Right Triangle Trigonometry for College Algebra


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1 Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 34) II. The Trigonometri Rtios (exerises on pges 67) III. Applitions (exerises on pges 910) IV. The Lw of Sines nd osines nd More Applitions (exerises on pges 141)
2 I. Bkground nd Definitions The word trigonometry is derived from the Greek words trigon, mening tringle nd metry, mening mesurement. For every tringle, we n mesure the sides nd the ngles. We use stndrd liner units of mesure for the sides. The mesure of n ngle is defined y the mount of rottion from its initil side to its terminl side. Terminl Side Vertex Initil Side For ngles, we will use the degree for our stndrd unit of mesure. One degree is the mesure of n ngle equivlent to rottion of of omplete revolution out its vertex. We n further sudivide one degree into minutes nd seonds s follows: 1 degree (1 ) = 60 minutes (60') 1 minute (1') = 60 seonds (60") EXAMPLE: Express 17.3 in degrees, minutes, nd seonds = 17 = 17 = (0.3 )(60'/1 ) ' = 17 13' + (0.8')(60"/1' ) = 17 13' + 48" = 17 13'48" EXAMPLE: Express 40 0'0" using degrees only. 40 0'0" = '(1 / 60') + 0"(1 / 3600")
3 At this point, we should remind ourselves wht we lredy know out right tringles, tringles with 90 ngle. B A First, we my rell tht A + B + = 180. So if B = 3 in the ove tringle, then A = 180 ( ) =. The seond thing we rell out right tringles is our oldest theorem, the Pythgoren Theorem. This theorem sttes tht the sums of the squres of the short sides of the tringle is equl to the squre of the ; i.e., +. So if = 13 = nd =, then = 13 = 1. EXERISES: 1. onvert the following degree mesures to degrees, minutes, nd seonds. () () 1.1. onvert the following mesures to degrees only. () 0 19' 18" () '9" 3. For eh tringle, find the unknown side. () () m 10 m 0 in 11 in 3
4 4. Why is irle divided into 360? This numer ppers to e ritrry; why not use 400 or for tht mtter 4? Use the internet to serh out the nswer. Be utious, s the internet is eoming fmous for flse informtion. II. The Trigonometri Rtios You my hve notied previously, tht if we know two ngles, we my find the third, nd tht if we know two sides, we my find the third. Wht if we only know two ngles nd one side? How might we find the other sides? This is where we egin our study of right tringle trigonometry. A B There re three trigonometri rtios tht we will use for the sis of our study (there re three others). These re lled sine (sin), osine (os), nd tngent (tn). In wht follows, the is lwys the longest side of the tringle, the side is the side the ngle, nd the djent side is the side next to the ngle. We hve the following definitions, sed on the ove tringle: sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent Mke speil note tht the nd djent sides depend on the ngle in question. 4
5 EXAMPLE: In the right tringle shown elow, find sin A, os A, tn A, sin B, os B, nd tn B. 4 A sin A = 3 os A = 4 tn A = sin B = 4 os B = 3 tn B = 4 3 B In the previous exmple, note the reltionship etween sin A nd os B. This similr reltionship lso ours in the other rtios. Now you my rell tht we erlier noted tht there re three more trigonometri rtios. These re sed on the reiprols of our three min rtios. They re lled sent (se), osent (s), nd otngent (ot) nd re defined s follows: 1 se θ = = os θ djent 1 s θ = = sin θ 1 ot θ = = tn θ djent We will fous our ttention on sine, osine, nd tngent in this ourse, ut you my find these identities useful in susequent ourses. At this point, we must disuss the role of lultors in our rief study of trigonometry. hnge your lultor to degree mode. To do this, press the mode utton nd highlight degree if rdin is urrently highlighted. Now test this y entering sin( 3). You should get out If you got , you re still in rdin mode. You should see three uttons leled sin, os, nd tn. We will use these uttons to find unknown sides, given n ngle nd side of tringle. Aove these uttons, you will find sin, os, nd tn. To use these, you will first press the nd or inverse utton. The key word here is inverse. These will undo wht sin, os, nd tn do, muh in the sme wy tht the ue root undoes wht ue does. This will llow us to find unknown ngles, given two sides of right tringle.
6 EXAMPLE: Find the length of the side leled x. EXAMPLE: Find the mesure of the ngle leled θ. x os3 = x = x os tnθ = 7 1 θ = tn EXERISES: 1. In eh of the right tringles elow, find sin A, os A, tn A, sin B, os B, nd tn B. () 1 () A A 13 3 B B 1. Find ll unknown sides nd ngles for eh of the following tringles. This is lled solving the tringle. () ()
7 () (d) III. Applitions The following prolem solving pproh will e helpful in solving the mny pplition prolems of right tringle trigonometry: Sketh piture. Lel your sketh with ll known sides nd ngles. Define vriles for the unknown sides nd ngles. Use the Pythgoren Theorem nd the trigonometri rtios to find unknown mesures. Before we jump into some prolems, we should disuss ngles of elevtion nd ngles of depression. Suppose you see n owl perhed on the rnh of tree. The ngle formed etween the ground nd your line of sight to the owl is lled n ngle of elevtion. When flying to Hwi i, you n see the Big Islnd in the distne. The ngle formed etween the horizontl line the plne is trveling on nd the line of sight down to the islnd is lled n ngle of depression. ngle of depression ngle of elevtion 7
8 EXAMPLE: A 100foot fire truk ldder is lening ginst wll. Find the distne the ldder goes up the wll if it mkes n ngle of 43 with the ground. First we mke helpful sketh nd lel the known nd unknown. 100 ft x Now, sin 43 =. x Thus x = 100sin feet. EXAMPLE: A surveyor is stnding 4 m from the se of redwood tree. She mesures the ngle of elevtion to the top of the tree s 33. Find the height of the tree. First we mke nd lel sketh. x Now, tn 33 =. 4 x 33 Thus x = 4 tn m tll. 4 m 8
9 EXAMPLE: A guy wire is nhored to the ground 0 ft from the se of telephone pole. If it is tthed to the telephone pole 30 ft ove the ground, find the ngle mde y the guy wire nd the ground. As usul, we first mke sketh. 30 Now, tn α =. 30 ft 0 30 Thus tn 1 α = ft 0 EXERISES 1. The most powerful lighthouse is on the ost of Brittny, Frne, nd is 0 meters tll. Suppose you re in ot just off the ost. How fr from the se of the lighthouse re you if your ngle of elevtion to the light soure is 1?. A hot ir lloon tkes off from the ground nd flots long n open field. If the ngle of elevtion from the initil tkeoff spot is 68 nd the lloon is 30 feet in the ir, wht is the lloon s distne from its tkeoff spot? 3. A peregrine flon perhed top tll uilding spots its lunh on the ground elow. If the prey is 1000 m from the se of the uilding, nd the uilding is 00 m tll, wht is the ngle of depression from the flon to the prey? 4. The sonr of nvy ruiser detets sumrine tht is 4000 feet from the ruiser. The ngle etween the wter line nd the sumrine is 34. How deep is the sumrine?. An eril photogrpher is in n irplne t n ltitude of 10 km nd sees two towns diretly est of the plne. The ngles of depression to the towns re nd 60. How fr prt re the two towns? 9
10 6. Thles, the first of the Seven Wise Men, is sid to hve omputed the height of the Gret Pyrmid of heops. If we were given distne wy nd knew our ngle of elevtion to the top, this would e simple trigonometri prolem. This is not how Thles determined the height. Use the internet to serh out the method used y Thles. IV. The Lw of Sines nd the Lw of osines Wht if we do not hve right tringle? Let us turn our ttention to two other types of tringles, ute nd otuse. Tringles with no right ngle re referred to s olique. An ute tringle is one in whih ll of the ngles re less thn 90. An otuse tringle is one in whih there is n ngle greter thn 90. To solve olique tringles, we will need to know the mesure of t lest one side nd ny two other prts of the tringle. Here re the four possile ses: 1. Two ngles nd ny side (AAS or ASA). Two sides nd n ngle one of them (SSA, the miguous se) 3. Three sides (SSS) 4. Two sides nd their inluded ngle (SAS) The first two ses n e solved using the Lw of Sines nd the seond two ses n e solved using the Lw of osines. 10
11 The Lw of Sines If AB is tringle with sides,, nd, then sin = =. sin sin A B B A A is ute A is otuse EXAMPLE (AAS): For tringle with = 130, B = 7, nd = 30, find the remining ngle nd sides. 30 A SOULUTION: 130 First we mke sketh. 7 Note, A = 180 ( ) = 3. B 30 30sin 3 By the Lw of Sines,.8. sin 3 = = sin 7 sin sin130 Also, 0.6. sin130 = = sin 7 sin 7 11
12 EXAMPLE (ASA): For tringle with A = 7, B = 4, nd = 10, A find the mesure of side First note tht = Using the Lw of Sines, we hve B 10 =. sin 7 sin 60 10sin 7 Thus = sin 60 EXAMPLE (SSA, the miguous se): onsider the tringle with B =, = 8., nd = Find the mesure of ngle A It is unler whether this is n ute tringle or n otuse tringle (or if it is not possile). We must sketh oth: B A B A sin Now, = sin A = sin A sin 7.3 This gives two ngles A 1 7. nd A Both of these ngles give tringles, s we do not exeed
13 Let us first tke the ute se, A 7.. Then Using the Lw of Sines, 7.3sin.48 we find tht the third side is sin Now we look t the otuse se, sin17. Lw of Sines, we hve sin 17. A. This gives. Agin using the Note: When the miguous se rises, it is possile to otin two tringles (s ove), one tringle, or no tringle. How do you know whih? Just work the prolem under the ssumption tht you will otin two tringles, nd the mthemtis will revel the truth. If we hve either SSS or SAS, we must use the Lw of osines s first step to solving nonright tringle. After this, we will finish with the Lw of Sines, s it is little esier to pply. The Lw of osines If AB is tringle with sides,, nd, then = = = os A os B. os Note tht the Lw of osines will e used to either find side or it s ngle. EXAMPLE: A ship trvels 60 miles due est, then djusts its ourse northwrd. After trveling 80 miles in tht diretion, the ship is 139 miles from its point of deprture. Desrie the hnge in ering from point B to point. 139 mi 80 mi A 60 mi B 13
14 Sine we do not know ny of the three ngles of this tringle, we must use the Lw of osines. Now, 139 This gives = (60)(80) os B os B = (60)(80) B os 1 ( ) So the ering mesured from due north from point B to point is given y = We write this s N 76.1 E, red 76.1 est of north. EXERISES 1. Sketh tringles for eh of the following nd solve eh tringle. () = 6, = 8, = 10 () A = 30, = 40, = 0 () A = 11, = 1, = 10 (d) B = 6.3, = 8.3, = 7. 6 (e) =, =, = 7 (f) A = 8, = 1, = (g) = 14, = 4, = 14 (h) A = 4.3, = 4.6, = A rnger loted t sttion A spots fire in the diretion 3 est of north. Another rnger, loted t sttion B, 10 miles due est of sttion A, spots the sme fire on line 48 west of north. Find the distne from eh rnger sttion to the fire. 3. A surveyor is ttempting to find the distne etween two points A nd B. A grove of trees is ostruting the view, so the surveyor sets stke on eh side of the grove t the points A nd B nd then moves to point. The distne from A to is 143 feet nd the distne from B to is 13 feet. Lstly, ngle AB mesures 78. Find the distne ross the grove. 4. Trigonometry is used extensively in eril photogrphy. Suppose mer lens hs n ngulr overge of 7. As piture is tken over level ground, the irplne s distne is 4800 feet from house loted on 14
15 the edge of the photogrph nd the ngle of elevtion of the irplne from the house is 48. Find the distne ross the photogrph.. A 10meter telephone pole sts 17meter shdow diretly down slope when the ngle of elevtion of the sun is 4. Find the ngle of elevtion of the ground. 6. Beuse of previling winds, tree grew so tht it ws lening 4 from the vertil. At point 3 meters from the tree, the ngle of elevtion to the top of the tree is 3. Find the height of the tree. 7. Suppose you re in hot ir lloon with your rew on the ground. Atthed to the lloon re two tether ords of length 00 ft nd 40 ft, whih your rew hs tthed to the ground. You note tht these ords form n ngle of 6 where they meet the lloon. Assuming these lines re tut nd the ground is level, with wht ngles do these ords meet the ground? 8. To pproximte the length of mrsh, surveyor wlks 0 meters from point A to point B, then turns 7 nd wlks 0 meters to point. Approximte the length A of the mrsh. 9. A 100foot vertil tower is to e ereted on the side of hill tht mkes 6 ngle with the horizontl. Find the length of eh of the two guy wires tht will e nhored 7 feet uphill nd downhill from the se of the tower. 10. The sell plyer in enter field is plying pproximtely 330 feet from the television mer tht is ehind home plte. A tter hits fly ll tht goes to the wll 40 feet from the mer. Approximte the numer of feet tht the enter fielder hs to run to mke the th if the mer turns 8 to follow the ply. (Note: Be very reful skething pitures for these prolems. There my e more thn one interprettion for ouple of them; solve them ll!) 1
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