REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute: x 10, y 4 r 10 2 4 2 r 116, or 2 29 b) Write the primry trigonometric rtios of u. sin U y cos U x r r 2 5 4, or 10, or 2 29 29 2 29 29 tn U y x 4, or 0.4 10 c) Wht is the vlue of u to the nerest degree? tn U 0.4 U tn 1 (0.4) U 21.8014... U is pproximtely 22. 2. Angle u is in stndrd position with its terminl rm in Qudrnt 1 2 nd sin u = 3. ) Determine cos u nd tn u. r 2 x 2 y 2 Substitute: r 3, y 2 3 2 x 2 2 2 x 2 5 x 5 cos U x r 5 3 tn U y x 2 5 b) Determine the vlue of u to the nerest degree. sin U 2 3 P DO NOT COPY. U sin 1 2 3 b U 41.8103... U is pproximtely 42. Chpter 6: Trigonometry Review Solutions 47
6.2 3. For ech ngle below: i) Sketch it in stndrd position. ii) Determine its reference ngle. ) 112 b) 286 i) Since the ngle is between i) Since the ngle is between 90 nd 180, the terminl 270 nd 360, the terminl rm is in Qudrnt 2. rm is in Qudrnt 4. y y O 112 x 286 O x ii) The ngle is in Qudrnt 2, ii) The ngle is in Qudrnt 4, so its reference ngle is: so its reference ngle is: 180 112 68 360 286 74 4. To the nerest degree, which vlues of u stisfy the eqution 3 sin u =- 8 for 0 u 360? The reference ngle is: sin 1 3 22 8 b sin U is negtive in Qudrnts 3 nd 4 so: U 180 22 nd U 360 22 202 338 6.3 5. In ABC, BC = 20 cm, AB = 25 cm, nd A = 45 Show tht is it possible to drw ABC, then determine if these mesurements illustrte n mbiguous cse. The rtio of the side opposite A to the side djcent to A is: BC 20, which is 0.8 AB 25 sin 45 0.7071... BC Since >sin 45, it is possible to drw tringle AB Since sin 45 <0.8<1, then this is n mbiguous cse 48 Chpter 6: Trigonometry Review Solutions DO NOT COPY. P
6.4 6. ) In ABC, AB = 18.7 cm, AC = 17.9 cm, nd B = 70 ; determine the mesure of BC to the nerest tenth of centimetre. Check for the mbiguous cse. The rtio of the side opposite B to the side djcent to B is: AC 17.9, which is 0.9572... AB 18.7 sin 70 0.9396... Since sin 70 <0.9572...<1, then this is n mbiguous cse, nd 2 tringles cn be constructed: ABC 1 is cute; ABC 2 is obtuse. Sketch digrm. B A 18.7 cm 17.9 cm 70 C 2 C 1 This digrm is not drwn to scle. In ABC 1 In ABC 2 Determine C 1. C 2 180 C 1 sin C 1 c sin B b Substitute: B 70, 100.9811... So, A 180 (70 100.9811... ) c 18.7, b 17.9 9.0188... sin C 1 sin 70 18.7 17.9 18.7 sin 70 sin C 1 17.9 C 1 sin 1 18.7 sin 70 b 17.9 C 1 79.0188... So, A 180 (70 79.0188... ) 30.9811... sin A b sin B Substitute: A 9.0188... B 70, b 17.9 17.9 sin 9.0188... sin 70 17.9 sin 9.0188... sin 70 2.9860... sin A b sin B Substitute: A 30.9811... B 70, b 17.9 17.9 sin 30.9811... sin 70 17.9 sin 30.9811... sin 70 9.8054... So, BC 9.8 cm or 3.0 cm P DO NOT COPY. Chpter 6: Trigonometry Review Solutions 49
b) In PQR, QR = 20 cm, PQ = 17 cm, nd P = 50 ; determine the mesure of R to the nerest degree. Check for the mbiguous cse. The rtio of the side opposite P to the side djcent to P is: QR 20, which is greter thn 1, so only 1 tringle is possible PQ 17 Sketch digrm. Q sin R r sin P p Substitute: P 50, r 17, p 20 17 cm 20 cm sin R sin 50 17 20 50 17 sin 50 P sin R 20 Since R is cute: R sin 1 17 sin 50 b 20 R 40.6274... R 41 R 7. Two tow trucks, on stright rod, re pulling vehicle from field. The cble from one truck is let out 47 m nd it mkes n ngle of 60 with the rod. The cble from the other truck is let out 50 m. To the nerest metre, how fr prt re the trucks? Sketch digrm. T 60 S Check for the mbiguous cse. The rtio of the side opposite T to the side djcent to T is: 47 m 50 m SV 50, which is >1, so only 1 tringle TV 47 V is possible Determine S first. Then determine TS. sin S v sin V t s sin T t sin T Substitute: T 60, s 47, t 50 Substitute: V 65.5050... T 60, t 50 sin S sin 60 47 50 47 sin 60 sin S 50 S sin 1 47 sin 60 b 50 v 50 sin 65.5050... sin 60 50 sin 65.5050... v sin 60 v 52.5387... Since S is cute: The trucks re pproximtely S 54.4949... 53 m prt. So, V 180 (60 54.4949... ) 65.5050... 50 Chpter 6: Trigonometry Review Solutions DO NOT COPY. P
6.5 8. Determine ech indicted mesure to the nerest tenth of unit. ) 30.0 cm f 2 d 2 e 2 2de cos F E F 74 Substitute: d 30, e 21.3, F 74 21.3 cm f 2 30 2 21.3 2 2(30)(21.3) cos 74 f D f 30 2 21.3 2 2(30)(21.3) cos 74 f 31.6453... f 31.6 cm b) G 18 cm 16 cm H J 15 cm j 2 g 2 h 2 2gh cos J Substitute: j 18, g 15, h 16 18 2 15 2 16 2 2(15)(16) cos J cos J 152 16 2 18 2 2(15)(16) J cos 1 152 16 2 18 2 b 2(15)(16) J 70.9081... U 70.9 9. In RST, R = 64, RS = 8.3 m, nd RT = 9.0 m Solve this tringle. Give the side lengths to the nerest tenth of metre nd the ngle mesures to the nerest degree. Since the given ngle is between the two given sides, only 1 tringle is possible. Sketch digrm. Determine ST first. 8.3 m r 2 s 2 t 2 2st cos R R S 64 Substitute: s 9, t 8.3, R 64 r 2 9 2 8.3 2 2(9)(8.3) cos 64 9.0 m r 9 2 8.3 2 2(9)(8.3) cos 64 r 9.1868... Determine S. T sin S s sin R r Substitute: R 64, s 9, r 9.1868... sin S sin 64 9 9.1868... 9 sin 64 sin S 9.1868... Since S is cute: S sin 1 9 sin 64 9.1868... b S 61.7049... T 180 (64 61.7049... ) 54.2950... So, TS 9.2 m, S 62, T 54 P DO NOT COPY. Chpter 6: Trigonometry Review Solutions 51
10. Two irplnes leve n irport on flight pths tht intersect t n ngle of 50. After one hour, one plne hs trvelled 550 km nd the other hs trvelled 310 km. ) To the nerest kilometre, how fr prt re the plnes? Sketch digrm. Since the given ngle is between the two given sides, only 1 tringle is possible. In ABC, determine BC. 2 b 2 c 2 2bc cos A Substitute: b 310, c 550, A 50 2 310 2 550 2 2(310)(550) cos 50 310 2 550 2 2(310)(550) cos 50 423.5674... The plnes re pproximtely 424 km prt. 550 km 50 A 310 km B C b) To the nerest degree, wht is the ngle between the line joining the plnes nd the course of the fster plne? Determine B. sin B sin A b Substitute: A 50, b 310, 423.5674... sin B sin 50 310 423.5674... 310 sin 50 sin B 423.5674... Since B is cute: B sin 1 310 sin 50 423.5674... b B 34.1008... The ngle between the line joining the plnes nd the course of the fster plne is pproximtely 34. 52 Chpter 6: Trigonometry Review Solutions DO NOT COPY. P