HPTER TRIGONOMETRI PPLITIONS n ocen is vst expnse tt cn e life-tretening to person wo experiences disster wile oting. In order for elp to rrive on time, it is necessry tt te cost gurd or sip in te re e le to mke n exct identifiction of te loction. distress signl sent out y te person in troule cn e nlyzed y tose receiving te signl from different directions. In tis cpter we will derive te formuls tt cn e used to determine distnces nd ngle mesures wen sufficient informtion is ville. 4 HPTER TLE OF ONTENTS 4- Similr Tringles 4-2 Lw of osines 4-3 Using te Lw of osines to Find ngle Mesure 4-4 re of Tringle 4-5 Lw of Sines 4-6 Te miguous se 4-7 Solving Tringles pter Summry Voculry Review Exercises umultive Review 547
548 Trigonometric pplictions 4- SIMILR TRINGLES In te study of geometry, we lerned tt if two tringles re similr, te corresponding ngles re congruent nd te corresponding sides re in proportion. If certin pirs of corresponding ngles nd sides re congruent or proportionl, ten te tringles must e similr. Te following pirs of congruent corresponding ngles nd proportionl corresponding sides re sufficient to prove tringles similr.. Two ngles (~) 2. Tree sides (SSS~) 3. Two sides nd te included ngles (SS~) 4. Hypotenuse nd one leg of rigt tringle (HL~) We cn use similr tringles to write te coordintes of point in terms of its distnce from te origin nd te sine nd cosine of n ngle in stndrd position. Let (x, y ) e point in te first y (x qudrnt of te coordinte plne wit, y ) (0, ) O 5 nd u te mesure of te ngle formed y O P(cos u, sin u) nd te positive ry of u te x-xis. Let P(cos u, sin u) e te point t wic O O Q (x, 0) x intersects te unit circle. Let Q(cos u, 0) e te point t wic perpendiculr line from P intersects te x-xis nd (x, 0) e te point t wic perpendiculr line from intersects te x-xis. Ten OPQ ~ O y ~. Terefore: O OQ 5 OP O x 2 0 cos u 5 x cos u 5 x 5 cos u QP 5 OP O y 2 0 sin u 5 y sin u 5 y 5 sin u Terefore, if is point units from te origin of te coordinte plne nd O is te terminl side of n ngle in stndrd position wose mesure is u, ten te coordintes of re ( cos u, sin u). Tis sttement cn e sown to e true for ny point in te coordinte plne.
Similr Tringles 549 Let O e n ngle in stndrd position wose mesure is u. If te terminl side, O, intersects te unit circle t P, ten te coordintes of P re (cos u, sin u). Te digrms elow sow O in ec qudrnt. y y O u P(cos u, sin u) x P(cos u, sin u) O u x y y u O x u O x P(cos u, sin u) P(cos u, sin u) diltion of wit center t te origin will stretc ec segment wose endpoint is te origin y fctor of. Under te diltion D, te imge of (x, y) is (x, y). If is te imge of P(cos u, sin u) under te diltion D, ten te coordintes of re ( cos u, sin u). Terefore, ( cos u, sin u) re te coordintes of point units from te origin on te terminl ry of n ngle in stndrd position wose mesure is u. ny tringle cn e positioned on te coordinte plne so tt ec vertex is identified y te coordintes of point in te plne. Te coordintes of te vertices cn e expressed in terms of te trigonometric function vlues of ngles in stndrd position.
550 Trigonometric pplictions EXMPLE Solution Point S is 2 units from te origin nd mkes n ngle of 35 wit te positive ry of te x-xis. Wt re te exct coordintes of S? Te coordintes of S re (2 cos 35, 2 sin 35 ). Since 35 5 80 2 45, cos 35 52cos 45 5 2!2 2 nd!2 sin 35 5 sin 45 5 2. OS Terefore, te coordintes of S re Q2 3 2!2!2 2, 2 3 2 R 5 26!2, 6!2. nswer y S(2 cos 35, 2 sin 35 ) 35 O x EXMPLE 2 Solution Te coordintes of re (25.30, 28.48).. Find O to te nerest undredt.. Find, to te nerest degree, te mesure of te ngle in stndrd position wose terminl side is O.. Let (25.30, 0) e te point t wic verticl line from intersects te x-xis. Ten O is te ypotenuse of rigt O, O 5 25.30 2 0 5 5.30 nd 5 28.48 2 0 5 8.48. Using te Pytgoren Teorem: O 2 5 O 2 2 O 2 5 5.30 2 8.48 2 O 5 "5.30 2 8.48 2 We reject te negtive root. (25.30, 0) y (25.30, 28.48) u O x ENTER: 2nd 5.30 x 2 DISPLY: (5.30 2 +8.48 2 ) 8.48 x 2 ) 0.00002 ENTER Write te mesure of O to te nerest undredt: O 5 0.00.
Similr Tringles 55. Let u e te mesure of te ngle in stndrd position wit terminl side O. is in te tird qudrnt. We cn use eiter coordinte to find te mesure of u, tird-qudrnt ngle. (O cos u, O sin u) 5 (25.30, 28.48) 0.00 cos u525.30 cos u520.530 ENTER: 2nd OS (-) 0.530 ) ENTER 0.00 sin u528.48 sin u520.848 ENTER: 2nd SIN (-) 0.848 ) ENTER DISPLY: cos - ( - 0.530) 22.0054548 DISPLY: sin - ( -.848) - 57.9948004 Te clcultor returns te mesure of second-qudrnt ngle. Use tis mesure to find te reference ngle to te nerest degree. R 5 80 2 22 5 58 Te clcultor returns te mesure of fourt-qudrnt ngle. Use tis mesure to find te reference ngle to te nerest degree. R 52(258 ) 5 58 Use te reference ngle to find te mesure of te tird-qudrnt ngle. nswers. O 5 0.00. u 5 238 u580 58 5 238 Exercises Writing out Mtemtics. In Exmple 2, is it possile to find te mesure of u witout first finding O? Justify your nswer. 2 cos 4p 3, 2 sin 4p 3 2. In wt qudrnt is point wose coordintes in rdin mesure re? Justify your nswer. Developing Skills In 3 4, write in simplest rdicl form te coordintes of ec point if is on te terminl side of n ngle in stndrd position wose degree mesure is u. 3. O 5 4, u545 4. O 5 2, u530 5. O 5 6, u590 6. O 5 8, u520 7. O 5 5, u535 8. O 5 0.5, u580
552 Trigonometric pplictions 9. O 5 9, u550 0. O 5 25, u520. O 5 2, u5270 2. O 5!2, u5225 3. O 5!3, u5300 4. O 5 2, u5 60 In 5 23, te coordintes of point re given.. Find te distnce of te point from te origin. Express pproximte distnces to te nerest undredt.. Find te mesure, to te nerest degree, of te ngle in stndrd position wose terminl side contins te given point. 5. (6, 8) 6. (25, 2) 7. (0, 7) 8. (2, 29) 9. (5, 0) 20. (28, 22) 2. (24, 7) 22. (6, 20) 23. (28, 8) In 24 29, for ec ORS, O is te origin, R is on te positive ry of te x-xis nd PS is te ltitude from S to OR g.. Find te exct coordintes of R nd S.. Find te exct re of ORS. p p 24. OR 5 5, m ROS 5 3, OS 5 3 25. OR 5 2, m ROS 5 2, OS 5 8 3p 26. OR 5 8, m ROS 5 4, OS 5 8 27. OR 5 20, OS 5 RS, PS 5 0 28. OR 5 9, ORS is equilterl p 29. OR 5 7, m ROS 5 6, PS 5 8 4-2 LW OF OSINES Wen we know te mesures of two sides nd te included ngle of tringle (SS), te size nd spe of te tringle re determined. Terefore, we sould e le to find te mesure of te tird side of te tringle. In order to derive formul to do tis, we will position te tringle in te coordinte plne wit one endpoint t te origin nd one ngle in stndrd position. s sown in te digrms, te ngle in stndrd position cn e eiter cute, otuse, or rigt. y ( cos, sin ) c O (0, 0) (c, 0) x ( cos, sin ) c O (0, 0) (c, 0) x y ( cos, sin ) 5 (0, sin ) c x O (0, 0) (c, 0) Let e tringle wit 5 c, 5, nd 5.Te coordintes of re (0, 0), of re (c, 0), nd of re ( cos, sin ). If, c, nd m re known mesures, ten te coordintes of ec vertex re known. We cn find, te lengt of te tird side of te tringle, y using te distnce formul.
Te distnce etween te two points P(x, y ) nd Q(x 2, y 2 ) is given y te formul: PQ 2 5 (x 2 x ) 2 (y 2 2 y ) 2 Let P(x, y ) 5 (c, 0) nd Q(x 2, y 2 ) 5 ( cos, sin ). 2 5 ( cos c) 2 ( sin 2 0) 2 5 2 cos 2 2 2c cos c 2 2 sin 2 5 2 cos 2 2 sin 2 c 2 2 2c cos 5 2 (cos 2 sin 2 ) c 2 2 2c cos 5 2 () c 2 2 2c cos 5 2 c 2 2 2c cos If we let 5, we cn write: Lw of osines 553 2 5 2 c 2 2 2c cos Tis formul is clled te Lw of osines. Te lw of cosines for cn e written in terms of te mesures of ny two sides nd te included ngle. 2 5 2 c 2 2 2c cos 2 5 2 c 2 2 2c cos c 2 5 2 2 2 2 cos We cn rewrite te Lw of osines in terms of te letters tt represent te vertices of ny tringle. For exmple, in DEF, side DE is opposite F so we let DE 5 f, side EF is opposite D so we let EF 5 d, nd side DF is opposite E so we let DF 5 e. We cn use te Lw of osines to write formul for te squre of te mesure of ec side of DEF terms of te mesures of te oter two sides nd te included ngle. D e f F d E d 2 5 e 2 f 2 2 2ef cos D e 2 5 d 2 f 2 2 2df cos E f 2 5 d 2 e 2 2 2de cos F
554 Trigonometric pplictions EXMPLE In, 5 8, 5 0, nd cos 5 8. Find. Solution 5 c 5 8, 5 5 0, nd cos 5 8. Use te Lw of osines to find 5. 2 5 2 c 2 2 2c cos 2 5 0 2 8 2 2 2(0)(8) 2 5 00 64 2 20 2 5 44 562 Since is te lengt of line segment, is positive numer. nswer 5 2 8 0 8 EXMPLE 2 Te digonls of prllelogrm mesure 2 centimeters nd 22 centimeters nd intersect t n ngle of 43 degrees. Find te lengt of te longer sides of te prllelogrm to te nerest tent of centimeter. Solution Let te digonls of prllelogrm PQRS intersect t T. Te digonls of prllelogrm isect ec oter. If PR 5 2, ten PT 5 6 nd if QS 5 22, ten QT 5. Let PQ e te longer side of te prllelogrm, te side opposite te lrger ngle t wic te digonls intersect. Terefore, m PTQ 5 43. Write te Lw of osines for PQ 2 5 t 2 in terms of QT 5 p 5, PT 5 q 5 6, nd cos T 5 cos 43. S P 6 t T 43 6 R Q t 2 5 p 2 q 2 2 2pq cos T 5 2 6 2 2 2()(6) cos 43 5 2 36 2 32 cos 43 t 5!57 2 32 cos 438 Note tt cos 43 is negtive so 232 cos 43 is positive. Using clcultor, we find tt t 6.9937923. nswer To te nerest tent, PQ 5 6.2 cm.
Lw of osines 555 Exercises Writing out Mtemtics. Explin ow te Lw of osines cn e used to sow tt in n otuse tringle, te side opposite n otuse ngle is te longest side of te tringle. 2. Explin te reltionsip etween te Lw of osines nd te Pytgoren Teorem. Developing Skills 3. In MR, express m 2 in terms of, r, nd cos M. 4. In NOP, express p 2 in terms of n, o, nd cos P. 5. In, if 5 3, 5 5, nd cos 5 5, find te exct vlue of c. 3 6. In DEF, if e 5 8, f 5 3, nd cos D 5 4, find te exct vlue of d. 7. In HIJ, if 5 0, j 5 7, nd cos I 5 0.6, find te exct vlue of i. In 8 3, find te exct vlue of te tird side of ec tringle. p 8. In, 5 4, c 5 4, nd m 5 3. p 9. In PQR, p 5 6, q 5!2, nd m R 5 4. p 0. In DEF, d 5!3, e 5 5, nd m F 5 6. 2p. In, 5 6, 5 4, nd m 5 3. 5p 2. In RST, RS 5 9, ST 5 9!3, nd m S 5 6. 3p 3. In, 5 2!2, 5 4, nd m 5 4. In 4 9, find, to te nerest tent, te mesure of te tird side of ec tringle. 4. In, 5 2.4, c 5 8.70, nd m 5 23. 5. In PQR, p 5 26, q 5 24, nd m R 5 42. 6. In DEF, d 5 3.25, e 5 5.62, nd m F 5 58. 7. In, 5 62.5, 5 44.7, nd m 5 33. 8. In RST, RS 5 0.375, ST 5.29, nd m S 5 67. 9. In, 5 2.35, 5 6.24, nd m 5 5.
556 Trigonometric pplictions pplying Skills 20. nn nd ill ekerede follow fmilir tringulr pt wen tey tke wlk. Tey wlk from ome for 0.52 mile long strigt rod, turn t n ngle of 95, wlk for noter 0.46 mile, nd ten return ome.. Find, to te nerest undredt of mile, te lengt of te lst portion of teir wlk.. Find, to te nerest undredt of mile, te totl distnce tt tey wlk. 2. Wen two forces ct on n oject, te resultnt force is te single force tt would ve produced te sme result. Wen te mgnitudes of te two forces re represented y te lengts of 2 two sides of prllelogrm, te resultnt cn e represented 75 y te lengt of te digonl of te prllelogrm. If forces of 2 pounds nd 8 pounds ct t n ngle of 75, wt is te 8 mgnitude of te resultnt force to te nerest undredt pound? 22. field is in te spe of prllelogrm. Te lengts of two djcent sides re 48 meters nd 65 meters. Te mesure of one ngle of te prllelogrm is 00.. Find, to te nerest meter, te lengt of te longer digonl.. Find, to te nerest meter, te lengt of te sorter digonl. 23. pole is rced y two wires tt extend from te top of te pole to te ground. Te lengts of te wires re 6 feet nd 8 feet nd te mesure of te ngle etween te wires is 0. Find, to te nerest foot, te distnce etween te points t wic te wires re fstened to te ground. 24. Two points nd re on te soreline of Lke George. surveyor is locted t tird point some distnce from ot points. Te distnce from to is 80.0 meters nd te distnce from to is 20.0 meters. Te surveyor determines tt te mesure of is 56.3. To te nerest tent of meter, wt is te distnce from to? 25. Two silots leve dock t te sme time siling on courses tt form n ngle of 2 wit ec oter. If one ot sils t 0.0 knots nd te oter sils t 2.0 knots, ow mny nuticl miles prt re te ots fter two ours? (nuticl miles 5 knots 3 time) Round to te nerest tent. 2 26. Use te Lw of osines to prove tt if te ngle etween two congruent sides of tringle mesures 60, te tringle is equilterl.
Using te Lw of osines to Find ngle Mesure 557 4-3 USING THE LW OF OSINES TO FIND NGLE MESURE EXMPLE Te mesures of tree sides of tringle determine te size nd spe of te tringle. If we know te mesures of tree sides of tringle, we cn use te Lw of osines to find te mesure of ny ngle of te tringle. For exmple, in, if 5 7, 5 5, nd c 5 8, use te Lw of osines to find cos. 2 5 2 c 2 2 2c cos 7 7 2 5 5 5 8 2 2 2(5)(8) cos 49 5 25 64 2 80 cos 8 80 cos 5 25 64 2 49 5 80 cos 5 40 40 cos 5 80 cos 5 2 Since is n ngle of tringle, 0,, 80. Terefore, 5 60. Te steps used to solve for cos in terms of te mesures of te sides cn e pplied to te generl formul of te Lw of osines to express te cosine of ny ngle of te tringle in terms of te lengts of te sides. 2 5 2 c 2 2 2c cos 2c cos 5 2 c 2 2 2 cos 5 2 c 2 2 2 2c Tis formul cn e rewritten in terms of te cosine of ny ngle of. cos 5 cos 5 cos 5 In, 5 2, 5 8, c 5 6. Find cos. 2 c 2 2 2 2c 2 c 2 2 2 2c 2 2 2 c 2 2 Solution How to Proceed () Write te Lw of osines in terms of cos : (2) Sustitute te given vlues: (3) Perform te computtion. Reduce te frctionl vlue of cos to lowest terms: nswer cos 5 43 48 cos 5 2 2 2 c 2 2 2 cos 5 2 8 2 2 6 2 2(2)(8) 44 64 2 36 cos 5 92 72 5 92 5 43 48
558 Trigonometric pplictions EXMPLE 2 Find, to te nerest degree, te mesure of te lrgest ngle of DEF if DE 5 7.5, EF 5 9.6, nd DF 5 3.5. Solution Te lrgest ngle of te tringle is opposite te longest side. Te lrgest ngle is E, te ngle opposite te longest side, DF. Let DE 5 f 5 7.5, EF 5 d 5 9.6, nd DF 5 e 5 3.5. Write te formul in terms of cos E. cos E 5 d 2 f 2 2 e 2 2df 5 9.62 7.5 2 2 3.5 2 2(9.6)(7.5) < 20.235 Terefore, E 5 rccos (20.235). Use clcultor to find te rccosine: D 7.5 E 3.5 9.6 F ENTER: 2nd OS (-) 0.235 DISPLY: cos - ( - 0.235) ) ENTER 03.596228 nswer m E 5 04 Exercises Writing out Mtemtics. Explin ow te Lw of osines cn e used to sow tt 4, 7, nd 2 cnnot e te mesures of te sides of tringle. 2. Sow tt if is n otuse ngle, 2 2, c 2. Developing Skills 3. In TUV, express cos T in terms of t, u, nd v. 4. In PQR, express cos Q in terms of p, q, nd r. 5. In KLM, if k 5 4, l 5 5, nd m 5 8, find te exct vlue of cos M. 6. In XYZ, if x 5, y 5 2, nd z 5!5, find te exct vlue of cos Z. In 7 2, find te cosine of ec ngle of te given tringle. 7. In, 5 4, 5 6, c 5 8. 8. In, 5 2, 5 8, c 5 8. 9. In DEF, d 5 5, e 5 2, f 5 8. 0. In PQR, p 5 2, q 5 4, r 5 5.. In MNP, m 5 6, n 5 5, p 5 8. 2. In, 5 5, 5 2, c 5 3.
re of Tringle 559 In 3 8, find, to te nerest degree, te mesure of ec ngle of te tringle wit te given mesures of te sides. 3. 2, 20, 22 4. 9, 0, 5 5. 30, 35, 45 6.,, 5 7. 32, 40, 38 8. 7, 24, 25 pplying Skills 9. Two ligtouses re 2 miles prt long strigt sore. sip is 5 miles from one ligtouse nd 20 miles from te oter. Find, to te nerest degree, te mesure of te ngle etween te lines of sigt from te sip to ec ligtouse. 20. tree is rced y wires 4.2 feet nd 4.7 feet long tt re fstened to te tree t te sme point nd to te ground t points 7.8 feet prt. Find, to te nerest degree, te mesure of te ngle etween te wires t te tree. 2. kite is in te spe of qudrilterl wit two pir of congruent djcent sides. Te lengts of two sides re 20.0 inces nd te lengts of te oter two sides re 35.0 inces. Te two sorter sides meet t n ngle of 5.. Find te lengt of te digonl etween te points t wic te unequl sides meet. Write te lengt to te nerest tent of n inc.. Using te nswer to prt, find, to te nerest degree, te mesure of te ngle t wic te two longer sides meet. 22. em 6.5 feet long supports roof wit rfters ec mesuring 2.4 feet long. Wt is te mesure of te ngle t wic te rfters meet? 23. wlking tril is lid out in te spe of tringle. Te lengts of te tree pts tt mke up te tril re 2,500 meters, 2,000 meters, nd,800 meters. Determine, to te nerest degree, te mesure of te gretest ngle of te tril. 24. Use te formul cos 5 2 2 2 c 2 2 to sow tt te mesure of ec ngle of n equilterl tringle is 60. 2.4 ft x 6.5 ft 2.4 ft 4-4 RE OF TRINGLE Wen te mesures of two sides nd te included ngle of tringle re known, te size nd spe of te tringle is determined. Terefore, it is possile to use tese known vlues to find te re of te tringle. Let e ny tringle. If we know te mesures of 5 c, 5, nd te included ngle,, we cn find te re of te tringle.
560 Trigonometric pplictions c c D D In let e n cute ngle, D e te ltitude from to g, nd opp D 5. In rigt D, sin 5 yp 5 D 5 c or 5 c sin. Terefore: Wt if is n otuse ngle? re of 5 2 5 2 c sin Let e n otuse ngle of, D e te ltitude from to g nd D 5. In rigt D, let m D 5u. Ten sin u5 opp yp 5 D 5 c or 5 c sin u. Terefore: re of 5 2 5 2 c sin u Since D nd re djcent ngles wose sum is strigt ngle, m D 5 80 2 m nd sin u5sin. Terefore, te re of is gin equl to 2 c sin. Tus, for ny ngle, we ve sown tt: re of 5 2 5 2 c sin Te re of tringle is equl to one-lf te product of te mesures of two sides of te tringle times te sine of te mesure of te included ngle.tis formul cn e written in terms of ny two sides nd te included ngle. re n 5 2 c sin 5 2 c sin 5 2 sin D c u Tringles in te oordinte Plne Wen tringle is drwn in te coordinte plne, te re formul follows esily. Let e ny tringle. Plce te tringle wit (0, 0) t te origin nd (, 0) on te positive ry of te x-xis, nd n ngle in stndrd position.
re of Tringle 56 y y y (c cos, c sin ) (c cos, c sin ) (c cos, c sin ) c x O (0, 0) (, 0) O (0, 0) (, 0) x x O (0, 0) (, 0) From Section 4-, we know tt te coordintes of re (c cos, c sin ). For ec tringle, is te lengt of te perpendiculr from to te x-xis nd 5 c sin. Terefore, re 5 2 5 2 c sin EXMPLE Find te re of DEF if DE 5 4, EF 5 9, nd m E 5 30. Solution DE 5 f 5 4, EF 5 d 5 9, nd m E 5 30. 63 re of DEF 5 2 df sin E 5 2 (9)(4) 2 5 5 3 2 2 nswer EXMPLE 2 Te djcent sides of prllelogrm D mesure 2 nd 5. Te mesure of one ngle of te prllelogrm is 35. Find te re of te prllelogrm. Solution Te digonl of prllelogrm seprtes te prllelogrm into two congruent tringles. Drw digonl D. In D, D 5 5 2, 5 d 5 5, nd m 5 35. D 2 35 5!2 re of D 5 2 d sin 5 2 (2)(5) sin 358 5 2 (2)(5)Q 2 R 5 45!2 re of D 5 re of D 5 45!2 re of prllelogrm D 5 re of D re of D 5 90!2 nswer 90!2 squre units
562 Trigonometric pplictions Note: Te sme nswer to Exmple 2 is otined if we use djcent ngle or D. onsecutive ngles of prllelogrm re supplementry. If m 5 35, ten m 5 80 2 35 5 45. Opposite sides of prllelogrm re congruent. If D 5 2, ten 5 2. Drw digonl. In, 5 5 2, 5 c 5 5, nd m 5 45. D 2 5 45!2 re of 5 2 c sin 5 2 (2)(5) sin 458 5 2 (2)(5)Q 2 R 5 45!2 re of D 5 re of 5 45!2 re of prllelogrm D 5 re of re of D 5 90!2 EXMPLE 3 Tree streets intersect in pirs enclosing smll tringulr prk. Te mesures of te distnces etween te intersections re 85.5 feet, 02 feet, nd 78.2 feet. Find te re of te prk to te nerest ten squre feet. Solution Let,, nd e te intersections of te streets, forming. Use te Lw of osines to find te mesure of ny ngle, for exmple,. Ten use te formul for te re of tringle in terms of te mesures of two sides nd n ngle. 85.5 ft 78.2 ft 02 ft cos 5 2 c 2 2 2 85.5 5 2 02 2 2 78.2 2 2c 2(85.5)(02) 0.6650 Use clcultor to find te mesure of. m 5 cos 0.6650 48.3 re of 5 2 c sin 5 2(85.5)(02)(sin 48.3 ) 3,256 nswer Te re of te prk is pproximtely 3,260 squre feet.
re of Tringle 563 Exercises Writing out Mtemtics. Ros found te re of prllelogrm D y using ()()(sin ). Riley found te re of prllelogrm D y using ()()(sin ). Explin wy Ros nd Riley ot got te correct nswer. 2. Jessic sid tt te re of romus PQRS is (PQ) 2 (sin P). Do you gree wit Jessic? Explin wy or wy not. Developing Skills In 3 8, find te re of ec. 3. 5 3, c 5 8, sin 5 4 4. 5 2, c 5 5, sin 5 3 5 3 5. 5 9, c 5 6, sin 5 6 6. 5 24, 5 2, sin 5 4 3 3 7. 5 7, c 5 8, sin 5 5 8. 5 0, c 5 8, sin 5 0 In 9 4, find te re of ec tringle to te nerest tent. 9. In, 5 4.6, c 5 2.8, m 5 56. 0. In, 5 326, c 5 57, m 5 72.. In DEF, d 5 5.83, e 5 5.83, m F 5 48. 2. In PQR, p 5 22, q 5 287, m R 5 24. 3. In RST, t 5 5.7, s 5 5.7, m R 5 98. 4. In DEF, e 5 336, f 5 257, m D 5 22. 5. Find te exct vlue of te re of n equilterl tringle if te lengt of one side is 40 meters. 6. Find te exct vlue of te re of n isosceles tringle if te mesure of leg is 2 centimeters nd te mesure of te vertex ngle is 45 degrees. 7. Find te re of prllelogrm if te mesures of two djcent sides re 40 feet nd 24 feet nd te mesure of one ngle of te prllelogrm is 30 degrees. pplying Skills 8. field is ordered y two pirs of prllel rods so tt te spe of te field is prllelogrm. Te lengts of two djcent sides of te field re 2 kilometers nd 3 kilometers, nd te lengt of te sorter digonl of te field is 3 kilometers.. Find te cosine of te cute ngle of te prllelogrm.. Find te exct vlue of te sine of te cute ngle of te prllelogrm. c. Find te exct vlue of te re of te field. d. Find te re of te field to te nerest integer.
564 Trigonometric pplictions 9. Te roof of sed consists of four congruent isosceles tringles. Te lengt of ec equl side of one tringulr section is 22.0 feet nd te mesure of te vertex ngle of ec tringle is 75. Find, to te nerest squre foot, te re of one tringulr section of te roof. 20. grden is in te spe of n isosceles trpezoid. Te lengts of te prllel sides of te grden re 30 feet nd 20 feet, nd te lengt of ec of te oter two sides is 0 feet. If se ngle of te trpezoid mesures 60, find te exct re of te grden. 2. In,m 5 30 nd in DEF,m E 5 50. Sow tt if 5 DE nd 5 EF, te res of te two tringles re equl. 22. ron wnts to drw wit 5 5 inces, 5 8 inces, nd n re of 40 squre inces.. Wt must e te sine of?. Find, to te nerest tent of degree, te mesure of. c. Is it possile for ron to drw two tringles tt re not congruent to ec oter tt stisfy te given conditions? Explin. 23. Let D e prllelogrm wit 5 c, 5, nd m 5u.. Write formul for te re of prllelogrm D in terms of c,, nd u.. For wt vlue of u does prllelogrm D ve te gretest re? D c u 4-5 LW OF SINES If we know te mesures of two ngles nd te included side of tringle (S), or if we know te mesures of two ngles nd te side opposite one of te ngles of tringle (S), te size nd spe of te tringle is determined. Terefore, we sould e le to find te mesures of te remining sides. In, let m nd m e two ngles nd 5 e te side opposite one of te ngles. Wen we know tese mesures, is it possile to find 5? D
Lw of Sines 565 Let D e te ltitude from to. Let D 5 nd 5. In rigt D, In rigt D, opp opp sin 5 yp sin 5 yp sin 5 sin 5 5 sin 5 sin Since sin nd sin re ec equl to, tey re equl to ec oter. Terefore, sin 5 sin. To solve for, divide ot sides of tis eqution y sin. sin 5 sin sin sin 5 sin sin 5 sin sin More generlly, we cn estlis proportionl reltionsip etween two ngles nd te sides opposite tese ngles in tringle. Divide ot sides of tis eqution y sin sin. 5 n lterntive derivtion of tis formul egins wit te formuls for te re of tringle. re 5 2 c sin 5 2 c sin 5 2 sin We cn multiply ec of te lst tree terms of tis equlity y 2. 2 2 c sin sin 5 sin sin sin sin 5 sin sin sin 5 5 c sin 5 c sin 5 sin Now divide ec side of te equlity y c. c sin c sin sin c 5 c 5 c sin sin sin 5 5 c Tese equl rtios re usully written in terms of teir reciprocls. sin sin sin 2 2 c sin 5 sin 5 Tis equlity is clled te Lw of Sines. c sin 2 2 sin
566 Trigonometric pplictions EXMPLE In, c 5 2, m 5 20, nd m 5 45. Find te exct vlue of. Solution How to Proceed c () Use te rtios of te sin 5 sin Lw of Sines tt use nd c: (2) Sustitute te given vlues: (3) Solve for, sustituting sine vlues: (4) Write te vlue of in simplest form: sin 208 5 2 sin 458 5 2 sin 208 sin 458 2Q!3 2 5 R!2 2 2!3 5 2 3 2!2 5 2!3!2 3!2!2 5 2!6 2 5 6!6 2 nswer 5 6!6 EXMPLE 2 In DEF,m D 5 50, m E 5 95, nd f 5 2.6. Find d to te nerest tent. E 95 F Solution Use te form of te Lw of Sines in terms of te side wose mesure is known, f, nd te side wose mesure is to e found, d. To use tis formul, we need to know m F. Terefore: d sin D 5 f sin F m F 5 80 2 (50 95) 5 35 d sin 508 5 sin 2.6 358 2.6 sin 508 d 5 sin 358 2.6 D 50
Lw of Sines 567 Use clcultor to evlute d. ENTER: ( 2.6 SIN 50 ) ) DISPLY: SIN 35 ) ENTER (2.6 sin(50))/si n(35) 6.82802739 nswer To te nerest tent, d 5 6.8. Exercises Writing out Mtemtics. If te sine of n ngle of tringle is known, is it possile to determine te mesure of te ngle? Explin wy or wy not. 2. If te cosine of n ngle of tringle is known, is it possile to determine te mesure of te ngle? Explin wy or wy not. Developing Skills p p 3. In, if 5 9, m 5 3, nd m 5 4, find te exct vlue of in simplest form. p p 4. In, if 5 24, m 5 6, nd m 5 2, find te exct vlue of in simplest form. 2p p 5. In, if c 5 2, m 5 3, nd m 5 6, find te exct vlue of in simplest form. p 5p 6. In, if 5 8, m 5 3, nd m 5 2, find te exct vlue of in simplest form. 7. In DEF, sin D 5 0.4, sin E 5 0.25, nd d 5 20. Find te exct vlue of e. 3 2 8. In PQR, sin P 5 4, sin R 5 5, nd p 5 40. Find te exct vlue of r. 9. In DEF,m D 5 47, m E 5 84, nd d 5 7.3. Find e to te nerest tent. 0. In DEF,m D 5 56, m E 5 44, nd d 5 37.5. Find e to te nerest tent.. In LMN,m M 5 2, m N 5 54, nd m 5 5.0. Find n to te nerest tent. 2. In,m 5 02, m 5 34, nd 5 25.8. Find c to te nerest tent. 3. In PQR,m P 5 25, m Q 5 4, nd p 5 22. Find r to te nerest integer. 4. In RST,m R 5 2, m S 5 75, nd r 5 3.52. Find t to te nerest tent. 5. In DE,m D 5 25, m E 5 28, nd d 5 2.5. Find c to te nerest undredt. 6. Te se of n isosceles tringle mesures 4.5 centimeters nd te vertex ngle mesures 0 degrees.. Find te mesure of one of te congruent sides of te tringle to te nerest undredt.. Find te perimeter of te tringle to te nerest tent.
568 Trigonometric pplictions 7. Te lengt of one of te equl sides of n isosceles tringle mesures 25.8 inces nd ec se ngle mesures 53 degrees.. Find te mesure of te se of te tringle to te nerest tent.. Find te perimeter of te tringle to te nerest inc. 8. Use te Lw of Sines to sow tt if of is rigt ngle, sin 5 c. pplying Skills 9. telepone pole on illside mkes n ngle of 78 degrees wit te upwrd slope. wire from te top of te pole to point up te ill is 2.0 feet long nd mkes n ngle of 5 degrees wit te pole.. Find, to te nerest undredt, te distnce from te foot of te pole to te point t wic te wire is fstened to te ground.. Use te nswer to prt to find, to te nerest tent, te eigt of te pole. 20. Tree streets intersect in pirs enclosing smll prk. Two of te ngles t wic te streets intersect mesure 85 degrees nd 65 degrees. Te lengt of te longest side of te prk is 275 feet. Find te lengts of te oter two sides of te prk to te nerest tent. 2. On te plyground, te 0-foot ldder to te top of te slide mkes n ngle of 48 degrees wit te ground. Te slide mkes n ngle of 32 degrees wit te ground.. How long is te slide to te nerest tent?. Wt is te distnce from te foot of te ldder to te foot of te slide to te nerest tent? 22. distress signl from sip, S, is received y two cost gurd sttions locted 3.8 miles prt long strigt costline. From sttion, te signl mkes n ngle of 48 wit te costline nd from sttion te signl mkes n ngle of 67 wit te costline. Find, to te nerest tent of mile, te distnce from te sip to te nerer sttion. 78 48 S 67 23. Two sides of tringulr lot form ngles tt mesure 29. nd 33.7 wit te tird side, wic is 487 feet long. To te nerest dollr, ow muc will it cost to fence te lot if te fencing costs $5.59 per foot?
Te miguous se 569 4-6 THE MIGUOUS SE If we know te mesures of two sides of tringle nd te ngle opposite one of tem, te Lw of Sines mkes it possile for us to find te sine of te ngle opposite te second side wose mesure is known. However, we know tt te mesures of two sides nd te ngle opposite one of tem (SS) is not sufficient to determine te size nd spe of te tringle in every cse. Tis is often clled te miguous cse. onsider te following cses in wic we re given,, nd ngle. For 0, sin,, tere re two vlues of in te intervl from 0 to 80. We will cll tese vlues nd 9. Since te sum of te degree mesures of te ngles of tringle is 80, te sum of te degree mesures of two ngles of tringle must e less tn 80. SE Two tringles cn e drwn. In, 5 8, 5 2, nd 5 30. We cn use te Lw of Sines to find sin. sin 5 sin 8 sin 308 5 sin 2 8 sin 5 2 sin 30 sin 5 2 2 8 3 sin 5 4 3 Wen sin 5 4,m 48.59 or m 9 80 2 48.59 5 3.4. s sown in te digrm, tere re two tringles, nd 9 in wic two sides mesure 8 nd 2 nd te ngle opposite te sorter of tese sides mesures 30. Two tringles cn e drwn. 30 9 8 2 8 SE 2 Only one tringle cn e drwn nd tt tringle is rigt tringle. In, 5 8, 5 6, nd 5 30. We cn use te Lw of Sines to find sin. sin 5 sin 8 sin 308 5 sin 6 8 sin 5 6 sin 30 6 8 sin 5 6 2 8 30 sin 5
570 Trigonometric pplictions Wen sin 5, m 5 90. Tis is te only mesure of tt cn e te mesure of n ngle of tringle. One tringle cn e drwn nd tt tringle is rigt tringle. Note: If m 5 50, sin 5 nd m 5 90. Tere is no tringle wit n otuse ngle nd rigt ngle. SE 3 Only one tringle cn drwn. In, 5 6, 5 8, nd 5 30. We cn use te Lw of Sines to find sin. sin 5 sin 6 sin 308 5 8 sin 6 sin 5 8 sin 30 8 6 8 2 sin 5 30 6 sin 5 4 Wen sin 5 4,m 5 4.48 or m 5 80 2 4.48 5 65.52. If we let e n cute ngle, m m 5 30 4.48, 80. Tere is tringle wit m 5 30 nd m 5 4.48. If we let e n otuse ngle, m m 5 30 65.52. 80. Tere is no tringle wit m 5 30 nd m 5 65.52. Only one tringle cn e drwn. SE 4 No tringle cn e drwn. In, 5 8, 5 20, nd 5 30. We cn use te Lw of Sines to find sin. sin 5 sin 8 sin 308 5 sin 20 8 sin 5 20 sin 30 20 8 sin 5 sin 5 20 2 8 5 4 30 Tere is no vlue of for wic sin.. No tringle cn e drwn.
Te miguous se 57 Tese four exmples sow tt for te given lengts of two sides nd te mesure of n cute ngle opposite one of tem, two, one, or no tringles cn e formed. To determine te numer of solutions given,, nd m in : Use te Lw of Sines to solve for sin. * If sin +, tere is no tringle. If sin 5, tere is one rigt tringle if is cute ut no tringle if is otuse. * * If is cute nd sin *, find two possile vlues of : * 0 * m, 90 nd m 9 580 2 m. * * * * 9 If m m 9,80, tere re two possile tringles, nd 9. If m m 9 $80, 9 is not tringle. Tere is only one possile tringle,. If is otuse nd sin *, must e cute: 0 * m * 90. * * * If m m, 80, tere is one tringle,. If m m $ 80, tere is no tringle.
572 Trigonometric pplictions lterntively, if we let 5 sin, te eigt of te tringle, we cn summrize te numer of possile tringles given,, nd m in : is: cute cute cute cute Otuse Otuse, 5,,. #. Possile None One, Two One None One tringles: rigt EXMPLE 9 Solution In, 5 9, c 5 2, nd m 5 45.. Find te exct vlue of sin.. For te vlue of sin in, find, to te nerest undredt, te mesures of two ngles, nd 9, tt could e ngles of tringle. c. How mny tringles re possile?. () Use te rtios of te Lw of Sines tt use nd c: (2) Sustitute te given vlues: (3) Solve for sin : (4) Sustitute te exct vlue of sin 45:. Use clcultor to find te pproximte mesure of. sin 45 9 5 c sin 9 sin 5 sin 2 458 2 sin 5 9 sin 458 sin 5 9 2 sin 458 3 sin 5 4 3!2 2 sin 5 3!2 8 2 ENTER: 2nd SIN ( 3 DISPLY: sin - 2 ((3 * (2))/8 2nd ) ) 32.027760 8 ENTER m 32.03 nd m 9 80 2 32.03 5 47.97 nswer c. m m 5 45 32.03, 80 nd is tringle. m m 9 545 47.97. 80 nd 9 is not tringle. Tere is one possile tringle. nswer
Te miguous se 573 EXMPLE 2 Solution How mny tringles cn e drwn if te mesures of two of te sides re 2 inces nd 0 inces nd te mesure of te ngle opposite te 0-inc side is 0 degrees? Q Let possile tringle e PQR wit 0 PQ 5 r 5 2 nd QR 5 p 5 0. Te ngle opposite QR is P nd m P 5 0. 2 R () We know p, r, nd P. Use te Lw of Sines to solve for sin R: (2) Use clcultor to find sin R: 0 P p r sin P 5 sin R 0 sin 08 5 sin 2 R 0 sin R 5 2 sin 0 2 sin 08 sin R 5 0 sin R.276345 Since sin R., tere is no tringle. lterntive Solution Let represent te eigt of te tringle. Ten 5 r sin P 5 2 sin 70.28 Q 0 nswer 0 Te eigt of te tringle, or te ltitude t Q, would e longer tn side QR. No suc tringle exists. 2 70 0 P R Exercises Writing out Mtemtics. Witout using formuls tt include te sine of n ngle, is it possile to determine from te given informtion in Exmple 2 tt tere cn e no possile tringle? Justify your nswer. 2. Explin wy, wen te mesures of two sides nd n otuse ngle opposite one of tem re given, it is never possile to construct two different tringles.
574 Trigonometric pplictions Developing Skills In 3 4:. Determine te numer of possile tringles for ec set of given mesures.. Find te mesures of te tree ngles of ec possile tringle. Express pproximte vlues to te nerest degree. 3. 5 8, 5 0, m 5 20 4. 5 5, 5 0, m 5 30 5. 5 2, c 5 0, m 5 49 6. 5 6, c 5 0, m 5 45 7. c 5 8, 5 0, m 5 20 8. 5 9, c 5 0, m 5 50 9. 5 4, 5 2, m 5 75 0. DE 5 24, EF 5 8, m D 5 5. PQ 5 2, PR 5 5, m R 5 00 2. 5 2, 5 2!2,m 5 35 3. RS 5 3!3, ST 5 3, m T 5 60 4. 5 8, 5 0, m 5 45 pplying Skills 5. ldder tt is 5 feet long is plced so tt it reces from level ground to te top of verticl wll tt is 3 feet ig.. Use te Lw of Sines to find te ngle tt te ldder mkes wit te ground to te nerest undredt.. Is more tn one position of te ldder possile? Explin your nswer. 6. Mx s tringulr grden. He mesured two sides of te grden nd te ngle opposite one of tese sides. He sid tt te two sides mesured 5 feet nd 8 feet nd tt te ngle opposite te 8-foot side mesured 75 degrees. n grden exist wit tese mesurements? ould tere e two grdens of different spes wit tese mesurements? Write te ngle mesures nd lengts of te sides of te grden(s) if ny. 7. Emily wnts to drw prllelogrm wit te mesure of one side 2 centimeters, te mesure of one digonl 0 centimeters nd te mesure of one ngle 20 degrees. Is tis possile? Explin wy or wy not. 8. Ross sid tt wen e jogs, is pt forms tringle. Two sides of te tringle re 2.0 kilometers nd 2.5 kilometers in lengt nd te ngle opposite te sorter side mesures 45 degrees. Ros sid tt wen se jogs, er pt lso forms tringle wit two sides of lengt 2.0 kilometers nd 2.5 kilometers nd n ngle of 45 degrees opposite te sorter side. Ros sid tt er route is longer tn te route Ross follows. Is tis possile? Explin your nswer.
Solving Tringles 575 4-7 SOLVING TRINGLES If te known mesures of ny tree prts of tringle include t lest one side, te mesures of te remining tree prts cn e determined. Te Rigt Tringle Wen te tringle is rigt tringle, te rtio of te mesures of ny two sides of te tringle is trigonometric function vlue of one of te cute ngles. In rigt wit m 5 90: opp sin 5 yp 5 cos 5 tn 5 dj yp 5 opp dj 5 Te eigt of uilding, tree, or ny similr oject is mesured s te lengt of te perpendiculr from te top of te oject to te ground. Te mesurement of eigt often involves rigt tringles. ngle of elevtion n ngle of depression is n ngle suc tt one ry is prt of orizontl line nd te oter ry represents line of sigt moved downwrd from te orizontl. To visulize te ngle of depression of uilding, tink of some point on te sme orizontl line s te top of te uilding. Te ngle of depression is te ngle troug wic our line of sigt would rotte from te top of te uilding to its se. In te digrm, D is te ngle of depression. n ngle of elevtion is n ngle suc tt one ry is prt of orizontl line nd te oter ry represents line of sigt rised upwrd from te orizontl. To visulize te ngle of elevtion of uilding, tink of some point on te sme orizontl line s te se of te uilding. Te ngle of elevtion is te ngle troug wic our line of sigt would rotte from te se of te uilding to its top. In te digrm, is te ngle of elevtion. Wen solving rigt tringles, we cn use te rtio of sides given ove or we cn use te Lw of Sines or te Lw of osines. D ngle of depression
576 Trigonometric pplictions EXMPLE Solution From point 2 feet from te foot of te tree, te mesure of te ngle of elevtion to te top of te tree is 57. Find te eigt of te tree to te nerest tent of foot. Te eigt of te tree is te lengt of te perpendiculr from te top of te tree to te ground. Use te rtio of sides of rigt tringle. We know te lengt of te side djcent to te ngle of elevtion nd we wnt to know te eigt of te tree, te lengt of te side opposite te ngle of elevtion. opp tn u5 dj tn 57 5 2 5 2 tn 57 5 8.478 57 2 ft lterntive Solution nswer We know te mesures of two ngles nd te included side. Find te mesure of te tird ngle nd use te Lw of Sines. m 5 80 2 (m m ) m 5 80 2 (57 90) m 5 33 Te tree is 8.5 feet tll. sin 5 sin sin 578 5 sin 2 338 2 sin 578 5 sin 338 5 8.478 Te Generl Tringle Te Lw of osines nd te Lw of Sines cn e used to find te remining tree mesures of ny tringle wen we know te mesure of side nd te mesures of ny two oter prts of te tringle. SE Given: Two sides nd te included ngle Use te Lw of osines to find te mesure of te tird side.
Solving Tringles 577 Use te Lw of Sines or te Lw of osines to find te mesure of noter ngle. Use te sum of te ngles of tringle to find te mesure of te tird ngle. SE 2 Given: Tree sides Use te Lw of osines to find te mesure of n ngle. Use te Lw of Sines or te Lw of osines to find te mesure of noter ngle. Use te sum of te ngles of tringle to find te mesure of te tird ngle. SE 3 Given: Two ngles nd side Use te sum of te ngles of tringle to find te mesure of te tird ngle. Use te Lw of Sines to find te remining sides. SE 4 Given: Two sides nd n ngle opposite one of tem Use te Lw of Sines to find te possile mesure(s) of noter ngle. Determine if tere re two, one, or no possile tringles. If tere is tringle, use te sum of te ngles of tringle to find te mesure(s) of te tird ngle. Use te Lw of Sines or te Lw of osines to find te mesure(s) of te tird side.
578 Trigonometric pplictions EXMPLE 2 Te Prks Deprtment is lying out nture tril troug te woods tt is to consist of tree strigt pts tt form tringle.te lengts of two pts mesure.2 miles nd.5 miles. Wt must e te mesure of te ngle etween tese two sections of te pt in order tt te totl lengt of te nture tril will e 4.0 miles? Solution () Drw digrm. Te tril forms tringle,, wit 5 c 5.2 nd 5 5.5:.2.5 nswer 56 (2) Te perimeter of te tringle is 4.0. Find 5 : (3) Use te Lw of osines to find te mesure of n ngle wen te mesures of tree sides re known. Find te mesure of : P 5 4.0 5.2.5 4.0 5 2.7.3 5 cos 5 2 c 2 2 2 2c.5 cos 5 2.2 2 2.3 2 2(.5)(.2) cos 5 0.5555... m 5 56.25 EXMPLE 3 Solution mn, stnding on cliff 85 feet ig t te edge of te wter, sees two sips. He positions imself so tt te sips re in strigt line wit point directly elow were e is stnding. He estimtes tt te ngle of depression of te closer sip is 75 degrees nd te ngle of depression of te frter sip is 35 degrees. How fr prt re te two sips? () Drw nd lel digrm. Let e te closer sip, te frter sip, te edge of te cliff were te mn is stnding, nd D te point directly elow t se level. Te ngle of depression is te ngle etween orizontl ry nd ry downwrd to te sip. Determine te mesure of ec ngle in te digrm: 35 35 75 40 05 75 D 85 ft
nswer (2) Use rigt D to find, te mesure of side of ot nd D. (3) In, we now know te mesures of two ngles nd te side opposite one of tem. Use te Lw of Sines to find, te required distnce. Te sips re 99 feet prt. Solving Tringles 579 opp sin 75 5 yp 85 sin 75 5 85 5 sin 758 87.998 sin / 5 sin / sin 408 5 sin 87.998 358 5 98.62 87.998 sin 408 sin 358 Exercises Writing out Mtemtics. Nvir sid tt in Exmple 3, it would ve een possile to solve te prolem y using D to find first. Do you gree wit Nvir? Explin wy or wy not. 2. Explin wy n ngle of depression is lwys congruent to n ngle of elevtion. Developing Skills In 3 0:. Stte weter ec tringle sould e solved y using te Lw of Sines or te Lw of osines.. Solve ec tringle, rounding nswers to te nerest tent. Include ll possile solutions. 3. 4. 5. 6. 2 7 63 7 33 2 5.8 9 x 0 30 x x x 6. 7.4 7. x 5 8. 6 x 3 9. x 0. 30 8 7 32 56 24 x 94 6.8 29 In 22, solve ec tringle, tt is, find te mesures of te remining tree prts of te tringle to te nerest integer or te nerest degree.. In, 5 5, 5 8, nd m 5 60. 2. In, 5 0, 5 2, nd m 5 30.
580 Trigonometric pplictions 3. In, 5 25, m 5 45, nd m 5 60. 4. In, 5 8, m 5 35, nd m 5 55. 5. In DEF, d 5 72, e 5 48, nd m F 5 0. 6. In PQR, p 5 2, m Q 5 80, nd m R 5 30. 7. In RST, r 5 38, s 5 28, nd t 5 8. 8. In, 5 22, 5 8, nd m 5 30. 9. In PQR, p 5 2, q 5 6, nd r 5 20. 20. In DEF, d 5 36, e 5 72, nd m D 5 30. 2. In RST, r 5 5, s 5 8, nd m T 5 90. 22. In, 5 5, 5 25, nd c 5 2. 23. In te digrm, D 5 25, D 5 0, D 5, nd m D 5 75. Find to te nerest tent. 75 0 25 pplying Skills 24. smll prk is in te spe of n isosceles trpezoid. Te lengt of te longer of te prllel sides is 3.2 kilometers nd te lengt of n djcent side is 2.4 kilometers. pt from one corner of te prk to n opposite corner is 3.6 kilometers long.. Find, to te nerest tent, te mesure of ec ngle etween djcent sides of te prk.. Find, to te nerest tent, te mesure of ec ngle etween te pt nd side of te prk. c. Find, to te nerest tent, te lengt of te sorter of te prllel sides. 25. From point on te ground 50 feet from te foot of verticl monument, te mesure of te ngle of elevtion of te top of te monument is 65 degrees. Wt is te eigt of te monument to te nerest foot? 26. verticl telepone pole tt is 5 feet ig is rced y two wires from te top of te pole to two points on te ground tt re 5.0 feet prt on te sme side of te pole nd in strigt line wit te foot of te pole. Te sorter wire mkes n ngle of 65 degrees wit te ground. Find te lengt of ec wire to te nerest tent. 27. From point t te top of cliff, two points, nd, re sited on level ground. Points nd re on strigt line wit D, point directly elow. Te ngle of depression of te nerer point,, is 72 degrees nd te ngle of depression of te frter point,, is 48 degrees. If te points nd re 20 feet prt, wt is te eigt of te cliff to te nerest foot? 28. Mrk is uilding kite tt is qudrilterl wit two pirs of congruent djcent sides. One digonl divides te kite into two unequl isosceles tringles nd mesures 4 inces. Ec leg of one of te isosceles tringles mesures 5 inces nd ec leg of te oter mesures 2 inces. Find te mesures of te four ngles of te qudrilterl to te nerest tent. D
pter Summry 58 HPTER SUMMRY Wen we know te mesures of tree sides, two sides nd te included ngle, or two ngles nd ny side of tringle, te size nd spe of te tringle re determined. We cn use te Lw of osines or te Lw of Sines to find te mesures of te remining prts of te tringle. Lw of osines: 2 5 2 c 2 2 2c cos cos 5 c 2 2 2 2c 2 5 2 c 2 2 2c cos cos 5 c 2 2 2 2c c 2 5 2 2 2 2 cos cos 5 2 2 c 2 2 Te Lw of osines cn e used to find te mesure of te tird side of tringle wen te mesures of two sides nd te included ngle re known. Te Lw of osines cn lso e used to find te mesure of ny ngle of tringle wen te mesures of tree sides re known. Lw of Sines: sin 5 sin 5 c sin Te Lw of Sines cn e used to find te mesure of side of tringle wen te mesures of ny side nd ny two ngles re known. Te Lw of Sines cn lso e used to find te numer of possile tringles tt cn e drwn wen te mesures of two sides nd n ngle opposite one of tem re known nd cn e used to determine te mesures of te remining side nd ngles if one or two tringles re possile. re of Tringle: re of 5 2 c sin 5 2 c sin 5 2 sin Te re of ny tringle cn e found if we know te mesures of two sides nd te included ngle. n ngle of elevtion is n ngle etween orizontl line nd line tt is rotted upwrd from te orizontl position. n ngle of depression is n ngle etween orizontl line nd line tt is rotted downwrd from te orizontl position. To determine te numer of solutions given,, nd m in, use te Lw of Sines to solve for sin. If sin., tere is no tringle. If sin 5, tere is one rigt tringle if is cute ut no tringle if is otuse. If is cute nd sin,, find two possile vlues of : 0, m, 90 nd m 9 580 2 m. If m m 9,80, tere re two possile tringles, nd 9. If m m 9 $80, 9 is not tringle. Tere is only one possile tringle,.
582 Trigonometric pplictions If is otuse nd sin,, must e cute: 0, m, 90. If m m, 80, tere is one tringle,. If m m $ 80, tere is no tringle. lterntively, if we let 5 sin, te eigt of te tringle, we cn summrize te numer of possile tringles given,, nd m in : 9 is: cute cute cute cute Otuse Otuse, 5,,.,. Possile None One, Two One None One tringles: rigt VOULRY 4-2 Lw of osines 4-5 Lw of Sines 4-6 miguous cse 4-7 ngle of elevtion ngle of Depression REVIEW EXERISES. In RST, RS 5 8, RT 5 27, nd m R 5 50. Find ST to te nerest integer. 2. Te mesures of two sides of tringle re 2.0 inces nd 5.0 inces. Te mesure of te ngle included etween tese two sides is 80 degrees. Find te mesure of te tird side of te tringle to te nerest tent of n inc. 3. In DEF, DE 5 84, EF 5 76, nd DF 5 94. Find, to te nerest degree, te mesure of te smllest ngle of te tringle. 4. Te mesures of tree sides of tringle re 22, 46, nd 58. Find, to te nerest degree, te mesure of te lrgest ngle of te tringle. 5. Use te Lw of osines to sow tt if te mesures of te sides of tringle re 0, 24, nd 26, te tringle is rigt tringle.
Review Exercises 583 6. In, 5 24, 5 40, nd m 5 30.. Find te re of.. Find te lengt of te ltitude from to. 7. Te lengts of te sides of tringle re 8,, nd 5.. Find te mesure of te smllest ngle of te tringle to te nerest tent.. Find te re of te tringle to te nerest tent. 8. In, 5 30.0, m 5 70, nd m 5 55. Find, to te nerest tent, nd. 9. Te mesures of two ngles of tringle re 00 nd 46. Te lengt of te sortest sides of te tringle is 2. Find, to te nerest integer, te lengts of te oter two sides. 0. In, 5 4, 5 6, nd m 5 48.. How mny different tringles re possile?. Find te mesures of nd of if is n cute tringle. c. Find te mesures of nd of if is n otuse tringle.. Sow tt it is not possile to drw PQR wit p 5 2, r 5 5, nd m P 5 66. 2. Te mesure of side of romus is 28.0 inces nd te mesure of te longer digonl is 50. inces.. Find, to te nerest degree, te mesure of ec ngle of te romus.. Find, to te nerest tent, te mesure of te sorter digonl. c. Find, to te nerest integer, te re of te romus. 3. Use te Lw of osines to find two possile lengts for of if 5 7, 5 8, nd m 5 60. 4. Use te Lw of Sines to sow tt tere re two possile tringles if 5 7, 5 8, nd m 5 60. 5. verticl pole is rced y two wires tt extend from different points on te pole to te sme point on level ground. One wire is fstened to te pole 5.0 feet from te top of te pole nd mkes n ngle of 6 degrees wit te ground. Te second wire is fstened to te top of te pole nd mkes n ngle of 66 degrees wit te ground. Find te eigt of te pole to te nerest tent. 66 6 5 ft
584 Trigonometric pplictions 6. ostgurd sttion is 2 miles west of costgurd sttion long strigt costline. sip is sited y te crew t sttion to te nortest of te sttion t n ngle of 35 degrees wit te costline nd y te crew t sttion to te nortwest of te sttion t n ngle of 46 degrees wit te costline. Find, to te nerest tent, te distnce from te sip to ec of te sttions. 7. In te digrm, is rigt tringle wit te rigt ngle t, 5 4.0, 5 3.0, nd m 5u. Side is extended to D nd m D 5 2u.. Find te exct vlue of sin 2u. (Hint: Use te doule-ngle formuls.). Find u to te nerest degree. c. Find D to te nerest tent. 35 2 mi 46 D 3.0 4.0 Explortion Prt Use softwre to drw ny tringle,. Trisect ec ngle of te tringle. Let D e te intersection of te trisection lines from nd tt re closest to. Let E e te intersection of te trisection lines from nd tt re closest to. Let F e te intersection of te trisection lines from nd tt re closest to. Mesure DE, EF, nd FD. Wt seems to e true out DEF? D E F Prt. Drw ny tringle,. Mesure te lengts of te sides nd te mesures of te ngles. 2. Drw te trisection lines nd lel DEF s in prt. 3. Use te Lw of Sines to find D nd D in D. 4. Use te Lw of Sines to find E nd E in E.
umultive Review 585 5. Use te Lw of Sines to find F nd F in F. 6. Use te Lw of osines to find DF in DF. 7. Use te Lw of osines to find DE in DE. 8. Use te Lw of osines to find EF in EF. 9. Is DEF n equilterl tringle? UMULTIVE REVIEW HPTERS 4 Prt I nswer ll questions in tis prt. Ec correct nswer will receive 2 credits. No prtil credit will e llowed.. Te sum of!225 nd 2!26 is () 23 (2) 3i (3) 28 (4) 8i 2. If f(u) 5 cot u, ten is () 0 (2) (3) ` (4) undefined 3 3. Te expression (k ) 2 is equl to k5 f p 2 () 3 (2) 4 (3) 8 (4) 29 4. If f(x) 5 x 2 nd g(x) 5 x 2, ten f(g(3)) is equl to () 8 (2) (3) 8 (4) 4 5. Te sum sin 50 cos 30 cos 50 sin 30 is equl to () sin 80 (2) sin 20 (3) cos 80 (4) cos 20 6. If log x 4 5 2, ten x equls () 6 (2) 2 (3) 2 (4) 6 7. Te solution set of x 2 2 2x 5 5 0 is () {2, 3} (3) { 2 2i, 2i} (2) {23, } (4) {2 2 2i, 2i} 8. In simplest form, te frction 2 is equl to 2 () (2) (3) 2( ) (4) 7p 9. n ngle of 4 rdins is congruent to n ngle of () 35 (2) 225 (3) 35 (4) 405
586 Trigonometric pplictions 0. Wic of te following is geometric sequence? (), 3, 5, 7,... (3), 0., 0.0, 0.00,... (2),3,6,0,... (4), 2, 3, 4,... Prt II nswer ll questions in tis prt. Ec correct nswer will receive 2 credits. lerly indicte te necessry steps, including pproprite formul sustitutions, digrms, grps, crts, etc. For ll questions in tis prt, correct numericl nswer wit no work sown will receive only credit.. Use te exct vlues of te sin 30, sin 45, cos 30, nd cos 45 to find te exct vlue of cos 5. 2!5 2. Write te frction s n equivlent frction wit rtionl!5 denomintor. Prt III nswer ll questions in tis prt. Ec correct nswer will receive 4 credits. lerly indicte te necessry steps, including pproprite formul sustitutions, digrms, grps, crts, etc. For ll questions in tis prt, correct numericl nswer wit no work sown will receive only credit. 3. Solve for x nd ceck: 3!2x 2 3 5 x 4. Find ll vlues of x in te intervl 0 # x, 360 tt stisfy te following eqution: 6 sin 2 x 2 5 sin x 2 4 5 0 Prt IV nswer ll questions in tis prt. Ec correct nswer will receive 6 credits. lerly indicte te necessry steps, including pproprite formul sustitutions, digrms, grps, crts, etc. For ll questions in tis prt, correct numericl nswer wit no work sown will receive only credit. 5. If log x 5 log 3 2 log 4 2 log 8, express x in simplest form. 6.. Sketc te grp of y 5 2 sin x in te intervl 0 # x # 2p.. On te sme set of xes, sketc te grp of y 5 cos 2x. c. For ow mny vlues of x in te intervl 0 # x # 2p does 2 sin x 5 cos 2x?