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1 REVIEW, pges 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 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 tn U y x 4, or c) Wht is the vlue of u to the nerest degree? tn U 0.4 U tn 1 (0.4) U U is pproximtely 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 x 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 b U U is pproximtely 42. Chpter 6: Trigonometry Review Solutions 47
2 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: To the nerest degree, which vlues of u stisfy the eqution 3 sin u = 8 for 0 u 360? The reference ngle is: sin b sin U is negtive in Qudrnts 3 nd 4 so: U nd U 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 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
3 ) 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 AB 18.7 sin Since sin 70 < <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 C 1 sin C 1 c sin B b Substitute: B 70, So, A 180 ( ) c 18.7, b sin C 1 sin sin 70 sin C C 1 sin sin 70 b 17.9 C So, A 180 ( ) sin A b sin B Substitute: A B 70, b sin sin sin sin sin A b sin B Substitute: A B 70, b sin sin sin sin So, BC 9.8 cm or 3.0 cm P DO NOT COPY. Chpter 6: Trigonometry Review Solutions 49
4 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 cm 20 cm sin R sin sin 50 P sin R 20 Since R is cute: R sin 1 17 sin 50 b 20 R 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 T 60, t 50 sin S sin sin 60 sin S 50 S sin 1 47 sin 60 b 50 v 50 sin sin sin v sin 60 v Since S is cute: The trucks re pproximtely S m prt. So, V 180 ( ) Chpter 6: Trigonometry Review Solutions DO NOT COPY. P
5 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 cm f (30)(21.3) cos 74 f D f (30)(21.3) cos 74 f 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 (15)(16) cos J cos J (15)(16) J cos b 2(15)(16) J U 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 (9)(8.3) cos m r (9)(8.3) cos 64 r Determine S. T sin S s sin R r Substitute: R 64, s 9, r sin S sin sin 64 sin S Since S is cute: S sin 1 9 sin b S T 180 ( ) So, TS 9.2 m, S 62, T 54 P DO NOT COPY. Chpter 6: Trigonometry Review Solutions 51
6 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 (310)(550) cos (310)(550) cos 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, sin B sin sin 50 sin B Since B is cute: B sin sin b B The ngle between the line joining the plnes nd the course of the fster plne is pproximtely Chpter 6: Trigonometry Review Solutions DO NOT COPY. P
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