Naming the sides of a right-angled triangle

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1 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 inlude surveying lnd, rhiteture, mesuring distnes nd determining heights of inessile ojets. In this hpter reltionships etween the sides nd ngles of right-ngled tringle will e eplored. Nming the sides of right-ngled tringle The longest side of right-ngled tringle (the side opposite the right ngle) is lled the hypotenuse. In order to nme the remining two sides nother ngle, lled the referene ngle, must e dded to the digrm. The side tht is ross from the referene ngle,, is lled the opposite side, nd the remining side (the side net to the referene ngle) is lled the djent side. Note: If there is no referene ngle mrked, only the hypotenuse n e nmed. Adjent WORKED EXAMPLE 1 Lel the sides of the right-ngled tringle shown using the words hypotenuse, djent nd opposite. 1 The hypotenuse is opposite the right ngle. 2 Lel the side net to ngle s djent nd the side opposite ngle s opposite. Adjent 176 Mths Quest 9

2 Trigonometri rtios Trigonometry is sed upon the rtios etween pirs of side lengths, nd eh one is given speil nme s follows. In ny right-ngled tringle: int-0744 sine () = opposite hypotenuse osine () = djent hypotenuse tngent () = opposite djent Adjent These rules re revited to: sin() = O H, os() = A H nd tn() = O A. The following mnemoni n e used to help rememer the trigonometri rtios. SOH CAH TOA O SOH H H A O A CAH TOA WORKED EXAMPLE 2 For this tringle, write the equtions for the sine, osine nd tngent rtios of the given ngle Lel the sides of the tringle Adjent 2 Write the trigonometri rtios. sin() = O H, os() = A H, tn() = O A 3 Sustitute the vlues of A, O nd H into eh formul. sin() = 5 12, os() = 13 13, tn() = Mths Quest 9

3 5 WE2 For eh of the following tringles, write the epressions for rtios of eh of the given ngles: i sine ii osine iii tngent i g h α 0.9 β d 7 γ e β f v u 6 WE3 Write the trigonometri rtio tht reltes the two given sides nd the referene ngle in eh of the following tringles. 25 t γ d 2.7 p e t 17 f α g 7 h i α 15 UNDERSTANDING 7 MC Wht is the orret trigonometri rtio for the tringle shown t right? γ A tn(γ) = B sin(γ) = C os(γ) = D sin(γ) = Trigonometry 181

4 6.3 Clulting unknown side lengths Vlues of trigonometri rtios The vlues of trigonometri rtios n e found using lultor. Eh lultor hs severl modes. For the following lultions, your lultor must e in degree mode. WORKED EXAMPLE 4 Evlute eh of the following, giving nswers orret to 4 deiml ples. sin(53 ) os(31 ) tn(79 ) 1 Set the lultor to degree mode. Write the first 5 deiml ples. TI CASIO WRITE sin(53 ) = Round orret to 4 deiml ples Write the first 5 deiml ples. os(31 ) = Round orret to 4 deiml ples Write the first 5 deiml ples. tn(79 ) = Round orret to 4 deiml ples Finding side lengths If referene ngle nd ny side length of right-ngled tringle re known, it is possile to find the other sides using trigonometry. WORKED EXAMPLE 5 Use the pproprite trigonometri rtio to find the length of the unknown side in the tringle shown. Give your nswer orret to 2 deiml ples. 1 Lel the given sides. TI CASIO Adjent 16.2 m m 58 2 These sides re used in TOA. Write the rtio. tn() = O A 3 Sustitute the vlues of, O nd A into the tngent rtio. tn(58 ) = Solve the eqution for tn(58 ) = = 16.2 tn(58 ) 5 Clulte the vlue of to 3 deiml ples, then round the nswer to 2 deiml ples. = m Trigonometry 183

5 WORKED EXAMPLE 6 Find the length of the side mrked m in the tringle t right. Give your nswer orret to 2 deiml ples. 1 Lel the given sides. TI CASIO Adjent 17.4 m m 17.4 m m 2 These sides re used in CAH. Write the rtio. os() = A H 3 Sustitute the vlues of, A nd H into the os(22 ) = 17.4 m osine rtio. 4 Solve for m: Multiply oth sides y m. Divide oth sides y os(22 ). 5 Clulte the vlue of m to 3 deiml ples, then round the nswer to 2 deiml ples. m os(22 ) = 17.4 m = 17.4 os (22 ) m = m m WORKED EXAMPLE 7 Benjmin set out on ushwlking epedition. Using ompss, he set off on ourse N 70 E (or 070 T) nd trvelled distne of 5 km from his se mp. N 5 km E 70 Bse mp How fr est hs he trvelled? How fr north hs he trvelled from the se mp? Give nswers orret to 2 deiml ples. 1 Lel the esterly distne. Lel the northerly distne y. Lel the sides of the tringle:,, Adjent. Adjent y 70 5 km 184 Mths Quest 9

6 2 To lulte the vlue of, use the sides of the tringle: = O, 5 = H. These re used in SOH. Write the rtio. 3 Sustitute the vlues of the ngle nd the pronumerls into the sine rtio. sin() = O H sin(70 ) = 5 4 Mke the sujet of the eqution. = 5 sin(70 ) 5 Evlute to 3 deiml ples, using lultor. = Round to 2 deiml ples km 7 Answer the question in sentene form. Benjmin hs trvelled 4.70 km est of the se mp. 1 To lulte the vlue of y, use the sides: y = A, 5 = H. These re used in CAH. Write the rtio. 2 Sustitute the vlues of the ngle nd the pronumerls into the osine rtio. os() = A H os(70 ) = y 5 3 Mke y the sujet of the eqution. y = 5 os(70 ) 4 Evlute y using lultor. = Round the nswer to 2 deiml ples km 6 Answer the question in sentene form. Benjmin hs trvelled 1.71 km north of the se mp. Eerise 6.3 Clulting unknown side lengths INDIVIDUAL PATHWAYS PRACTISE 1 3, 4, 5, 6 f, 7 9, 11, 12 CONSOLIDATE 1 3, 4 d, 5 d, 6d i, 7 10, MASTER 1 3, 4d f, 5d f, 6g i, 7 10, 11, REFLECTION Wht does sin(60 ) tully men? Individul pthwy intertivity int-4499 FLUENCY 1 WE4 Evlute the following orret to 4 deiml ples. i sin(55 ) ii sin(11.6 ) Copy nd omplete the tle elow. Use your lultor to find eh vlue of sin() orret to 2 deiml ples sin() Summrise the trend in these vlues. 2 Evlute the following orret to 4 deiml ples. i os(38 ) ii os(53.71 ) do do do Trigonometry 185

7 Copy nd omplete the tle elow. Use your lultor to find eh vlue of os() orret to 2 deiml ples os() Summrise the trend in these vlues. 3 Evlute the following orret to 4 deiml ples. i tn(18 ) ii tn(51.9 ) Copy nd omplete the tle elow. Use your lultor to find eh vlue of tn() orret to 2 deiml ples tn() Find the vlue of tn(89 ) nd tn(89.9 ). d Wht do you notie out these results? 4 WE5 Use the pproprite trigonometri rtios to find the length of the unknown side in eh of the tringles shown. Give the nswers orret to 2 deiml ples. eles m m 27 y mm z d p m e 29.5 m z f s m 22 5 WE6 Use the pproprite trigonometri rtio to find the length of the unknown side in eh of the tringles shown. Give the nswers orret to 2 deiml ples. k m 16 m 52 s 16.1 m q 5 d 5.72 km 66 e e f t p 7.7 km m 186 Mths Quest 9

8 6 Find the length of the unknown side in eh of the following tringles, orret to 2 deiml ples. (Note: In some ses the unknown will e in the numertor nd in other ses it will e in the denomintor.) 13 l mm m km d km e m 8 d f m y y g 63.2 m 30 m h n m i z mm 7 Find the lengths of the unknown sides in the tringles shown, orret to 2 deiml ples UNDERSTANDING 8 MC The vlue of orret to 2 deiml ples is: A B C D The vlue of orret to 2 deiml ples is: A mm B mm C mm D mm The vlue of y orret to 2 deiml ples is: A B 7.94 C 1.37 D 0.23 d The vlue of y orret to 2 deiml ples is: A 0.76 km B 1.79 km C 3.83 km D 3.47 km 25 y y 1.62 km mm Trigonometry 187

9 9 WE7 A ship tht ws to trvel due north veered off ourse nd trvelled N 80 E (or 080 T) for distne 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 resue heliopter spots missing surfer drifting out to se on his dmged ord. The heliopter desends vertilly to height of 19 m ove se level nd drops down n emergeny rope, whih the surfer grips. Due to the wind the rope swings t n ngle of 27 to the vertil, s shown in the digrm. Wht is the length of the rope? 11 Wlking long the ostline, Mihelle (M) looks up through n ngle of 55 nd sees her friend Helen (H) on top of the liff t the lookout point. How high is the liff if Mihelle is 200 m from its se? (Assume oth girls re the sme height.) m H m M 188 Mths Quest 9

10 Finding the ngle when 2 sides re known If the lengths of ny 2 sides of right-ngled tringle re known, it is possile to find n ngle using inverse sine, inverse osine or inverse tngent. WORKED EXAMPLE 10 Determine the vlue of in the tringle t right. Give your nswer orret to the nerest degree Lel the given sides. These re used in CAH. Write the rtio Adjent os() = A H 2 Sustitute the given vlues into the osine rtio. os() = is the inverse osine of = os Evlute. = Round the nswer to the nerest degree. 79 WORKED EXAMPLE 11 Roert enjoys wter skiing nd is out to try new rmp on the Hwkesury River. The inlined rmp rises 1.5 m ove the wter level nd spns horizontl distne of 6.4 m. Wht is the mgnitude (size) of the ngle tht the rmp mkes with the wter? Give the nswer orret to the nerest degree. 6.4 m 1.5 m 1 Drw simple digrm, showing the known lengths nd the ngle to e found. 6.4 Adjent Lel the given sides. These re used in TOA. Write the rtio. 3 Sustitute the vlues of the pronumerls into the tngent rtio. tn() = O A tn() = is the inverse inverse tngent of = tn Mths Quest 9

11 5 Evlute. = Round the nswer to the nerest degree Write the nswer in words. The rmp mkes n ngle of 13 with the wter. Eerise 6.4 Clulting unknown ngles INDIVIDUAL PATHWAYS PRACTISE 1, 2, 3 f, 4 10 CONSOLIDATE 1d f, 2, 3d h, 4 11 MASTER 1g i, 2, 3e i, 4 13 REFLECTION Why does os(0 ) = 1? Individul pthwy intertivity int-4500 FLUENCY 1 WE8 Evlute eh of the following, orret to the nerest degree. sin 1 (0.6294) os 1 (0.3110) tn 1 (0.7409) d tn 1 (1.3061) e sin 1 (0.9357) f os 1 (0.3275) g os 1 (0.1928) h tn 1 (4.1966) i sin 1 (0.2554) 2 WE9 Determine the size of the ngle in eh of the following. Give nswers orret to the nerest degree. sin() = sin() = sin(β) = d os(β) = e os(α) = f os(α) = g tn() = h tn() = 1 i tn() = j sin() = k os() = l tn(α) = WE10 Determine the vlue of in eh of the following tringles. Give nswers orret to the nerest degree. d e f do g 26 h i Trigonometry 193

12 4 MC If os() = , the vlue of orret to 2 deiml ples is: A B C D If sin() = , the vlue of orret to 2 deiml ples is: A B C D The vlue of in the tringle shown, orret to 2 deiml ples, is: A B C D d The vlue of in the tringle shown, orret to 2 deiml ples, is: A B C D Copy nd fill in the tle elow y = os 1 () Plot the ove tle on grph pper or with spredsheet or suitle lultor. UNDERSTANDING 6 A piee of fri mesuring 2.54 m y 1.5 m hs design onsisting of prllel digonl stripes. Wht ngle does eh digonl mke with the length of the fri? Give your nswer orret to 2 deiml ples m 1.5 m 7 WE11 Dnny Dingo is perhed on top of liff 20 m high wthing n emu feeding 8 m from the se of the liff. Dnny hs purhsed flying ontrption, whih he hopes will help him pture the emu. At wht ngle to the liff must he swoop to th his prey? Give your nswer orret to 2 deiml ples. 20 m 8 m 194 Mths Quest 9

13 6.5 Angles of elevtion nd depression When looking up towrds n ojet, n ngle of elevtion is the ngle etween the horizontl line nd the line of vision. When looking down t n ojet, n ngle of depression is the ngle etween the horizontl line nd the line of vision. Line of vision Angle of elevtion Horizontl Horizontl Angle of depression Line of vision Angles of elevtion nd depression re mesured from horizontl lines. WORKED EXAMPLE 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 entimetre? 1 Drw simple digrm. The ngle of elevtion is 38 from the horizontl. h 2 Lel the given sides of the tringle. These sides re used in TOA. Write the rtio Adjent tn(38 ) = h 10 3 Multiply oth sides y tn(38 ) = h 4 Clulte orret to 3 deiml ples. h = Round to 2 deiml ples Write the nswer in words. The tree is 7.81 m tll. WORKED EXAMPLE 13 A lighthouse, 30 m tll, is uilt on top of liff 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 liff. Angle of depression 30 m 180 m 3700 m Trigonometry 197

14 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. T 210 S 3700 Adjent C 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 7 Write the nswer in words. The ngle of depression of the ship from the top of the lighthouse is 3. Note: In Worked emple 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 euse the ngle of depression nd the ngle of elevtion re lternte (or Z ) ngles. Angle of depression Angle of elevtion This n e generlised s follows: For ny two ojets, 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. Angle of depression of A from B B A Angle of elevtion of B from A Eerise 6.5 Angles of elevtion nd depression INDIVIDUAL PATHWAYS REFLECTION Why does the ngle of elevtion hve the sme vlue s the ngle of depression? PRACTISE 1 6, 8, 10, 12 CONSOLIDATE 1 6, 8, Individul pthwy intertivity int-4501 MASTER 1 3, 6, 7, Mths Quest 9

15 FLUENCY 1 WE12 Building speifitions require the ngle of elevtion of ny rmp onstruted for puli use to e less thn 3. 1 m 7 m do Rmps eing onstruted t new shopping entre re eh mde in the rtio 7 m horizontl length to 1 m vertil height. Find the ngle of elevtion of these rmps nd, hene, deide whether they meet uilding speifitions. 2 A lifesver stnding on his tower 3 m ove the ground spots swimmer eperiening 12 diffiulty. The ngle of depression of the swimmer from the lifesver is 12. How fr is the swimmer from the lifesver s tower? (Give your nswer orret to 2 deiml ples.) 3 m 3 From the top of lookout 50 m ove the ground, the ngle of depression of mp site tht is level with the se of the lookout is 37. How fr is the mp site from the se of the lookout? 50 m UNDERSTANDING 4 From resue heliopter 80 m ove the oen, the ngles of depression of two shipwrek survivors re 40 nd 60 respetively. If the two silors nd the heliopter re in line with eh other: drw lelled digrm to represent the sitution lulte the distne 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. Trigonometry 199

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