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8.3 Trigonometric Ratios sine, cosine, tangent io| - c*T1 - ro A . ^ opposite side '^ SmA== hypotenuse COSA_ a_ c- = 'oçroj b c- adjacent side hypotenuse q. opposite side _ -b tanA- adjacent side Y 3 4 Z _t_ f tanY: + sinX- If sin A sin45o 5 l,6 then tan A -- + sin 30o : LL tanMcosN- _t IL 5 /3 (r' = st+ x' - 36 - zl+\z ll=x2 171 tan 45" - tan 60o - =x 5in lp f$^'l 28 = ¿utk¡0")As sL= # Å X = tL(ces tr¡ /(t X--l(,P2 T6 x- X- 7.51 l. zt t^(=T E x- 72_.31 o cosx =T ?l sin-l - 'ï1"{ån "oí' ' '!{''!: Sin X -- -t 2l X=s¡rí'(+) The grade of the road is 7Yo. What angle does the road make with the horizonÍ,al? -7 tj^^X -- f.ô tol'(ft) f, = \.oo f, = loo A rhombus has diagonals that arc 4 meters and 10 meters long. Find the measures of the angles of the rhombus. ,l tztvry?l v V LI = - -t r s\ G8. Lo" Z- (tr tl = tZ zo) l3G.¿1 " = 43. G" 'fi CoS Lo5 = 2- IL lL (cos 6r) zLl(cos =Av -- * cs) = \ z4 (ø" es) lÇ.to =!z 7= J!x Xl0. tq 2 = 21,36 24 ?, X: l1.LlE K-y 31'2:6 -- K - lb'to 24 CosS =X 5=-Ez1 þ'- za(a"ss) F - SLl.Zt Assignment: p 510 #1 l, 13, 15, 16, 18, 20,21,,23,24, 26,27,29, 29,34,36,38, 48, 53,54 Problems Using Angles of Elevation and Depression The angle of elevation from a sailboat to the top of sailboat from the shore? lighthouse on the shore is 16". How far is the a ún tb crl?l -H t4,¡-lb 'N= A sonar operator on a cruiser detects a submarine at a distance How deep is the sub? 5öÐ --Ët ¿tLl .19 +t of 50O meters and an angle of depression of 37" x trrr S7 = .;6 x = (û" nYÍu) The grade of the road is SVo. What angle does the road make with the horizontal? too Galileo used the "Leaning Tower of Pisa" to conduct his experiments on the laws of gravity. When he dropped objects from the top of the 55-meter tower, they landed 4.8 meters from the base of the tower, How many degrees does the tower lean off vertical? 5f- &^'¡ e -é {.8 ê= square based pyramid was uncovered in Egypt.The base is intactand measures 120 meters on a side. The top portion has eroded away, but what remains of each face of the pyramid forms an angle of 65" with the ground. V/hat was the original height of the pyramid? A þ^^ (t'f x '- bo ¡d.il lY= tz l---..-- g looks ngle of part. A pers Two buildings are 100 feet apa building. lf th e angle of elevat 'ation to the reight of the the building is 30', find the hei eet at the other the bottom x åqn z5=tæ l. = 4b. *¡^ 3e -- T IQ 7l V= I lo'f I . U3 of Law of Sines b a sin A X: sin B sin C G6.a1 2s sin (nnn) = 2o (stnx) zs (sin *) = gi Ì4 X z-s srlt x: x (tt ßnv) \ \Ã-/ =X l.4.13 t0 5ìn'|L ^. K St,n 1 1t I ro-þ,vrr) = X to (stnt) 3M 'lz -_ (srrt W) X o 42; tul f,; l{ 7L = l,^ro (3M æ) = 2z'(stnx) l&l (Srvr fo) x- 4- 36.81 2L = s jr1 K lo ut' (* L::'') = x A surveyor has the job of determining the distance across the Rio Grande Gorge in northern New Mexico. Standing at one side of the ridge, she measures the angle formed by the edge of the ridge and the line of sight to a tree on the other side of the ridge. She then walks along the ridge 3 15 north and measures the angle formed by the edge of the ridge and the new line of sight to the same tree. If the first angle is 80o, and the second angle is 85o, find the distance across the gorge. ( V 850 I É-[ 315 -- 80"- J Sin 3tç s¡a Í5 = tllt-,s7 lS ar (srnæ) = x Sts ("tnrc) (rt ø) ("in ts) à-- (s;nwXn1rs7) 1t14.Ôl =X lllÍ.f7 --x Law of Cosines Notes Filled In.pdf - Page þ.e law oF cB: 1 cosINE$ q'[03 'q h= q.ri fvt \.Jt 54Iq: I t¡1 = b-s LD3ü4 = F1 n4a{ =t lr i = (t/ @S 44 t cß: { C-B' cb nL + 8,bvL Law of Cosines Notes Filled ln.pdf - Page 2 \i iì Bi ----{ C.-Find the value of --------ô a. L. I a- = (c-*f+ht LC-)LJ T- n / / / i t K= b,l'r,¡ { I LL a- = (,-- & =ú \-, n tL LcK+x'+h- Lctô'sa) *L7- I LL t +L - 1Á.@A Law of Cosines Notes Filled ln.pdf - Page 3 [,aw of Cosines MN: La\ Jll É= ¡nL -t-4L .!t -- (zr)t * fr)t- L(Lilffi)cos(o+) .f1e551 4' =- 3S\8 -11 A -- [,01, Law of Cosines Notes Filled ln.pdf - Page 4 -rL L- K=V ++. - LY? cos Å L- [r t)t+ (r, Ðt- LFtXt¿) cpS{ (q .L ¿) q5 ..L ù' - (l L)' æ (ro- ¿\L = l LO-l)(a'¿) c.,âsf t L.l ¿)L - (-1)t - Lr,-r) =- acsx - L(r t)(o'¿) Law of Cosines Notes Filled ln.pdf - Page 5 *\ P <.ifl _r. t û'C rf srn 3L \.L t3.w Ì .33 Law of Cosines Notes Filled ln.pdf - Page 6 An air fraffic controller is tracking a plane 2.1 kilometers due south of the radar tower. A second plane is located 3.5 kilometers form the tower at a heading of N 75'E (75'east of north). Tc the nearest tenth of a kilometer, how far apart are the two planes? Plane D B Tow er b t 2.1 km = f+ ** L',bcos D Plane A - L(3+lX.^lÓs) Å1 = M,4trqN'" LAW OF'SINES v: x lÔ. ,lL fv\=¡[.?{ y æ- 97= Sin r1 1 ')( r -- n/t+P = ll- 14 + 1'tl YÉ tqn 47 = 1'bz -- l0,t1z [ = ll 5,rn t|l 1'?2 f' = ânqz ry_ sin ^/-A sin ^/B - órn 33 = x r = ?=1.,1 h (ssn zs) = J. 6L (os33=fr ' ?.fl t^-- tl (cos P) = ll.1f C h h- b'slnA b h- a-'sin B I, 4_ -bsinA sin A sin B Name: Út" Geometry: Angles of Elevation and Depression Solve each problem using trigonometric ratios. Show all work. I , A twenty-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What angle does the ladder make with the ground? A with a cable secured at the top of the tower and tied 30 meters from the base, V/hat angle does the cable from with the vertical tower? SO-meter vertical tower is braced - \ Ito.rn x - lD to^-' 5ù H, / /.\,J, i ,l/, I ,.; j 3o A water slide is 25 meters long and stafts at the top of a tower that is 2l elevation between the slide and the ground. S,, meters n'/"| i r' high. Find the angle of .z- 5l"t'10 ) .) riangle. pc"i0O! 7 [; sì UJI)r\¿" vo , ì A,^^ot D4 !r]-.,ry ^' - =tV'.('J),t2'.) g- = [os r-J ':-..--, c'ru "P'1Ù" Findtheareaof aparallelogramif onesideis l0cm.,anothersideisScm,andtheanglebetweenthrose twosidesis{7"' ,-J8-- ,--, lÙ i,ud tn.y',Afl* ¡0.¡:ffrc^ x ll ll-l ¡ à lr\o5"85 lÙ'5,85 r{-'þ6,Í^2 x=5'65'h*J* For the right triangle at right, find the following values: a) sinA= 3 d)sinB= \ b) cosA= e) cosB= Ð (si cc c) (sin Þ A)2 (o.)' Find the me +' i B), = (?Y in right triangle ABC 8(6, - -fnn^A'å ù-qjÐ 8. If the diagonals of a rhombus have lengths of 12 meters and 8 meters,find the measures of the angles of the rhombus. Äþ, ,fttr <{ \ V \ =9i '2 *o^)'9u "L= ^o = I l' ' Ue 'Òbfua'îû¿ -ta^ I.rl,3E'crr,rl\(t'¡n(rc To be used safely a ladder should make an angle of about 75' with the ground. a) b) How far should the foot of a 20-foot ladde v How long should a ladder be in order to reach a window that is 36 feet above the ground? 1. Jb(K $lrya5 , fu 10. From the top of a 575-foot high bluff, a rescue scout looks down upon a hot-air balloon accident. The angle of depression is 7o. How f is the accident from the rescue team at the foot of the bluff? 'ua11 a I u 1,:'j o I (, _ .') 1. A road climbs at a 8o angle. What is the grade of the road? lcr,u- i -E' ! tü\) 2 12, (Give an exact answer) If tan A (/\a -I , then sin ìr\ s,fã A = ¿.î" ?¡n,-Þ.^' "()5'Tn v-x \F , ,r ,'r \/ ; ') -- \/ "/ .-6r {-: 13. A raked stage is a stage that is slanted toward the audience, like a ramp. In general , a raked stage that slants much more than 5o is too steep for the actors to move around on easily. Suppose a theater company builds a raked stage on a stage that is 29 fee¡ deep. If the raked stage is supported at the back by a post that is 4 feet long and perpendicular to it, at what angle does the stage slant? Is the stage acceptable? Si^' ï, 14. 4' :l I' A jet takes off at a 30' angle. The jet's air speed is 200 feet per second. a. After l0 seconds what horiz /.,r \13 ?,05Q* . co__Úo' I __Ð_) X b. After 10 seconds what is the jet's altitude?--1) (+ lfro J= vlL'( -1. r)ll )t) ;'n'jt) -----ì . )'.i.\ ¡) i) 15. At a point on the ground 50 feet from the foot of a tree, the angle Find the height of the tree. of elevation to the top of the tree is 53". ..-""':-1' '/, ô. .íi' I C)tl-)L'¡'t ì) -r t,J' ill (. r 'j\) t6. \ i^ -l i n r' laì, , )'-t-\'t --' From the top of a lighthouse 210 feet high, ,nr of a boat is 27o . Find the distance from the boat to the foot of the lighthouse if the lighthouse was built at sea level. @n*trrr,--ù, 0nr x 17. Richard is flying a kite. The kite string makes an angle of 57o with the ground. If Richard is standing 100 feet from the point on the ground directly below the kite , find the length of the kite string. /, .tL,7t I ( oi ..'-i i i -. ---.f-,1 'f. I $ ),bl0.r') ìú\) ç.lr 18, Anairplanerisesvertically- l000feetoverahorizontal distanceof I mile.-Whatistheangleof of the äirplane's path? =Gr$Qt r t¿i!} ¿;f;\) lfiÐ (. --/ elevation ( lb ,- t;)' | '' ) 5l6D (r 19, The steepest raílway in the world is the Katoomba Scenic Railway in the Blue Mountains of New South Wales, Australia. A single passenger car is pulled by two steel cables up the length of the railway. The gradient (or grade) of the railway is l22Vo. What angle does the Katoomba Scenic Railway make with the horizontal? -t -tatt \?, ', /, (. t/ ) W /N \ ì,) \ ' 'r ¡\), t, r )" )' ¡¡l.tt t Nrme: 'Geometry: Law of Sinos Krr/ Show all work. 1 . If mZT = 45o , mZE = 63o , x and RE :22, then RT : ?7 "7L 27- gnllf 5þ'A b3 = N 2, = If RT =3.2,m1T:52o,m.Æ,:'l0o,thenRE stf' 5z XÈ lf mZT = 2 . Cg . 3.2 sIYì ?t _ð_ 3. (sn æ). zz stn ¿r5 S. 7 10 (s;n sz) 3í" S¡^'7Ô = 62o,mZE:26o, and ET =2,6,then RE = 2.0 8 7?- = 2.30 À_ slvt uL Y!= If mlR: 7lo;ET = 7 ,4, and mZT : mZE: (n8'- ER: Å rh t{l 5.13 RT= 7.26 4lo, != ú = sin -71 _-y-bT sJh 7'1 (svtt\ er^ f= 1-'l "l s,X 7l If mZR = 25o, mZE: 52o, RE = 15.6 mlT: lO3' ET= G.77 F(T: 17 , GZ _y cn6 ts.b 5'h lô3 x-{= t5. C (s'a zs) x 5l\\52 The longest side of a triangle is 34 feet. The measures of t\a'o angles of the hiangle are 40o and 65o Find the lenglh of the other two sides. g1 _L{o 5M = "U ,oS )/\ C ' lìYlÛ,l X* BI(s"Ò= lof - sln (s¡ qo) Y-- 31Slrr lof GRAM is a parallelogram. MG: 32 ll. 37 Gó MR= ?-Sl X G Y tño= 7^u = ¿ t f,= 6(s¡ntt) stvtzz = ll '32 v liY1 3 z Vince is planning a triangular garden. He wants to put a fence around it. The length of one side of the garden is 30 feet. If the angles at each end of this side a¡e 44o and 58", find the length of the fence needed to enclose the garden. 5inq4 ? -- 3o + z6.ot + zt.3t l: 3o ínÍ) si¡ 1t = zl-gl Two ranger stations 5 miles apart Spot a forest fire, Station A determines that the angle formed by the line of sight to the fire and the line connecting the two stations is 37o, Station B determines that the angle formed by tl¡e líne of sight ûo the fire and the line connecting the two stations is 52o, How far is each station from the fire? Jn3z1 = '--I slrt áh ql K sin 3.11 Y= r,.s,!4 SMcl 3.01 52 I 0. An isosceles lriangle has a base of 22 centimeters and a vertex angle of 36o. Find the perimeter of the triangle. xzz 5tn1l lL N = Zz (strt zz) 5M 3t 1L ?L ?= I l, lly\ 3U z(ss.æ) + - Sf. 6O : zz '-l?3-Zou, Scientists are facking polar bears that have been fitted with radio collars. The scientists have two stations that are 9 km apart along a straight road, At Station l, the sígnal from one of the collars comes from a direction of 49o from the road. At Station 2,thesignal from the same collar comes from a direction 65o from the road. How far is the polar bear from station 2? starion I Polr f,= 1(srrnrt) Sin 12. bcar Qb A plume of smoke is spotted from two different frre towers that are 5 mi apart, From Tower A, the angle between the smoke plume and Tower B is 80o, From Tower B, the angle between the smoke plume and Tower A is 70o. rühich tower is closer to the smoke plume? How far ís the smoke from the tower? Towø , þ- srV -- SiYl tÞ ì J sM-to = são a- A þ -f (s'u Ð 5r'z\ 3Ò Smokc E0 1tr 1. rto I 13. The maximum height that a basketball reaches after being shot is given by the formula H: , where V represents the initial velocity, x represents the degree measure of the angle which the path of the basketball makes with the ground, and g represents the acceleration due to gravity, which is 32 feet per second t , Find the maximum height reached by a free-throw if it is shot with an initial velocity of 25 feet per second at an angle of 650. H Fl 14. (s¡n cç)' z ( sz #/sù = (zç l")" _^ Dee and Bill are standing 100 feet apart and in a straight line with the W\MV television tower, The angle of elevation from Bíll to the tower is 30o and the angle of elevation from Dee is 20o, Find the height of the television tower, Dcc 100 fcct Bill Sin L0 x-- l1c.7G {+ 11c.lc ++ Name: Geometry: Law of Sines V.-"'t T- TheLAW OF SINES is o powerful trlongle îool which is used To f lnd missing ¡ldcs or ongles of ANY triongle. By motching up ongles with their opporltc rldes, the equotion is: abc Exomolc: Find the rrissing side x: sinl9' _ sin40" L6x 326 _.643 16x .326x = 10,288 x=3[.56cm Flnd the tnlssfng sldc x ln eoch tnlongle: x ,{o 20 9n Sh X (rtacr) = lo (tu, tt) f, = tlo (su a) 'zç = ìíñtt _;þ X (crn zr K ,- ) = Öe 5(^ bZ zc (stnso) Zt' (snæ) <rVì ¿1 ). z Lll.OS Xzf S,h (s,rael) 73 t'rr 83 - X a= zr(snß) sln Find thø t1-tz lb sùr l\ Y-V= = \= J J-L srn 11 l{O X (stn zr) = t.f") g^l loo úta zl = Gy:!. sln zl lll.37 À a 5l4lo tu Gn 1o) x-- |- t* 31, 11 tôo (srn no) Th¿ LAW OF SINES con olso bc us¿d to flnd mlsslng onglcs. Exonplc: Find th¿ missing sinxo _ sin75o 36 sin ¡' 36 x: ongle c 50 .966 50 50(sinx') =34.776 sinxo =.69532 v - (!,o A (from trig toble) Flnd thc nlsslng onglc x ln coch frfongle: 7'., ¿11 uo âin sln x + sin 1o (sM os ) u4 (sn x) ¡ ,5,V 5'l .lo " ît X -- sin lzl , e SrVr X -- X sin -t ( loç .(si" st) \ 8s- l5ó sîn Z8 ¡zr (stn¿s) \ lsô / 3t _þ --= 51rl x 5;^ lzf ---l 10) Find the oreo 3l af(snx) = lol(iust. s,hX= þ5 (6lY1z't) Llo (st" æ) sl'n X è .E tÔt G3 oÍ circleC. î'çL Tt =A þtu)' = A ItrsE, to = A (-ù GEOMETRY Review for Test on Trigonometry Show all work, t2 B t2l. sinA= J7 2, g cosQ= t x ranR- 7f tz tanC= zL7 t7 35 3, If sinx= I,then cosx= tanx= 88 sÉ-. 31 -Is1 st+ nL = EZ x O"= (tol-25 ,-''- 3l ø.= fs1 Solve for each variable, cosst = \= # þMst= * tS(c"s st) x= úÌr5/ -f^ x 4, x = \Z.fuØ x = c,rlB 2+ þn =# ¿t, X = z1 (t^n zc) 15 rectangle 6. x= G+1'1 G'+" x= \\it -LZl X Sìn cþs sìn loI y 3l= TtL = tz(cos zs) fl.t.ES Î.rs ú,Yr g. * = 9^ b5=þ* f,= 11. x = ZO'l\ 12. ABCD is a rhombus 18 lr,.,,a Q" nZABC =78 AC= LL,bb BD = Cos ?\,q$ 13. *= ?.tO y= \\. 2Ø =*$ + ]-: j,?l g I âwr.=l tt **o-- ' t4 AB = x \2,45 X sm tlo 10. \b.q tt = ¿l - I(>LSv-.,8o" 9r^6t )L (x sq:t t! 14, The equal sides of an isosceles triangle are 16 inches. If each base angle is 32 degrees, find the length of the altitude. Y tb 15. A tree casts a 4O-foot shadow at a time when the measure of the angle of elevation of the !r'nt4Ro, x 16, j¡a-r\ X' tlo gL1'qzç vel of ths surrouncling land, a file is sighted at From a lookout towet 800 How far is the fire from a point directly underneath angle of of 10 degrees. an depression the tower and on the same level as the surrounding ground? \oo 17 , At a point on the ground 80 feet from tl¡e base of a building, the measure of the angle of elevation to the top of the building is 5l degrees. Find the height of the building, tu 19. A rhombus has a l2 cm side and a76 degree angle, Find the lcrrgths ol Srnt'"=# ,Þñ v-- t.b1 eo53fr.-+ \ çÕ t- q.,qb l1 tr 20. A road climbs at a 5o angle. What is the grade of the road? Y B-'lQ{$^.":nù \L--"- --=*-J -- lsò I ì 21. You are flying a kite and have let out 80 meters of string. The kite's angle of elevation with the ground is 4O degrees. If the string is stretched straight, how high is the kite above the ground? Sr". Hb= l- bt,4 Z$+ rÒ I rx 22. Use what you know about a 30-60-90 triangle to find the exact value for tan 30o. .-E ¡1^ão' \3 23. A helicopter is hovering over a landing pad 100 meters from you are standing. The helicopter'sangleofelevationwithth_¡:grou¡disl2degrees, Whatisthealtitudeofthe helicopter? A. S,rn r2" 1 =. toÒ An airplane climbs at an angle of 18 degrees with the ground, Find the ground distance the plane travels as it moves 2500 meters through the air. r <oS 18 =.7sø 25. You are driving toward a mountain. At a rest stop you use a clinometer (a pocket device that can rneasure the angle of elevation) to measure the angle of inclination from your eye to the top of the mountain. The angle measures 11 .5o. You then drive toward the mountain on level ground for 5 miles, You stop and measure the angle of inclination to 27 .5". Find the height of the mountain, lt, / slrl zt.s = *,, 26. Giventhe following triangle. Explain wny I e-ao = tiÏB *{.TEST QUESTI ON'll.,|. * st',r l'+ = I 3in g= þ l^ = sufi' b Ja = sdn8'< T-rans;'þr- ?r'7.rlY 2t. o= süitffi p'fl .zB to, ñ s1 = Sòn 6î.oa' = mZA = 10.51 t7' 3 sìnÙ o , -t ( tL.3 'sfrttf I ß-_ SrÌr 28. z L A-= \øt s4. 10" L, .l .1 - + .3' - Z (q,t[z.3 )<or s n- = IlZ.llolsqt 12 o- = lO, S\ mlB = mZC- lO,'lî' 8#=+ ry .5T -- ¿c's A 29, 1r=zz+ -- 23 + lôz - z(z)(to) ' ent A ' = - Z(¿Xp) cot A E( 4-toe D( - 1-'Cro + = êos A --ccs¡ T--eosA @'-' Two airplanss leave an airport, and the angle between their flight paths is 40o' An hour later, onL plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time? g X'- @ùt*(u)"- z(,þ)(::), X' = 3 8074-çcctE Name: Geometry Quiz: TrÍgonometry KeV a-2 1. If sin Q =5 /r,then cos Q = 5'+X- =7 24 xz/= zí Solve for each variable. \."ïlï;'i'** dn Lt--ä | = tz(sm u) strt zs= {= JE sin 2€ I z, ^= I .3Ö 3. x= Y 3G.81 The equal sides of an isosceles triangle are23 inches long. If each base angle is 56 degrees, find the length of the altitude. q l1 5, The grade of the rodd is lZ%o. Find the angle the road makes with the horizontal. /.\-- t2- lan e = IOO loo A water slide e- 6;t ffi) = 00 meters long and starts at the top of a tower that is 45 meters high, Find the angle of elevation between the slide and the ground. \5/\ is /r ¿Jf f,= sfn 6 >--t- != sí,ì ' (Iã) =[78] .ot ,^ 7 . x Find the measures of the angles of a7-?L25 right triangle. (= # sr^ f = .Jr'h -7e ,q -' (!t) = -73 -J',{ 2rl 8. lG . If the diagonals of a rhombus have lcngths of the angles of the rhombus. of l2 meters and lrt. zt' ', za' l6 meters,find the measures &'' 8= * D-nA Q = (¡^-l 7'e '- G) = sÇ.t7' z-3c-87 < 1 loc 3 .-?1' =^LA = ¡¿ C za mLD -^L B A jet takes off at a 4Oo angle. The jet's air speed is 300 feet per second. a. After 10 seconds what horizontal distance Afte the jet's has the jet traveled? altitu 3 10. From the top of a6?.Çfoot high bluff, a rescue scout looks down upon a hot-air balloon accident, The angle of depression is 8o. How far is the accident from the rescue team at the foot of the blufflr )1 - Eoi \l n \ \\ GZ'l I fartt = + 1 1 s1 ,11r