etion 6. Lw of osines 59 etion 6. Lw of osines If is ny oblique tringle with sides, b, nd, the following equtions re vlid. () b b os or os b b (b) b os or os b () b b os or os b b You should be ble to use the Lw of osines to solve n oblique tringle for the remining three prts, given: () Three sides () (b) Two sides nd their inluded ngle () Given ny tringle with sides of length, b, nd, the re of the tringle is re ss s bs, where s b. (Heron s Formul) Vobulry hek. osines.. Heron s re b os. Given: 7, b 0, 5 os b b sin b sin 49 00 5 70 0 sin.88 5 80 4.05.88.07 0.549.88 0.5599 4.05. Given: 8, b, 9 os b b sin sin 8 9 8 8 sin 99.59 9 80 6. 99.59 9.9 0.6667 99.59 0.8765 6.. Given: 0, b 5, 0 b b os 5 900 50 os 0 45.577 8.59 os b b 8.59 5 0 8.595 80 0 6..79 0.5907 6.
540 hpter 6 dditionl Topis in Trigonometry 4. Given: 05, 0, b 4.5 b b os 0 4.5 04.5 os 05 4.547.98 os b 0.0 4.5 0.0 80 05.7 5.7 0.987.7 5., b 4, 0 os b b sin b sin 96 400 4 4 sin 05.6 0 80 4.8 05.6.99 0.695 05.6 0.674 4.8 6. Given: 55, b 5, 7 os b b os b b 55 5 7 555 5 7 55 57 80.9 9.5 6.74 0.5578.9 0.77 9.5 7. Given: 75.4, b 5, 5 os b b sin b sin 4.5 50.9987 75.4 5 5 75.4 55 8. Given:.4, b 0.75,.5 os b b os b 80 86.68.8 6.50 0.68876 4.5 0.75.5.4 0.75.5.4.5 0.75.4.5 0.055 9.94 0.0579 86.68 0.84969.8 9. Given: sin b sin 5, b 4, 9 b b os 6 8 49os 5 47.97.6 4 sin 5.6 80 5.45.55 0.6.45 0. Given: 55, b, 0 b b os 0 0 os 55 74.585 8.64 sin b sin sin 55 8.66 80 6.5 55 08.47 0.846 6.5
etion 6. Lw of osines 54. Given: sin sin b 0 5, 40, 0 b os 600 900 400os 0 5 40.868 b.87 0 sin 0 5 0.464 7.66 7 40.87 80 0 5 7 40 4 45. Given: sin sin b 75 0, 6., 9.5 b os 6. 9.5 6.9.5 os 75 0 6. sin 75 0 0.604 7., or 7 6 9.94 80 75 0 7 6 67 4 98.866 b 9.94. Given: 5 40,, b os os 5 40 4.888 b 56.94 80 5 40 54 0 7 0 4. Given: 5 5, 6.5, b.5 b b os 6.5.5 6.5.5 os 5 5 7.756 4. os b b.5 4.8 6.5.54.8 80 5 5 57.04 7.7 or 7 4 0.908 57.04 or 57 5. 4, 4 9, b 7 9 b b os 4 9 7 9 4 9 7 os 4 0.968 0.54 9 sin sin 49 sin 4 0.5448 80 4.8 0.0 0.5564.80 6. Given: 0, 8, b 4 b b os 8 4 8 os 0 0.897 0.9 4 os b b 4 0.9 8 4 0.9 80.65 0 5.5 0.960.65 7. d 5 8 58os 45.45 d 5.69 60 45 70 5 5 8 58os 5 45.5685.07 5 45 8 φ d 8 5
54 hpter 6 dditionl Topis in Trigonometry 8. 5 0 5 9. 5 d 0 φ 4 0 d 0 5 5 5 55 os 0 75 5.0 60 0 60 d 5 5 55 os 60 975 d. 4 os 0 4 0 04.8º 60.8 68. d 0 4 04 os 68. d.86 0. os 40 60 80 4060 4 60 04.5 75.5 04.5 40 60 4060 os 75.5 998 40 φ 60 80 40 6. 60. os.5 5 0.55 os 0 5.5 05 z 80 8.8 u 80 z 97.8 0.75 4.4 0.565 55.77 55 b.5 0.50os 97.8 87.4967 b 6.96 β δ φ 0 b u b z 0.5.5 β x os.5 6.96 0.56.96 0.8 5.80 4.4 5.80 77. 60 60 77. 0.8. os 5 7.5 5 57.5 69.5 80 0.488 7.5 5 7.55 os 0.488 5.8 z 80 40.975 os 5 5.8 7.5 55.8 7.775 z 68.7 80 4.78. µ z 5 ω 5 7.5 β 7.5 5 5
etion 6. Lw of osines 54. 5, b 7, 0 s b 4., b 5, 9 s 5 9 8 re ss s bs 64 6.5 re 869 54 5. b.5, b 0., 9 s 0.85 re ss s bs 0.858.50.65.85 0.4 6. Given: s 75.4, b 5, 5 75.4 5 5 89.7 re ss s bs 89.74.7.77.7 50. 7. b., b 8.46, 5.05 s 7.95 re ss s bs 7.955.5959.455.865 5. 8. Given: s.05, b 0.75,.45.05 0.75.45.5 re ss s bs.50.075.750.675 0.6 9. os 700 700 000 5.9 700700 ering: 90 5.9 N 7. E os 700 000 700 700000 00. 700 m N 000 m E ering: 90 6.9 6. E 700 m 0. Distne from Frnklin to Rosemount: d 80 648 80648 os7 57.8 miles ering from Frnklin to Rosemount: N 75 E os 57.8 80 648 57.880 9.0 0.9456 ering from Frnklin to Rosemont: N 56.0 E N E 75 80 miles Frnklin Rosemount 648 miles enterville. b 0 50 050os 05 b 7. meters 75 0 m 05 50 m b
544 hpter 6 dditionl Topis in Trigonometry.. 76 9 650 575 5 75 os 5 76 9 576 os 5 9 76 59 0.608 5.9 0.750 4. The lrgest ngle is ross from the lrgest side. os 650 575 75 650575 os 7. 4. os 4.5 7. 0.6047 5. 80 5 67 60 b b os 6 48 6480.5 87 4. mi 48 mi 6 mi 5 60 67 N E 6. The ngles t the bse of the tower re 96 nd 84. The longer guy wire is given by: The shorter guy wire g g 75 00 7500 os 96 7,9.9 g. feet g is given by: g 75 00 7500 os 84 4,057. g 8.6 feet 7. () (b) Denver os 7 78 5 778 58.4 ering: N 58.4 os 5 78 7 578 8.5 ering: 8.5 Nigr Flls 7 5 φ Orlndo N 78 E 8. 59.7 6 miles 7.8. 65, b 6, 68 os 65 68 6 6568 7. os 6 68 65 668. 65 miles 7. 68 miles N E 0.955 0.974 () ering of Minnepolis () from Phoenix () N 90 7.. E N 59.7 E (b) ering of lbny () from Phoenix () N 90 7. E N 7.8 E
etion 6. Lw of osines 545 9. d 60.5 90 60.590 os 45 4059.857 d 6.7 ft T d P F 90 60.5 45 H 40. d 0 40 040 os 8 0.9 feet 4. 5 0 50os 4 4. miles 4. 0 0 00 os 4..8 miles R 8 0 64 4.8 ft PQ 6 0 56 89 9.4 ft tn P 0 Q Q 6 P 8 Q 5 44. () (b) x x os 46.75 0 49.5 x x os x os ± os 446.75 x os 9 os 87 7.5 x.5 x os () 0 0 0 (d) Mximum: 8.5 inhes 45. d 0 7 07 os ros 0 7 d 60 s 60 07 60 r 45 d (inhes) 9 0 4 5 6 (degrees) 60.9 69.5 88.0 98. 09.6.9 9.8 s (inhes) 0.88 0.8 8.99 8.8 7.48 6.55 5.7 46. x 40 0 0 0 x 0 sin 0 sin 0 x 0 sin 0 sin 0.95 feet 47. 00 b 500 600 s 00 500 600 650 re 6504505050 46,87.5 squre feet
546 hpter 6 dditionl Topis in Trigonometry 48. re 7000 sin 70 49. 6577.8 squre meters (The re of the prllelogrm is the sum of the res of two tringles.) s 50 840 0 5 re 55 505 8405 0 0,674 squre yrds ost 0,674 4840 000 $8,6.6 50. re ss s bs s b 490 860 50 446.0 ft 8.669 re 4560 ft re 8.669 re$00re $6,40.7 850 re 85060990500 446.0 ft 5. Flse. The verge of the three sides of tringle is b b, not s. 5. Flse. To solve n tringle, the Lw of ines is needed. 5. Flse. If 0, b 6, nd 5, then by the Lw of osines, we would hve: os 6 5 0 65.5 > This is not possible. In generl, if the sum of ny two sides is less thn the third side, then they nnot form tringle. Here 0 5 is less thn 6. 54. () orking with OD, we hve This implies tht R. os ine we know tht sin b sin sin, we n omplete the proof by showing tht os The solution of the system 80 is R sin. 90. β os β R O Therefore: os90 D R os R. sin. (b) y Heron s Formul, the re of the tringle is re ss s bs. e n lso find the re by dividing the re into six tringles nd using the ft tht the re is the bse times the height. Using the figure s given, we hve re xr xr yr yr zr zr Therefore: rx y z rs. x rs ss s bs r s s bs. s x z r z y y
etion 6. Lw of osines 547 55. 5, b 55, 7 () re of tringle: s 5 55 7 76 (b) re of irumsribed irle: re 7654 570.60 () re of insribed irle: r s s bs s 54 76 re r 77.09 7.5 (see #54 os 5 55 7 555 R 4.7 sin re R 5909. 0.5578.9 (see #54) 56. Given: s 00 ft, b 50 ft, 5 ft 00 50 5 87.5 Rdius of the insribed irle: r s s bs (see #54) s 87.57.56.5 64.5 ft 87.5 irumferene of n insribed irle: r 64.5 405. ft 57. b os b b b 58. b os b b b b b b b b b b b 4 b b b 4 b b 4 b b b b b 4 b b b b 59. rsin 60. ros 0 6. rtn 6. rtn rtn 6. rsin 64. ros ros 6 5 6 rsin x, 65. Let then 66. Let u ros x sin x x nd x os u x x. 9x se 4x. 4x tnros x tn u u 9x x x
548 hpter 6 dditionl Topis in Trigonometry rtnx, 67. Let then tn x x ot x. nd x 68. Let u rsin x sin u x. x os rsin x os u 4 x u 4 ( x ) 69. 5 5 x, x 5 sin 5 5 5 sin 5 5 sin 5 5 os os se os s is undefined. 70. x os, < 4 x 4 sin sin sin os se os s sin 4 os 4 4 os 4 os < 7. x 9, x se se 9 9se tn tn se tn 7. x 6 tn, < < 6 x 6 6 tn 6 6 tn 6 tn 6 se 6 se ot tn se s ot os sin sin 4 4 sin ± 4 ± s sin ± ± ±