Right Triangle Trigonometry 8.7

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1 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd metri (mesure), n e tred to the nient Egyptin, Bylonin, nd Indin iviliztions more thn 3000 yers go. The nme trigonometry, however, first ppered in 1595 s the title of the ook Trigonometri pulished y Brtholomeo Pitisus. There re mny pplitions of trigonometry: stronomy, geogrphy, stellite nvigtion, nd T sns nd ultrsound, to nme few. In this setion we will onentrte on three importnt trigonometri rtios involving the lengths of the sides of right tringle: sine (revited sin), osine (revited os), nd tngent (revited tn). The reson for this is tht when one of the ute ngles nd the length of one of the sides of right tringle re known, we n solve for the length of the other two sides using these rtios. Thles of Miletus my hve used similr tringles to lulte the height of the pyrmids nd solve simple prolems. For exmple, how fr from shore is the ship? d e Sine the two tringles re similr, their sides re proportionl, nd we n solve for e in d e y ross-multiplying to otin e ( )d. Dividing y, we find the nswer e. Sine,, nd d re known, the vlue of e from the eqution is e ( ) d

2 304470_Bello_h08_se7_we /8/06 2:35 PM Pge R2 R2 8 Geometry. Solving Tringles nd the Trigonometri Rtios The sme prolem n e solved using trigonometry. We define the trigonometri rtios using right tringle with ute ngle nd right ngle s shown in the figure elow. is the length of the side opposite (ross) ngle. B is the length of the hypotenuse. is the length of the side djent (next to) ngle. is sometimes lled the referene ngle. Trigonometri Rtios opposite side sin hypotenuse djent side os hypotenuse opposite side tn djent side Online Study enter To red the funny story of hief Sohhto, go nd visit link on this textook s Online Study enter. You n rememer these definitions if you rememer SOHHTO (pronouned so-h-tow-h ): S OH H T O sin opp hyp os dj hyp tn opp dj Oh Hek nother Hour Of lger Online Study enter EXMPLE 1 Solving Tringles Find the sine, osine, nd tngent of the given tringle. If you wnt to solve ny tringle when you know 1. Two sides or 2. One ngle nd one side go to link on this textook s Online Study enter. B 5

3 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R3 8.7 Right Tringle Trigonometry R3 Solution To find sin, os, nd tn, we first hve to find. Using the Pythgoren theorem, Next, we use the definition of the trigonometri rtios. Note tht the side opposite ngle is 5 units, nd the side djent to is units. Now sustitute the numers 5,, nd 13 in the rtios. sin os tn opposite side hypotenuse 5 13 djent side hypotenuse 13 opposite side djent side 5 B 5 13 If we re given the mesure of one of the ute ngles in right tringle nd the length of one of the sides, we n find the length of the other two sides. To do this, we need to use sientifi lultor or grpher to find pproximtions for the trigonometri rtios. For exmple, to find the sine, osine, nd tngent of 50 ngle rounded to four deiml ples, we use the following keystrokes with sientifi lultor (left) nd with grpher (right). Using Grpher Using Sientifi lultor (Press MODE nd Selet Degrees) 50 sin SIN 50 ENTER os OS 50 ENTER tn TN 50 ENTER

4 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R4 R4 8 Geometry EXMPLE 2 Finding Missing Side Find (to the nerest whole numer) in the right tringle in. Solution First, omplete the piture y leling ll sides nd ngles. Note tht the 50 ngle is inside the tringle nd must e leled euse it is opposite side. Lel the remining ngle B s shown elow. B in. The tringle dels with (opposite ) nd 20 in. (djent to ), whih mens tht we should use the tngent of. Now, opposite side tn 50 djent side 20 Sustituting for tn 50, Multiply y (1.1918) Thus, 24 We lredy know how to find the length of the hypotenuse of right tringle y using the Pythgoren theorem. In doing so, we need to know the length of oth legs (sides) of the tringle. Wht out the se in whih we only hve the length of one side of the tringle nd the mesure of one of the ngles? We n still do it if we use the trigonometri rtios we hve just lerned! Let us see how. EXMPLE 3 Finding the Hypotenuse When the Length of One Side nd n ngle re Given In the right tringle B, the length of B is, nd the mesure of ngle B is 40. Find the length of the hypotenuse B to the nerest whole numer.

5 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R5 8.7 Right Tringle Trigonometry R5 Solution First drw tringle B (left), nd then sustitute B, B h, nd B 40 (right). B B 40 B = h ngle B is 40, B is, nd B h. Sine B is djent to B nd h is the hypotenuse, we need the rtio involving the djent side nd the hypotenuse; tht is, the osine of B 40. Using lultor will tell us tht Thus, os os 40 Sustituting for os 40, h ross-multiplying, h Dividing oth sides y , h 13 ft n we hek the nswer? By the Pythgoren theorem, if , we re orret. To find the length of, we use the ft tht the sum of the ngles on tringle is 180. Sine the mesure of ngle B is 40 nd m is 90, m With this informtion, we n find the length of y using tn 50 djent side hypotenuse opposite side djent h Sustituting for tn 50, Our result will e orret if , ut , so our nswer is right!

6 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R6 R6 8 Geometry B. pplitions of the Trigonometri Rtios 50 m 67 EXMPLE 4 Finding the Height of the lexndri Lighthouse. The figure ove shows one of the Seven Wonders of the World, the Gret Lighthouse t lexndri, Egypt, whose onstrution strted in 290 B.. The pltform on whih the lighthouse stnds is out 100 m wide, nd the ngle of elevtion from the orner of the pltform to the top of the lighthouse is 67. To the nerest meter, how high is the lighthouse? Solution We strt y mking piture. Sine the pltform is 100 m wide, the distne from the enter of the pltform to the orner is 50 m, the ngle of elevtion is 67, nd the height we would like to find is x, s shown in the digrm. x Using ngle s our referene ngle, we re looking for the length x of the opposite side when we know the length of the djent side, so we hve to use tn 67 opposite djent x m Using lultor, ross-multiplying, x (2.3559) x 118 x (to the nerest meter)

7 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R7 8.7 Right Tringle Trigonometry R7 If we know the length of two sides of right tringle, the mesures of the orresponding ngles n e found y using the inverse trigonometri keys on lultor. For exmple, suppose tht tn We n find the mesure of ngle y using the inverse tn key usully leled tn 1. Using Grpher Using Sientifi lultor (Press MODE nd Selet Degrees) tn 1 tn INV tn 2nd TN ENTER In oth ses, the nswer is , whih mens tht the mesure of ngle is out We will use this ide in the next exmple. EXMPLE 5 Inlintion of Rmp When uilt to provide n essile hndip entrne, the slope of rmp should e s smll s possile. The mximum slope in new onstrution is 1: (every inh of rise will require 1 ft of run). Wht is the mximum ngle of inlintion of the rmp shown in the figure? Level lnding Level lnding Slope 1: mx. Solution Mke digrm using the given informtion nd leling the unknown ngle s. We re given the length of the sides opposite (1) nd djent () to, nd we re looking for the mesure of ngle. 1 We know opposite tn djent 1 Thus, tn Now we use the inverse tn key to find. Using sientifi lultor or grpher, tn INV tn tn 1 INV TN ENTER Thus, the mesure of ngle is pproximtely This mens tht the mximum inlintion for the rmp is You n round the nswer down to 4 (rememer, is the mximum inlintion, so we round down).

8 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R8 R8 8 Geometry EXERISES 8.7 Solving Tringles nd the Trigonometri Rtios In prolems 1 8, use the Pythgoren theorem to find the length of the missing side. 1. right tringle hs side lengths 3 nd 4, s shown in the figure. Find the length of the hypotenuse. 6. The length of one side of right tringle is, nd the length of the hypotenuse is 20, s shown in the figure. Find the length of the other side The length of one side of right tringle is 24, nd the length of the hypotenuse is 26, s shown in the figure. Find the length of the other side. 2. right tringle hs side lengths 5 nd, s shown in the figure. Find the length of the hypotenuse right tringle hs side lengths 8 nd 15, s shown in the figure. Find the length of the hypotenuse The length of one side of right tringle is 20, nd the length of the hypotenuse is 25, s shown in the figure. Find the length of the other side right tringle hs side lengths 6 nd 8 s shown in the figure. Find the length of the hypotenuse. For prolems 9, refer to the right tringle elow The length of one side of right tringle is 9, nd the length of the hypotenuse is 15, s shown in the figure. Find the length of the other side If 4, 3, nd 5, find. os.. sin.. tn. Express your nswers s redued frtions. 10. If, 5, nd 13, find. os.. sin.. tn. Express your nswers s redued frtions.

10 304470_Bello_h08_se7_we R10 11/8/06 7:08 PM Pge R10 8 Geometry Use the following tle to nswer prolems 19 nd 20. sin os tn The Wshington Monument is 555 ft tll. If the Sun s rys nd the monument form n ngle of 27 (see figure), how long is the shdow to the nerest whole numer? (Use tn ) 19. The length of one side of right tringle is 24 in., s shown in the figure.. Find the length of side.. Find the length of hypotenuse. Round your nswers to the nerest whole numer ft 20 oris 24 in. 20. The length of one side of right tringle is 40 ft, s shown in the figure.. Find the length of side.. Find the length of hypotenuse. Round your nswers to the nerest whole numer ft When flying ojet (suh s n irplne or ird) sends, its trjetory forms n ngle with the ground, lled the ngle of inlintion (see figure elow). ory jet Tr ltitude ngle of inlintion B pplitions of the Trigonometri Rtios 21. tree sts shdow 32 ft long. hiker estimtes the ngle etween the Sun s rys nd the ground to e 72 (see figure). Using tn 72 3, estimte the height of the tree to the nerest whole numer. In exerises 23 26, use trigonometry to find the ltitudes. 23. n irplne sends with 30 ngle of inlintion. If the irplne is flying t rte of 5 mi/min, wht is the ltitude fter 5 min? (Use sin , nd round the nswer to the nerest whole numer.) 24. roket lsts of with 67 ngle of inlintion. If the roket trvels with speed of 25 mi/min, wht is the ltitude fter 10 min? (Use sin , nd round the nswer to the nerest whole numer.) ft 25. ird tkes off with ngle of inlintion. If the ird is flying t rte of 200 ft/min, how high is the ird fter 7 min? (Use sin 0.21, nd round the nswer to the nerest whole numer.)

11 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R11 R Right Tringle Trigonometry 26. heliopter sends with 45 ngle of inlintion. If the heliopter trvels t speed of 40 mph, wht is the heliopter s ltitude fter 15 min? (Hint: 15 min 14 hr.) (Use sin , nd round the nswer to the nerest whole numer.) 27. The plne shown is sending t 10 ngle. If the plne is 40 ft high, how fr is the end of the runwy? (Use tn , nd round the nswer to the nerest whole numer.) 29. Pedro Mendiet opertes the onveyor-elt mhine used to hul mterils to the top of the onrete mixing ontiner. If the elt mkes 20 ngle with the horizontl nd ends 40 ft ove the ground, how fr do the mterils trvel to get to the top of the onveyor elt? How long is the elt? (Use sin , nd round the nswer to the nerest whole numer.) 40 ft ft. Emm Lee/Life File/PhotoDis/Getty Imges Roger Ressmeyer/oris 28. The Dmes Point Bridge spns the St. John River in Jksonville, Florid. The longest le supporting the ridge is 720 ft long nd mkes 25 ngle with the rod. Wht is the height h of the pole? (Use sin , nd round the nswer to the nerest whole numer.) 30. nother onveyor-elt mhine operted y John Tylor mkes n 18 ngle with the horizontl nd ends 45 ft ove the ground. How fr do the mterils trvel to get to the top of the onveyor elt? How long is the elt? (Use sin , nd round the nswer to the nerest whole numer.) h ft. 720 ft ft. 18 roline/zef/oris In Other Words ourtesy HNTB 31. We use the memory devie SOHHTO to rememer the formuls for the sine, osine, nd tngent. n you give other words for rememering those three trigonometri rtios? 32. Explin in your own worlds wht it mens when we sy to solve tringle.

12 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R R 8 Geometry 33. Wht hppens to the tngent of n ngle s the mesure of the ngle gets lose to 90? (Hint: Try tn 88, tn 89, tn 89.5, nd tn 90 with your lultor.) 34. Explin in your own words wht the ngle of elevtion is. Wht would the ngle of depression e? Using Your Knowledge Do you rememer how to solve the eqution x 2? You extrt the squre root of oth sides to get the nswers x 1, where is positive. Wht out simplifying 2? (See Setion 5.7.) Use the knowledge of these two fts to solve the following prolems. right tringle hs ngles 30 nd 60, s shown in the figure. Use the following tle to nswer prolems 35 40: os 30 sin os 60 sin If 2, find the ext vlues of nd. 36. If 4, find the ext vlues of nd. 37. If 23, find the ext vlues of nd. 38. If 5, find the ext vlues of nd. 39. Wht is the ext vlue of tn 30? 40. Wht is the ext vlue of tn 60? Disovery n eqution relting ertin trigonometri vlues is lled trigonometri identity. For exmple, n you disover if the eqution (sin ) 2 (os ) 2 1 is true for every numer? This eqution is lled the Pythgoren identity. If one trigonometri vlue is known, you n use this identity to disover the others. 1 For exmple, using the ft tht os 30 2, we n sustitute in the Pythgoren identity to disover the ext vlue of sin 30. (sin 30 ) 2 (os 30 ) 2 1 (sin 30 ) (sin 30 ) 2 (sin 30 ) 2 1 sin B Note: The ext vlue usully involves rdil expressions. 41. Use the lultion ove to find the ext vlue of tn 30. ompre with prolem Use the ft tht os 45 2 to find the ext vlues of sin 45 nd tn Use the pproximtion os to pproximte sin 15 nd tn 15 (Round your nswers to two deiml ples). 44. Use the pproximtion sin to pproximte os 23 nd tn 23. (Round the nswer to two deiml ples.)

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