 m i 0.6 m 21 m 34 m 2.3 m 0.95 m 13 m z 7.4 m 206 Mths Quest 9

34 mesurement nd geometry Activities 6.1 overview video The story of mthemtics: Secret society (eles-1693) 6.2 Wht is trigonometry? Digitl docs SkillSHEET (doc-10830): Rounding to given numer of deciml plces SkillSHEET (doc-10831): Mesuring ngles with protrctor Interctivities Investigtion: Trigonometric rtios (int-0744) IP interctivity 6.2 (int-4498) Wht is trigonometry? 6.3 Clculting unknown side lengths elesson Using n inclinometer (eles-0116) Digitl docs SkillSHEET (doc-10832): Solving equtions of the type = x to fi nd x SkillSHEET (doc-10833): Solving equtions of the type = to fi nd x x SkillSHEET (doc-10834): Rerrnging formuls WorkSHEET 6.1 (doc-10835): Trigonometry Interctivity IP interctivity 6.3 (int-4499) Clculting unknown side lengths to ccess ebookplus ctivities, log on to 6.4 Clculting unknown ngles Digitl doc SkillSHEET (doc-10836): Rounding ngles to the nerest degree Interctivity IP interctivity 6.4 (int-4500) Clculting unknown ngles 6.5 ngles of elevtion nd depression Digitl docs SkillSHEET (doc-10837): Drwing digrm from given directions WorkSHEET 6.2 (doc-10838): Trigonometry using elevtion nd depression Interctivity IP interctivity 6.5 (int-4501) Angles of elevtion nd depression 6.6 review Interctivities Word serch (int-0889) Crossword (int-0703) Sudoku (int-3206) Digitl docs Topic summry (doc-10784) Concept mp (doc-10797) Topic 6 Trigonometry 207

35 mesurement AND geometry Answers TOPIC 6 Trigonometry Exercise 6.2 Wht is trigonometry? 1 hyp dj c e opp hyp dj opp opp hyp dj d f opp dj opp 2 DE = hyp DF = opp E = GH = hyp IH = dj H = c JL = hyp KL = opp J = 3 dj hyp dj opp hyp hyp sin cos tn D 5 i sin1 2 = 4 5 i sin(α) = i g ii cos() = 3 5 ii cos(α) = h g iii tn( ) = 4 3 iii tn(α) = i h c i sin(β) = 0.8 ii cos(β) = 0.6 iii tn(β) = 1.3 d i sin(γ ) = e i sin(β) = c f i sin1γ2 = v u 6 sin() = d tn() = 2.7 p g sin(15 ) = 7 x 7 D B 8 H α O 41 A ii cos(γ ) = 7 25 ii cos(β) = c ii cos( γ) = t u cos() = e sin(35 ) = 17 t h tn() = iii tn(γ ) = 24 7 iii tn(β) = iii tn1γ2 = v t c tn() = 4 5 f sin(α) = i cos(α) = O = 33 mm A = 38 mm H = 50 mm c i sin(41 ) = 0.66 ii cos(41 ) = 0.75 iii tn(41 ) = 0.87 d α = 49 e i sin(49 ) = 0.75 ii cos(49 ) = 0.66 iii tn(49 ) = 1.15 f They re equl. g They re equl. h The sine of n ngle is equl to the cosine of its complement. 9 The missing ngle is lso 45, so the tringle is n isosceles tringle, therefore = Provided n is positive vlue, (m + n) would e the hypotenuse, s it hs greter vlue thn oth m nd (m n). 11 Ground Ldder c Brick wll 12 The rtio of the length of the opposite side to the length of the hypotenuse will increse. The rtio of the length of the djcent side will decrese, nd the rtio of the opposite side to the djcent will increse. c i 1 ii 1 iii Exercise 6.3 Clculting unknown side lengths 1 i ii sin() c As increses, so does sin(), strting t 0 nd incresing to 1. 2 i ii cos() c As increses, cos() decreses, strting t 1 nd decresing to 0. 3 i ii tn() Undefined c tn(89 ) = 57.29, tn(89.9 ) = d As increses, tn() increses, strting t 0 nd ecoming very lrge. There is no vlue for tn(90 ) m 7.04 m c mm d 2.79 cm e 6.27 m f m cm cm c cm d km e 8.43 km f m mm cm c 0.84 km d 0.94 km e 5.59 m f m g m h cm i mm 7 = 17.95, = = 15.59, = 9.00, c = c = 12.96, = 28.24, c = D B c A d D km km m m 12, Answers will vry. c AC = AB tn () 13 Answers will vry. 14 x = m h = 3.00 m c x = 2.60 m 15 w = 41 mm m 5.2 m 17 4 m Chllenge km; 0.97 km 208 Mths Quest 9

36 mesurement AND geometry Exercise 6.4 Clculting unknown ngles c 37 d 53 e 69 f 71 g 79 h 77 i c 55 d 21 e 49 f 80 g 35 h 45 i 41 j 23 k 58 l c 24 d 43 e 45 f 18 g 26 h 12 i 76 4 D B c D d C 5 x y = cos 1 (x) y = cos 1 (x) x m Answers will vry. 9 Answers will vry. 10 = 6.50 = c x = The roof would not e high enough Answers will vry. Chllenge Lrge squre: 5 cm 5 cm Lrge tringles: 5 cm 5 cm Smll squre:!50 2 Smll tringles:!50 2 cm!50 cm 2 cm!50 cm 2 Prllelogrm:!50 cm 5 cm Exercise 6.5 Angles of elevtion nd depression No, the rmps do not meet specifictions m m m 49 m 42 m m m , 3 m 9 Con: 34, John: B 11 Smi 4 m tn() = 4 30 = tn 1 Q 4 30 R c 4 m Smi m 20 m Wter s edge Adjcent 10 m 20 m Wter s edge Mrker Opposite Mrker As the dolphins swim towrds Smi, the djcent length decreses nd the opposite remins unchnged. tn() = opposite djcent Therefore, will increse s the djcent length decreses. If the dolphins re t the wter s edge, tn() = 10 = tn 1 Q 4 10 R Answers will vry. 13 Answers will vry m 14 Answers will vry. 15 Answers will vry. 16 Answers will vry. 37 c i Yes ii m Investigtion Rich tsk The Gret Pyrmid of Giz m m m Wll rces 1 Answers will vry. 2 Answers will vry requires the shortest rce nd 37 requires the longest rce m m m; 4.52 m 7 Answers will vry. Code puzzle A room with stomch Mud with the juice squeezed out A ird s home town The mn who mrried mudder Topic 6 Trigonometry 209

37 ict ctivity Lerning or erning? serchlight ID: Pro-0086 scenrio Yer 9 Students t Progressive High School hve seen the pper Are young people lerning or erning? produced y the Austrlin Bureu of Sttistics. They strt the week discussing their thoughts nd then decide to do their own reserch. Simon decides tht he will use this reserch to demonstrte to his prents tht life is very different to when they were his ge nd tht he is cple of undertking prt-time jo nd keeping up with his studies. tsk Red the rticle, summrise it nd design survey to reserch the topic. Your fi ndings will refl ect the work study lnce of the students in your school nd demogrphic. You will produce presenttion for your prents outlining the rticle nd detiling their reserch nd the conclusions you hve drwn. Process Open the ProjectsPLUS ppliction for this chpter in your ebookplus. Wtch the introductory video lesson, click the Strt Project utton nd then set up your project group. You cn complete this project individully or invite other memers of your clss to form group. Sve your settings nd the project will e lunched. Nvigte to your Medi Centre. Red the rticles nd complete the tsk elow. Answer the questions provided in the Are Young People Lerning or Erning fi le in the Medi Centre. Summrise the thoughts of the rticle in 200 words or less. Wordle is site tht cretes n imge of the words in n rticle ccording to the frequency of their usge. Use the Wordle welink in your ebookplus nd select the crete t. Copy the summry of your rticle into the text ox. Keep selecting the rndomise utton until you re hppy with the result. Print your fi nl choice. Tke screenprint of your fi nl choice. Tke it into Pint for use s slide in your presenttion. Sve your Pint fi le. Use Surveymonkey to survey 100 students t your school to determine who hs prt-time jos nd how long they work t these ech week. You re le to sk only 10 questions per survey. Record your 10 questions in Word. When plnning your survey think crefully out the types of questions you wnt to sk. For exmple, do you wnt to know why they hve prt-time jo? 210 Mths Quest 9

38 Your 100 students need to represent the school s popultion. How will you mke your smple representtive of this? Will it e rndom smple or strtifi ed smple? Explin. Wht out the gender lnce? Justify your decision. How will you notify the chosen students tht they need to do the survey? Wht instructions will you give to the students completing your survey? How long will they hve to complete it? Wht will you do to ensure they hve ll completed it? Complete the survey tle provided in the Medi Centre. Type your instructions to ech person completing the survey. Copy this into the Survey instructions templte in the Medi Centre. Anlysis. Record the results from your survey in frequency distriution tle. Use the Results tle provided in the Medi Centre. Write prgrph summrising your fi ndings. Include men, medin, mode nd rnge for ech yer group. Wht percentges of the students hve prt-time jos? Is there trend s the students get older (re more or less students working)? Include sttement s to why you still wnt to get prt-time jo. Represent your fi ndings in frequency histogrm nd frequency polygon. Use the Excel templte for your results nd include your grphs on tht sheet. If you were to do this reserch gin, is there nything tht you would chnge? Why? Are there ny etter resources for your reserch? Wht is the men, medin nd mode nd rnge of your results? Reserch. Visit the Medi Centre in your ebookplus nd open the Suur sttistics lour force y ge welink. Type in your postcode. Follow the prompts to downlod the lour force sttistics y ge, nd y sex for your postcode, nd sve. If you live in n urn re repet for rurl region, if you live in rurl region, repet for n urn re. Sve the Excel fi les columns for the rurl nd urn postcodes. Use Excel to crete column grph with series of columns. See Smple spredsheet fi le. Include the grph in your Prezi fi le with n explntion of wht you did. Visit the Medi Centre nd downlod the Prezi smple nd the Prezi plnning templte to help suggested softwre ProjectsPLUS microsoft Word PowerPoint, Prezi, Keynote or other presenttion softwre microsoft Excel Surveymonkey Wordle you prepre your presenttion. Your Medi Centre lso includes imges tht cn help to liven up your presenttion. As you rrnge your imges on your Prezi pge mke them form lrge circle so tht they fl ow smoothly when they re linked nd presented. Use the Prezi templte to develop your presenttion. Rememer tht you re trying to convince your prents tht you should e le to undertke prt-time jo. Mke sure you include ll the results of your reserch, nd tht your presenttion will gr their ttention. To include tles in Prezi you need to tke them into pint nd sve the fi le s jpeg in order to uplod them. Use Word to type up your dilogue to your prents when you present your cse ( words). Topic 6 Trigonometry 211