Section 55 Solving Right Triangles*


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1 55 Solving Right Tringles Geometry. The re of retngulr nsided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml ples. (B) Wht numer does A seem to pproh s n? (Wht is the re of irle with rdius?) 80. Geometry. The re of regulr nsided polygon insried in irle of rdius is given y A n 360 sin 2 n n 8 r (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml ples. (B) Wht numer does A seem to pproh s n? (Wht is the re of irle with rdius?) 8. Angle of Inlintion. Rell (Setion 2) the slope of nonvertil line pssing through points P (x, y ) nd P 2 (x 2, y 2 ) is given y slope m (y 2 y )/(x 2 x ). The ngle tht the line L mkes with the x xis, 0 80, is lled the ngle of inlintion of the line L (see figure). Thus, Slope m tn, 0 80 (A) Compute the slopes to two deiml ples of the lines with ngles of inlintion 88.7 nd (B) Find the eqution of line pssing through (4, 5) with n ngle of inlintion 37. Write the nswer in the form y mx, with m nd to two deiml ples. L y 82. Angle of Inlintion Refer to Prolem 8. (A) Compute the slopes to two deiml ples of the lines with ngles of inlintion 5.34 nd (B) Find the eqution of line pssing through (6, 4) with n ngle of inlintion 06. Write the nswer in the form y mx, with m nd to two deiml ples. L x Setion 55 Solving Right Tringles* FIGURE In the previous setions we hve pplied trigonometri nd irulr funtions in the solutions of vriety of signifint prolems. In this setion we re interested in the prtiulr lss of prolems involving right tringles. A right tringle is tringle with one 90 ngle. Referring to Figure, our ojetive is to find ll unknown prts of right tringle, given the mesure of two sides or the mesure of one ute ngle nd side. This is lled solving right tringle. Trigonometri funtions ply entrl role in this proess. To strt, we lote right tringle in the first qudrnt of retngulr oordinte system nd oserve, from the definition of the trigonometri funtions, six trigonometri rtios involving the sides of the tringle. [Note tht the right tringle is the referene tringle for the ngle.] *This setion provides signifint pplition of trigonometri funtions to relworld prolems. However, it my e postponed or omitted without loss of ontinuity, if desired. Some my wnt to over the setion just efore Setions 7 nd 72.
2 380 5 TRIGONOMETRIC FUNCTIONS TRIGONOMETRIC RATIOS (, ) 0 90 sin os tn s se ot Side is often referred to s the side opposite ngle, s the side djent to ngle, nd s the hypotenuse. Using these designtions for n ritrry right tringle removed from oordinte system, we hve the following: RIGHT TRIANGLE RATIOS Hyp Opp sin Opp Hyp os Adj Hyp s Hyp Opp se Hyp Adj Adj 0 90 tn Opp Adj ot Adj Opp Explore/Disuss For given vlue, 0 90, explin why the vlue of eh of the six trigonometri funtions is independent of the size of the right tringle tht ontins. TABLE Angle to Nerest Signifint Digits for Side Mesure 2 0 or 0. 3 or or The use of the trigonometri rtios for right tringles is mde ler in the following exmples. Regrding omputtionl ury, we use Tle s guide. (The tle is lso printed inside the k over of this ook for esy referene.) We will use rther thn in mny ples, relizing the ury indited in Tle is ll tht is ssumed. Another word of ution: when using your lultor e sure it is set in degree mode.
3 55 Solving Right Tringles 38 EXAMPLE Right Tringle Solution Solve the right tringle with 6.25 feet nd Solution First drw figure nd lel the prts (Fig. 2): FIGURE ft 32.2 Solve for nd re omplementry. Solve for sin Or use s. sin sin feet Solve for os Or use se. os os feet MATCHED PROBLEM Solve the right tringle with 27.3 meters nd In the next exmple we re onfronted with prolem of the type: Find given sin We know how to find (or pproximte) sin given, ut how do we reverse the proess? How do we find given sin? First, we note tht the solution to the prolem n e written symolilly s either rsin or rsin nd sin oth represent the sme thing. sin Both expressions re red is the ngle whose sine is
4 382 5 TRIGONOMETRIC FUNCTIONS CAUTION It is importnt to note tht sin does not men /(sin 0.496). The supersript is prt of funtion symol, nd sin represents the inverse sine funtion. Inverse trigonometri funtions re developed in detil in Setion 59. Fortuntely, we n find diretly using lultor. Most lultors of the type used in this ook hve the funtion keys sin, os, nd tn or their equivlents (hek your mnul). These funtion keys tke us from trigonometri rtio k to the orresponding ute ngle in degree mesure when the lultor is in degree mode. Thus, if sin 0.496, then we n write either rsin or sin We hoose the ltter nd proeed s follows: sin or To the nerest hundredth degree To the nerest minute Chek sin Explore/Disuss 2 Solve eh of the following for to the nerest hundredth of degree using lultor. Explin why n error messge ours in one of the prolems. (A) os (B) tn.438 (C) sin.438 EXAMPLE 2 Right Tringle Solution Solve the right tringle with 4.32 entimeters nd 2.62 entimeters. Compute the ngle mesures to the nerest 0. Solution Drw figure nd lel the known prts (Fig. 3): FIGURE m 4.32 m Solve for tn tn or [(0.2)(60)] 2 0 to nerest 0
5 Solve for Solving Right Tringles 383 Solve for sin 2.62 Or use s entimeters sin 3.2 or, using the Pythgoren theorem, entimeters Note the slight differene in the vlues otined for (5.05 versus 5.06). This ws used y rounding to the nerest 0 in the first lultion for. MATCHED PROBLEM 2 Solve the right tringle with.38 kilometers nd 6.73 kilometers. EXAMPLE 3 Solution FIGURE 4 Geometry If pentgon ( fivesided regulr polygon) is insried in irle of rdius 5.35 entimeters, find the length of one side of the pentgon. Sketh figure nd insert tringle ACB with C t the enter (Fig. 4). Add the uxiliry line CD s indited. We will find AD nd doule it to find the length of the side wnted. C B D 5.35 A Angle ACB Angle ACD sin (ngle ACD) AD AC sin 36 Ext Ext AD AC sin (ngle ACD) 3.4 entimeters AB 2AD 6.28 entimeters MATCHED PROBLEM 3 If squre of side 43.6 meters is insried in irle, wht is the rdius of the irle?
6 384 5 TRIGONOMETRIC FUNCTIONS EXAMPLE 4 FIGURE 5 Arhiteture In designing house n rhitet wishes to determine the mount of overhng of roof so tht it shdes the entire south wll t noon during the summer solstie (Fig. 5). Minimlly, how muh overhng should e provided for this purpose? Winter solstie sun (noon) Summer solstie sun (noon) 8 ft 32 Solution FIGURE 6 From the figure we drw the following relted right tringle (Fig. 6) nd solve for x: 8 x tn x x tn 9.7 feet MATCHED PROBLEM 4 With the overhng found in Exmple 4, how fr will the shdow of the overhng ome down the wll t noon during the winter solstie? Answers to Mthed Prolems. 42.2, 20.2 m, 8.3 m 2. 40, 78 20, 6.87 km m 4.. ft
7 55 Solving Right Tringles 385 EXERCISE 55 A For the tringle in the figure for Prolems 2, write the rtios of sides orresponding to eh trigonometri funtion given in Prolems 6. Do not look k t the definitions.. sin 2. ot 3. s 4. os 5. tn 6. se Prolems 3 36 give geometri interprettion of the trigonometri rtios. Refer to the figure, where 0 is the enter of irle of rdius, is the ute ngle AOD, D is the intersetion point of the terminl side of ngle with the irle, nd EC is tngent to the irle t D. E F ot D s sin tn Eh rtio in Prolems 7 2 defines trigonometri funtion of (refer to the figure for Prolems 2). Indite whih funtion without looking k t the definitions. 7. / 8. / 9. / 0. /. / 2. / In Prolems 3 8, find eh ute ngle in degree mesure to two deiml ples using lultor. 3. os sin tn os sin tn.993 B Solve eh tringle in Prolems 9 30 using the informtion given nd the tringle leling in the figure for Prolems O os A se 3. Explin why (A) os OA (B) ot DE (C) se OC 32. Explin why (A) sin AD (B) tn DC (C) s OE 33. Explin wht hppens to eh of the following s the ute ngle pprohes 90. (A) os (B) ot (C) se 34. Explin wht hppens to eh of the following s the ute ngle pprohes 90. (A) sin (B) tn (C) s 35. Explin wht hppens to eh of the following s the ute ngle pprohes 0. (A) sin (B) tn (C) s 36. Explin wht hppens to eh of the following s the ute ngle pprohes 0. (A) os (B) ot (C) se B C , , 22.4 C , , , , , , , , Show tht d h ot ot h , , 65 d
8 386 5 TRIGONOMETRIC FUNCTIONS 38. Show tht d h ot ot d h time, no timing devies existed to mesure the veloity of freeflling ody, so Glileo used the inlined plne to slow the motion down.) A steel ll is rolled down glss plne inlined t 8.0. Approximte g to one deiml ple if t the end of 3.0 seonds the ll hs mesured veloity of 4.2 meters per seond. APPLICATIONS 39. Surveying. Find the height of tree (growing on level ground) if t point 05 feet from the se of the tree the ngle to its top reltive to the horizontl is found to e Air Sfety. To mesure the height of loud eiling over n irport, serhlight is direted stright upwrd to produe lighted spot on the louds. Five hundred meters wy n oserver reports the ngle of the spot reltive to the horizontl to e How high (to the nerest meter) re the louds ove the irport? 4. Engineering. If trin lims t onstnt ngle of 23, how mny vertil feet hs it limed fter going mile? ( mile 5,280 feet) 42. Air Sfety. If jet irliner lims t n ngle of 5 30 with onstnt speed of 35 miles per hour, how long will it tke (to the nerest minute) to reh n ltitude of 8.00 miles? Assume there is no wind. 43. Astronomy. Find the dimeter of the moon (to the nerest mile) if t 239,000 miles from Erth it produes n ngle of 32 reltive to n oserver on Erth. 44. Astronomy. If the sun is 93,000,000 miles from Erth nd its dimeter is opposite n ngle of 32 reltive to n oserver on Erth, wht is the dimeter of the sun (to two signifint digits)? 45. Geometry. If irle of rdius 4 entimeters hs hord of length 3 entimeters, find the entrl ngle tht is opposite this hord (to the nerest degree). 46. Geometry. Find the length of one side of ninesided regulr polygon insried in irle of rdius 4.06 inhes. 47. Physis. In ourse in physis it is shown tht the veloity v of ll rolling down n inlined plne (negleting ir resistne nd frition) is given y v gt sin where g is grvittionl onstnt (elertion due to grvity), t is time, nd is the ngle of inlintion of the plne (see the figure). Glileo ( ) used this eqution in the form g v t sin to estimte g fter mesuring v experimentlly. (At tht 48. Physis. Refer to Prolem 47. A steel ll is rolled down glss plne inlined t 4.0. Approximte g to one deiml ple if t the end of 4.0 seonds the ll hs mesured veloity of 9.0 feet per seond. 49. Engineering Cost Anlysis. A le television ompny wishes to run le from ity to resort islnd 3 miles offshore. The le is to go long the shore, then to the islnd underwter, s indited in the ompnying figure. The ost of running the le long the shore is $5,000 per mile nd underwter, $25,000 per mile. City 20 miles Resort Islnd 3 miles Shore (A) Referring to the figure, show tht the ost in terms of is given y C() 75,000 se 45,000 tn 300,000 (B) Clulte tle of osts, eh ost to the nerest dollr, for the following vlues of : 0, 20, 30, 40, nd 50. (Notie how the osts vry with. In ourse in lulus, students re sked to find so tht the ost is minimized.) 50. Engineering Cost Anlysis. Refer to Prolem 49. Suppose the islnd is 4 miles offshore nd the ost of running the le long the shore is $20,000 per mile nd underwter, $30,000 per mile. (A) Referring to the figure for Prolem 49 with pproprite hnges, show tht the ost in terms of is given y C() 20,000 se 80,000 tn 400,000 (B) Clulte tle of osts, eh ost to the nerest dollr, for the following vlues of : 0, 20, 30, 40, nd 50.
9 56 Grphing Bsi Trigonometri Funtions Geometry. Find r in the ompnying figure (to two signifint digits) so tht the irle is tngent to ll three sides of the isoseles tringle. [Hint: The rdius of irle is perpendiulr to tngent line t the point of tngeny.] 52. Geometry. Find r in the ompnying figure (to two signifint digits) so tht the smller irle is tngent to the lrger irle nd the two sides of the ngle. [See the hint in Prolem 5.] 30 r 2.0 meters 30 r 2.0 in. Setion 56 Grphing Bsi Trigonometri Funtions Periodi Funtions Grphs of y sin x nd y os x Grphs of y tn x nd y ot x Grphs of y s x nd y se x Grphs on Grphing Utility Consider the grphs of sunrise times nd sound wves shown in Figure. Wht is ommon feture of the two grphs? Both represent repetitive phenomen; tht is, oth pper to e periodi. Trigonometri funtions re prtiulrly suited to desrie periodi phenomen. Sunrise time F A J A O D F A J A O D Pressure t erdrum Month Sunrise times nd time of yer () FIGURE Periodi phenomen. Time (seonds) Soundwve rriving t erdrum () In this setion we disuss the grphs of the six trigonometri funtions with rel numer domins introdued erlier. We lso disuss the domins, rnges, nd periodi properties of these funtions. The irulr funtions introdued in Setion 52 will prove prtiulrly useful in this regrd. It ppers there is lot to rememer in this setion. However, you only need to e fmilir with the grphs nd properties of the sine, osine, nd tngent funtions. The reiprol reltionships disussed erlier will enle you to determine the grphs nd properties of the other three trigonometri funtions from the grphs nd properties of the sine, osine, nd tngent funtions.
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