Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.


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1  Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help For eh tringle, find () the length of the leg l nd () the length of the leg djent to l.. 9; ; " New Voulr sine osine identit 9 0; "9 esson Pln Ojetives o use sine nd osine to determine side lengths in tringles Emples Writing Sine nd osine tios elworld onnetion Using the Inverse of osine nd Sine Mth kground Using Sine nd osine in ringles he tngent rtio, s ou hve seen, involves oth legs of right tringle. he sine nd osine rtios involve one leg nd the hpotenuse. sine of & osine of & leg / hpotenuse leg djent to / hpotenuse Hpotenuse eg djent to eg unit irle hs rdius nd enter (0,0) in the oordinte plne. For ll rel vlues of u, the point tht is rehed trveling u rdins from point (,0) in ounterlokwise diretion hs oordintes (os u, sin u). More Mth kground: p. D elworld onnetion For n ngle of given sie, the sine nd osine rtios re onstnt, no mtter where the ngle is loted. Quik hek hese equtions n e revited: djent sin hpotenuse os hpotenuse EXMPE Writing Sine nd osine tios Use the tringle to write eh rtio.. sin sin hpotenuse. os djent os hpotenuse. sin sin hpotenuse d. os djent os hpotenuse. Write the sine nd osine rtios for &X nd &Y. See right.. ritil hinking When does sin X os Y Eplin. sin X os Y when lx nd ly re omplementr. X Z 0 6 esson  Sine nd osine tios 9 Y 6. sin X 0 ; os X 0 ; sin Y 0 ; os Y 6 0 esson Plnning nd esoures See p. E for list of the resoures tht support this lesson. ell inger Prtie hek Skills You ll Need For intervention, diret students to: Writing ngent tios esson : Emple Etr Skills, Word Prolems, Proof Prtie, h. Speil Needs Present the mnemoni devie SOHHO for the definition of the three trigonometri rtios: Sine is Opposite over Hpotenuse; osine is djent over Hpotenuse; ngent is Opposite over djent. lerning stle: verl elow evel Hve students drw nd mesure right tringles to mke tle of sine nd osine vlues for the ngles in the set {0, 0,..., 0 }. lerning stle: visul 9
2 . eh uided Instrution Visul erners Hve students displ poster listing the trigonometri rtios. dditionl Emples Use the tringle to find sin, os, sin, nd os. 6 sin 0, os 0, 6 sin 0, os 0 0ft wire supporting flgpole forms ngle with the flgpole. o the nerest foot, how high is the flgpole 0 ft 0 6 For: Sine nd osine tivit Use: Intertive etook,  Quik hek One w to desrie the reltionship of sine nd osine is to s tht sin os (90  ) for vlues of etween 0 nd 90. his tpe of eqution is lled n identit euse it is true for ll the llowed vlues of the vrile. You will disover other identities in the eerises. EXMPE elworld onnetion stronom he trigonometri rtios hve een known for enturies peoples in mn ultures. he Polish stronomer Niolus opernius () developed method for determining the sies of orits of plnets loser to the sun thn Erth. he ke to his method ws determining when the plnets were in the position shown in the digrm, nd then mesuring the ngle to find. If. for Merur, how fr is Merur from the sun in stronomil units (U) One stronomil unit is defined s the verge distne from Erth to the enter of the sun, out 9 million miles. sin. Use the sine rtio. sin. Solve for Use lultor. Merur is out 0. U from the sun.. If 6 for Venus, how fr is Venus from the sun in U. out how mn miles from the sun is Venus Merur 66,960,000 mi;,0,000 mi When ou know the leg nd hpotenuse lengths of right tringle, ou n use inverse of sine nd inverse of osine to find the mesures of the ute ngles. EXMPE Erth Using the Inverse of osine nd Sine U Sun not to sle out 0. U 6 ft right tringle hs leg. units long nd hpotenuse.0 units long. Find the mesures of its ute ngles to the nerest degree., 6 esoures Dil Notetking uide  Dil Notetking uide  dpted Instrution losure right tringle whose hpotenuse is m long ontins 6 ngle. Find the lengths of its legs to one deiml ple. 6. m,.6 m Prolem Solving Hint hink of os . (.0) s the ngle whose. osine is.0, nd sin . (.0) s the ngle whose sine is. the quotient.0. Quik hek Find m& to the nerest degree. 0 hpter ight ringles nd rigonometr. F. O Method Method os..0 Find the trigonometri rtio. sin..0 m& os Q..0 Use the inverse. m& sin  Q..0 OS ..0 Use lultor. SIN m& < m& < Find the vlue of. ound our nswer to the nerest degree dvned erners Enourge students to mke onjetures out the vlues of sin 0, os 0, sin 90, nd os 90, defend their onjetures, nd then hek the vlues on lultor. lerning stle: verl English nguge erners E Help students distinguish etween sine nd inverse sine. he sine of n ngle is rtio, or numer. he inverse sine of numer, or rtio, is n ngle mesure. So, inverses re used to find ngle mesures. lerning stle: verl
3 EXEISES For more eerises, see Etr Skill, Word Prolem, nd Proof Prtie. Prtie nd Prolem Solving Prtie Emple for Help Emple (pge 9) Emple (pge 0) Emple (pge 0) Write the rtios for sin M nd os M... M M. 9 " ; 9 ; 9 Find the vlue of. ound nswers to the nerest tenth Esltors n esltor in the suw sstem of St. Petersurg, ussi, hs vertil rise of 9 ft 9. in., nd rises t n ngle of 0.. How long is the esltor ound our nswer to the nerest foot. 0 ft Find the vlue of. ound nswers to the nerest degree ; " M. Prtie ssignment uide 0 hllenge 6 est Prep 0 Mied eview  Homework Quik hek o hek students understnding of ke skills nd onepts, go over Eerises,,,,. Error Prevention! Eerises 6, Some students m need help solving equtions with the vrile in the denomintor. eview tehniques suh s rossmultiplition nd tking the reiprol of eh side. Eerise ell students tht there is lso otngent rtio. sk: Wht do ou think is the otngent rtio djent ppl Your Skills. onstrution rlos is plnning to uild grin in with rdius of ft. He reds tht the reommended slnt of the roof is. He wnts the roof to overhng the edge of the in ft. Wht should the length e ive our nswer in feet nd inhes. out ft in. ft overhng PS Enrihment uided Prolem Solving etehing dpted Prtie Prtie Nme lss Dte Use wht ou know out trigonometri ft rtios (nd other identities) to show tht eh eqution is n identit. 0. See mrgin.. tn X sin X 9. sin X os X tn X 0. os X sin X os X tn X Prtie  lger Find the geometri men of eh pir of numers.. nd. nd 6. nd. nd. 0 nd nd 0 lger efer to the figure to omplete eh proportion h h 0.. h. lger Find the vlues of the vriles. h h Similrit in ight ringles elworld onnetion orn tht fills the in in Eerise would mke,00 gllons of ethnol.. Error nlsis student sttes tht sin. sin X euse the lengths of Y the sides of # re greter thn the lengths of the sides of #XYZ. Is the student orret Eplin. Z X No; the > re M nd the sine rtio for is onstnt esson  Sine nd osine tios. he ltitude to the hpotenuse of right tringle divides the hpotenuse into segments 6 in. nd 0 in. long. Find the length h of the ltitude.. sin X os X hp. dj. hp. dj. tn X 9. os X tn X dj. hp. dj. hp. sin X 0. sin X tn X hp. dj. dj. hp. os X
4 . ssess & eteh esson Qui Use this figure for Eerises nd.. Write the rtios for sin nd 6 0 sin. sin, sin. Write the rtios for os nd 0 6 os. os, os Use this figure for Eerises nd.. Find to the nerest tenth..0. Find to the nerest tenth..6 Use this figure for Eerises nd P N Find to the nerest degree. 6. Find to the nerest degree. 6 lterntive ssessment Hve students write two mesurement prolems involving distnes in our shool. Students lso should show how to solve one prolem using the sine rtio nd the other prolem using the osine rtio. M nline Homework Help Visit: PHShool.om We ode: ue00. he re equl; es; he sine nd osine of omplementr ' re.. Smple: osine of l sine of the ompl. of l.. Yes; use n trig. funtion nd the known mesures to find one other side. Use the Pthgoren hm. to find the rd side. Sutrt the ute l mesure from 90 to get the other l mesure. Find the vlues of w nd then. ound lengths to the nerest tenth nd ngle mesures to the nerest degree.... PS w 0 w 6 0 w w ;. w ;. w 6.;.6.. In #, how does sin ompre to os Is this true for the ute ngles of other right tringles. eding Mth he word osine is derived See left. 6 from the words omplement s sine (see pge 69). Whih ngle in # is the omplement of & 0 Of & l; l. Eplin wh the derivtion of the word osine mkes sense. See left. 6. Find eh rtio. P. sin P ". os P.. " ". d. ". sin d. os e. Mke onjeture out how the sine nd osine of ngle re relted. he re equl.. Writing eon sid tht if she hd digrm tht showed the mesure of one ute ngle nd the Q length of one side of right tringle, she ould find the mesure of the other ute ngle nd the lengths of the other sides. Is she orret Eplin. See left.. Find eh rtio.. sin S. os S. sin.. " " d. os e. Mke onjeture out how the sine nd e. os 0 " sin 0 osine of 0 ngle re relted. See left. f. sin 60 " os 60 f. Mke onjeture out how the sine nd 60 osine of 60 ngle re relted. See left. S Proof 9. For right # with right &, prove eh of the following.. sin, no mtter how lrge & is.. See mrgin.. os, no mtter how smll & is. 0 d. nswers m vr. Smples re given. 0. sin X for X 9.9; no PHShool.om For: rphing lultor proedures We ode: ue hllenge hpter ight ringles nd rigonometr 0. rphing lultor Use the E feture of our grphing lultor to stud sin X s X gets lose (ut ) to 90. In the Y sreen, enter Y sin X.. Use the SE feture so tht X strts t 0 nd hnges. ess the E. From the tle, wht is sin X for X Perform numeril oom in. Use the SE feture, so tht X strts with 9 nd hnges 0.. Wht is sin X for X 9.9. ontinue to numerill oom in on vlues lose to 90. Wht is the gretest vlue ou n get for sin X on our lultor How lose is X to 90 Does our result ontrdit wht ou re sked to prove in Eerise 9 See left. d. Writing Use right tringles to eplin the ehvior of sin X found ove. See mrgin. Show tht eh eqution is n identit showing tht eh epression on the left simplifies to.. See mrgin.. (sin ) + (os ). (sin ) + (os ).  (tn ).  ( os ) ( sin ) ( tn ). Show tht (tn )  (sin ) (tn ) (sin ) is n identit. 0 N See mrgin. 9. nswers m vr. Smples re given.. Sine sin hp., if sin, then hp., whih is impossile. dj.. Sine os hp., if os, then dj. hp., whih is impossile. 0. d. For ' tht pproh 90, the side gets lose to the hp. in length, so hp. pprohes.
5 elworld onnetion Polnd honored opernius with this 000lot note, lst used in 99. est Prep 6. stronom opernius devised method different from the one in Emple in order to find the sies of the orits of plnets frther from the sun thn Erth. His method involved noting the numer of ds etween the times tht plnet ws in the positions leled nd in the digrm. Using this time nd the numer of ds in eh plnet s er, he lulted nd d.. For Mrs,. nd d 0.. How fr is Mrs from the sun in stronomil units (U) out. U. For Jupiter,.9 nd d 00.. How fr is Jupiter from the sun in stronomil units Outer plnet s orit Erth s orit d Sun not to sle U out. U est Prep esoures For dditionl prtie with vriet of test item formts: Stndrdied est Prep, p. 6 estking Strtegies, p. 60 estking Strtegies with rnsprenies Eerise 6 Point out tht opernius s method depends on the sun, Erth, nd outer plnets eing in line t one point in time nd forming right ngle t the other point in time. Multiple hoie Short esponse Mied eview. Wht is the vlue of to the nerest whole numer... D. 6. Wht is the vlue of to the nerest tenth F.... H..6 J.. 9. Wht is the vlue of to the nerest whole numer... D. 0. Use the figure t the right.. Find m&. Show our work.. See mrgin.. Find m& two different methods. Show our work. H (sin ) ± (os ) Q ± ± Q. (os ) (tn ) ( ). (sin ) (tn ) Q Q for Help esson  esson  esson 6 lesson qui, PHShool.om, We ode: u00. (sin ) ± (os ) Q ± ± Q Find the vlue of. ound nswers to the nerest tenth he wll of room is in the shpe of golden retngle. If the height of the wll is ft, wht re the possile lengths of the wll to the nerest tenth.9 ft or.9 ft Find the vlue of for eh prllelogrm esson  Sine nd osine tios. (tn ) (sin ) ( ) () () (tn ) (sin ) () () 0. []. os 0 ml os ( 0) N 6. ml N 90 6 O ml sin ( 0) N [] one ngle found orretl
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