If ABC is any oblique triangle with sides a, b, and c, the following equations are valid. 2bc. (a) a 2 b 2 c 2 2bc cos A or cos A b2 c 2 a 2.

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1 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 sin Given: 8, b, 9 os b b sin sin sin Given: 0, b 5, 0 b b os os os b b

2 540 hpter 6 dditionl Topis in Trigonometry 4. Given: 05, 0, b 4.5 b b os os os b , b 4, 0 os b b sin b sin sin Given: 55, b 5, 7 os b b os b b Given: 75.4, b 5, 5 os b b sin b sin Given:.4, b 0.75,.5 os b b os b Given: sin b sin 5, b 4, 9 b b os os sin Given: 55, b, 0 b b os 0 0 os sin b sin sin

3 etion 6. Lw of osines 54. Given: sin sin b 0 5, 40, 0 b os os b.87 0 sin Given: sin sin b 75 0, 6., 9.5 b os os sin , or b Given: 5 40,, b os os b Given: 5 5, 6.5, b.5 b b os os os b b or or , 4 9, b 7 9 b b os os sin sin 49 sin Given: 0, 8, b 4 b b os os os b b d os d os φ d 8 5

4 54 hpter 6 dditionl Topis in Trigonometry d 0 φ 4 0 d os d os d. 4 os º d os 68. d os os φ os os z u 80 z b os b 6.96 β δ φ 0 b u b z β x os os os z os z µ z 5 ω β

5 etion 6. Lw of osines 54. 5, b 7, 0 s b 4., b 5, 9 s re ss s bs re b.5, b 0., 9 s 0.85 re ss s bs Given: s 75.4, b 5, re ss s bs b., b 8.46, 5.05 s 7.95 re ss s bs Given: s.05, b 0.75, re ss s bs os ering: N 7. E os m N 000 m E ering: E 700 m 0. Distne from Frnklin to Rosemount: d os miles ering from Frnklin to Rosemount: N 75 E os ering from Frnklin to Rosemont: N 56.0 E N E miles Frnklin Rosemount 648 miles enterville. b os 05 b 7. meters 75 0 m m b

6 544 hpter 6 dditionl Topis in Trigonometry os os The lrgest ngle is ross from the lrgest side. os os os b b os mi 48 mi 6 mi 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 os 96 7,9.9 g. feet g is given by: g os 84 4,057. g 8.6 feet 7. () (b) Denver os ering: N 58.4 os ering: 8.5 Nigr Flls 7 5 φ Orlndo N 78 E miles , b 6, 68 os os miles miles N E () ering of Minnepolis () from Phoenix () N E N 59.7 E (b) ering of lbny () from Phoenix () N E N 7.8 E

7 etion 6. Lw of osines d os d 6.7 ft T d P F H 40. d os feet os 4 4. miles os 4..8 miles R ft PQ ft tn P 0 Q Q 6 P 8 Q () (b) x x os x x os x os ± os x os 9 os x.5 x os () (d) Mximum: 8.5 inhes 45. d os ros 0 7 d 60 s r 45 d (inhes) (degrees) s (inhes) x x 0 sin 0 sin 0 x 0 sin 0 sin 0.95 feet b s re ,87.5 squre feet

8 546 hpter 6 dditionl Topis in Trigonometry 48. re 7000 sin squre meters (The re of the prllelogrm is the sum of the res of two tringles.) s re ,674 squre yrds ost 0, $8, re ss s bs s b ft re 4560 ft re re$00re $6, re 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 > 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 () 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

9 etion 6. Lw of osines , b 55, 7 () re of tringle: s (b) re of irumsribed irle: re () re of insribed irle: r s s bs s re r (see #54 os R 4.7 sin re R (see #54) 56. Given: s 00 ft, b 50 ft, 5 ft Rdius of the insribed irle: r s s bs (see #54) s ft 87.5 irumferene of n insribed irle: r 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 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

10 548 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 ) x, x 5 sin 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 ± ± ±