Trignmetry, 8th ed; Lial, Hrnsby, Schneider Trignmetry Final Exam Review: Chapters 7, 8, 9 Nte: A prtin f Exam will cver Chapters 1 6, s be sure yu rewrk prblems frm the first and secnd exams and frm the Exam 1 and Exam Reviews. Directins: Wrk these prblems with n resurces ther than the departmental frmula sheet and a graphing calculatr. Yur bk, ntebk, hmewrk, slutins manual, etc. shuld be clsed. Read and carefully fllw all directins. Shw all f yur wrk n all prblems. Anytime yu are asked t perfrm a calculatin manually r t give an exact answer, yu may nt use a calculatr. Chapter 7: Applicatins f Trignmetry and Vectrs 1. Determine the remaining sides and angles f the triangle ABC. Shw all wrk and / r supprt yur answer.. Determine the remaining sides and angles f the triangle ABC. Shw all wrk and / r supprt yur answer. C = 71.8, B = 4.57, a =.614. A ship is sailing due nrth. At a certain pint the bearing f a lighthuse 1.5 km distant is N 8.8 E. Later n, the captain ntices that the bearing f the lighthuse has becme S 44. E. Hw far did the ship travel between the tw bservatins f the lighthuse? Shw all wrk and / r supprt yur answer. 4. Mark ntices that the bearing f a tree n the ppsite bank f a river flwing nrth is 115.45. Lisa is n the same bank as Mark, but 48. m away. She ntices that the bearing f the tree is 45.47. The tw banks are parallel. What is the distance acrss the river? Shw all wrk and / r supprt yur answer. 5. Find the area f the triangle ABC. Shw all wrk and / r supprt yur answer. A = 0.50, b = 1.00 cm, C = 11.60 6. A real estate agent wants t find the area f a triangular lt. A surveyr takes measurements and finds that tw sides are 5.1 m and 1. m, and the angle between them is 4.. What is the area f the triangular lt? Shw all wrk and / r supprt yur answer. 7. Determine the number f triangles ABC pssible with the given parts. Shw all wrk and / r supprt yur answer. a) b = 60, a = 8, B = 100 b) a = 5, b = 0, A = 40 c) B = 54, c = 8, b = 8. Find the unknwn angles in triangle ABC, if the triangle exists. Shw all wrk and / r supprt yur answer. a) C = 41 0', b = 5.9 m, c = 8.4 m b) B = 48., a = 890 cm, b = 697 cm c) B = 74., a = 859 m, b = 78 m 9. Slve the triangle ABC, if pssible. Shw all wrk and / r supprt yur answer. C = 5., a =.5 yd, c = 59.8 yd 1
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 10. Slve the triangle ABC, if pssible. Shw all wrk and / r supprt yur answer. A = 8 40', a = 9.7 km, b = 11.8 km 11. Slve each triangle and then find the area f the triangle. Apprximate values t the nearest tenth. Shw all wrk and / r supprt yur answer. a) b) 1. Slve each triangle and then find the area f the triangle. Shw all wrk and / r supprt yur answer. a) C = 8., b = 5.71 in., a = 4.1 in. b) a = 189 yd, b = 14 yd, c = 5 yd 1. The sides f a parallelgram are 4.0 cm and 6.0 cm. One angle is 58 while anther is 1. Find the lengths f the diagnals f the parallelgram. Shw all wrk and / r supprt yur answer. 14. A weight is supprted by cables attached t bth ends f a balance beam, as shwn in the figure. What angles are frmed between the beam and the cables? Shw all wrk and / r supprt yur answer. 15. Using the vectrs in the figure belw, draw a sketch t represent: a) c + d b) d c c) c 16. Use the vectrs in the adjacent figure t find a) a + b, b) a b, c) a. Shw all wrk and / r supprt yur answer. 17. Fr vectrs u and w with angle θ between them, sketch the resultant, u + w. u = 8, w = 1, θ = 0 18. Find the magnitude and directin angle fr vectr v = 15, -8. Shw all wrk and / r supprt yur answer. 19. Vectr v has magnitude v = 15. 4 and directin angle α = 7 0. Find the hrizntal (a) and vertical (b) cmpnents f v. Shw all wrk and / r supprt yur answer.
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 0. Write vectr v in cmpnent frm, a, b. Shw all wrk and / r supprt yur answer. 1. Tw frces act at a pint in the plane. One is 19 newtns, the ther is newtns. The angle between the frces is 118. Find the magnitude f the resultant frce. Shw all wrk and / r supprt yur answer.. Use the parallelgram rule t find the magnitude f the resultant frce fr the tw frces shwn in the adjacent figure. Rund yur answer t the nearest tenth. Shw all wrk and / r supprt yur answer.. Given u =, 5 and v = 4,, find u + 4v. Write yur answer in the frm ai + bj. Shw all wrk and / r supprt yur answer. 4. Find the dt prduct f, - and 6, 5. Shw all wrk and / r supprt yur answer. 5. Find the angle between the vectrs -5i + 1j and i + j. Shw all wrk and / r supprt yur answer. 6. Given u =, 1, v =, 4, and w = 5, 1, evaluate u ( v w). Shw all wrk and / r supprt yur answer. 7. Determine whether the vectrs 1, and 6, are rthgnal. Shw all wrk and / r supprt yur answer. 8. Tw tugbats are pulling a disabled speedbat int prt with frces 140 lb and 1480 lb. The angle between these frces is 8.. Find the magnitude and directin f the equilibrant. Shw all wrk and / r supprt yur answer. 9. Tw frces f 18 lb and 5 lb act at a pint. The equilibrant frce is 0 lb. Find the angle between the frces. Shw all wrk and / r supprt yur answer. 0. A frce f 176 lb makes an angle f 78 50' with a secnd frce. The resultant f the tw frces makes an angle f 41 10' with the first frce. Find the magnitudes f the secnd frce and the resultant. Shw all wrk and / r supprt yur answer. 1. A 10-lb frce keeps a 600-lb bject frm sliding dwn an inclined ramp. Find the angle f the ramp. Shw all wrk and / r supprt yur answer.. Find the magnitude f frce necessary t keep a 500-pund car frm sliding dwn a ramp f 5. Shw all wrk and / r supprt yur answer.. Tw peple are carrying a bx. One persn exerts a frce f 150 lb at an angle f 6.4 with the hrizntal. The ther persn exerts a frce f 114 lb at an angle f 54.9. Find the weight f the bx. Shw all wrk and / r supprt yur answer.
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 4. A ship leaves prt n a bearing f 4.0 and travels 10.4 miles. The ship then turns due east and travels 4.6 miles. Hw far is the ship frm prt, and what is its bearing frm prt? Shw all wrk and / r supprt yur answer. 5. A pilt is flying at 168 mph. She wants her flight path t be n a bearing f 57 40'. A wind is blwing frm the suth at 7.1 mph. Find the bearing the pilt shuld fly, and find the plane s grundspeed. Shw all wrk and / r supprt yur answer. 6. A pilt wants t fly n a bearing f 74.9. By flying due east, he finds a 4-mph wind, blwing frm the suth, puts him n curse. Find the grundspeed and airspeed. Shw all wrk and / r supprt yur answer. Chapter 8: Cmplex Numbers, Plar Equatins, and Parametric Equatins 7. Write 88 as the prduct f a real number and i. Shw all wrk and / r supprt yur answer. 8. Slve x + = 4x and express all nnreal cmplex slutins in terms f i. Shw all wrk and / r supprt yur answer. 9. Perfrm the indicated peratins and then simplify. Shw all wrk and / r supprt yur answer. a) 10 40 b) 1 6 8 40. Perfrm the indicated peratin and then simplify. Write yur answers in standard frm. Shw all wrk and / r supprt yur answer. a) ( 4 i) ( + i) + ( 4 + 5i) b) ( + i)( 4 i) c) + i 1 5i 41. Simplify each pwer f i. Shw all wrk and / r supprt yur answer. a) 5 i b) 1 i 4. Find the sum f 5 6i and + i by graphing each number in the cmplex plane and finding their resultant. Shw all wrk and / r supprt yur answer. 4. Write the cmplex number [cs 150 + i sin 150 ] in rectangular frm. Shw all wrk and / r supprt yur answer. 44. Write the cmplex number 5 5i in trignmetric frm r(cs θ + i sin θ), with θ in the interval [0, 60 ). Shw all wrk and / r supprt yur answer. 45. Find each prduct and write it in rectangular frm. On part b, rund t fur decimal places. Shw all wrk and / r supprt yur answer. a) [ 8( cs 00 i sin 00 )] [ 5( cs10 + i sin 10 )] + b) (1 cis 18.5 ) ( cis 1.5 ) 4
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 4( cs10 + i sin 10 ) 46. Find the qutient ( cs150 + i sin 150 ) & write it in rectangular frm. Shw all wrk and / r supprt yur answer. i 47. Cnvert the numeratr and denminatr f t trignmetric frm. Then find the qutient and write it in 1 i rectangular frm. Shw all wrk and / r supprt yur answer. 48. Find each pwer. Write each answer in rectangular frm. Shw all wrk and / r supprt yur answer. a) [ 100 ] cis b) ( i) 5 49. Find all cube rts f each cmplex number. Leave answers in trignmetric frm. Shw all wrk and / r supprt yur answer. a) 7 cis 00 b) i 50. Find all cmplex slutins f each equatin. Leave answers in trignmetric frm. Shw all wrk and / r supprt yur answer. a) x 4 i = 0 b) x 4 + 16 = 0 51. Fr each pair f plar pints, plt the pint; give tw ther pairs f plar crdinates (including ne with a negative r); and give the rectangular crdinates fr the pint. Shw all wrk and / r supprt yur answer. 5π b), a) (, 15 ) 5. Given the rectangular crdinates 1,, give tw pairs f plar crdinates fr the pint, including ne with a negative r. Select θ s it is in the interval [0, 60 ). Shw all wrk and / r supprt yur answer. 5. Fr each rectangular equatin, give its equivalent plar equatin and sketch its graph. Shw all wrk and / r supprt yur answer. a) x y = 6 b) x + y = 9 5
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 54. Match each equatin with its crrespnding graph. a. r = a cs θ A. b. r = a sin θ B. a c. r = a ± b sin θ with < 1 b C. a d. r = a ± b cs θ with = 1 b D. e. r = a cs 4θ E. f. r = a sin 5θ F. g. r = c acs θ b sin θ G. h. r = 4 sin θ H. 6
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 55. Identify the type f plar equatin and draw a cmplete graph. Include a t-chart with at least 5 key pints with yur graph. Shw all wrk and / r supprt yur answer. a) r = + cs θ b) r = 4 cs θ 56. Fr each plar equatin, give its equivalent rectangular equatin and sketch its graph. Include a t-chart with at least 5 key pints with yur graph. Shw all wrk and / r supprt yur answer. a) r = sin θ b) r = 1 sinθ Chapter 9: Expnential and Lgarithmic Functins 57. Graph the fllwing expnential functin. Give the equatin f the hrizntal asymptte, a 5-pint t-chart with the key pint at the center, and an accurate graph with ne ntch n each axis labeled. Shw all f yur wrk. ( x 4 y = ) + 1 58. Slve the fllwing expnential equatins algebraically. Shw all yur wrk and give an exact answer. x 9 x + 4 x 6 a. = b. = 8 c. x 4 = 15 4 59. Find the balance A fr P dllars invested at rate r fr t years and cmpunded n times. a. P = $000, r = 6.5%, t = 10 years, cmpunded quarterly. b. P = $000, r = 7%, t = 8 years, cmpunded cntinuusly. 60. Find the required annual interest rate t the nearest tenth f a percent fr $48,000 t grw t $78,186.94 when cmpunded semiannually fr 5 yr. Rund yur decimal answer t fur decimal places and cnvert t a percent. Shw all f yur wrk. 61. Slve the fllwing lgarithmic equatins algebraically. Shw all yur wrk and give an exact answer. 1 a. lg x = b. lg x = 5 c. lg 4 = x 16 16 6. Graph the fllwing lgarithmic functin. Give the equatin f the vertical asymptte, the expnential frm f the equatin, a 5-pint t-chart with the key pint at the center, and an accurate graph with ne ntch n each axis labeled. Shw all f yur wrk. y = ( x + ) 4 lg 6. Use the prperties f lgarithms t expand r cndense the fllwing lg expressins. Shw all f yur wrk. a. y x 1 lg b b. lg lg y lg z 4 a x a a z 14 64. Find the ph fr lye sap which has a hydrnium in cncentratin f. 10 mles per liter. + 65. Find the [ O ] H fr drinking water which has a ph f 6.5. 7
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 66. Use the change f base therem t find an apprximatin fr lg 1 15. Rund yur answer t fur decimal places. 67. Given f(x) = x, evaluate the fllwing. Shw all f yur wrk. a. f (lg 7) b. f [lg (ln )] c. f [lg ( ln )] 68. a. Find the decibel rating f a sund having an intensity f 100,000 I 0. b. If the intensity f the sund is dubled, by hw much is the decibel rating increased? 69. a. Find the Richter scale rating fr an earthquake having an amplitude f 1,000,000 I 0. b. Express the magnitude f a 6.7 earthquake (n the Richter scale) in terms f I 0. 70. Slve. x 1 a. 4 = 6 b. x+ 5 x = 4x+ c. 6e 1 = 11 4t.06 d. 1000 1 + = 5000 4 71. a. lg (x 1) + lg 10x = lg 10 b. ln x + ln (x 1) = c. 4x lg ( x ) lg 1 lg6 6 = 6 d. lg (11x + 9) = + lg (x + ) 7. Elena McDuff wants t buy an $8,000 painting. She has saved $6,000. Find the number f years (t the nearest tenth) it will take fr her savings t grw t $8,000 at 5.7% cmpunded mnthly. 7. At what interest rate will $,500 grw t $4,671.04 if invested fr 10 years and interest is cmpunded quarterly? Answers Chapter 7: Applicatins f Trignmetry and Vectrs 1. A = 7., a = 178 m, c = 44 m. A = 65.60, b = 1.94 cm, c =.77 cm. 7.8 km 4. 9.4 m 5. 65.94 cm 6. 7 m 7. a) 0; b) 1; c) 8. a) B 6 0', A 11 10'; b) A 1 7., C 1 59.6, A 107.8, C 4.0 ; c) n such triangle 9. A 5.5, B 10., b 7.9 yd 10. B 1 49 0', C 1 9 00', c 1 15.5 km, B 10 40', C 10 40', c.88 km 8
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 11. a) A., B 108., C 49.5 ; area = 1 15. units b) A 56.7, C 68., b 88.; area = 1547079 680.6 units 1. a) c.8 in., A 44.9, B 106.8 ; area = b) C 107 0', B 9 00', A 40'; area = 1. 5. cm, 8.8 cm 14. 6.4 and 6..5688517 5.7 units 7645000 1904.0 units 15. a) b) c) 16. a) 0, 10 b) 8, c) 4, - 6 17. 18. v = 17 ; θ = 1.9 19. a 1.7, b 7.11 0. v = 4, 4 1. 9 newtns. 4.4 lb. 0i + j 4. 5. 78.9 6. 4 7. rthgnal 8. 640 lb at an angle f 167. with the 1480-lb frce 9. 70.1 0. magnitude f secnd frce: 190 lb; magnitude f the resultant: 8 lb 1. 1.5. 05 lbs. 6 lb 4. 1.5 mi; 50.4 5. bearing: 65 0'; grundspeed: 181 mph 6. grundspeed: 161 mph; airspeed: 156 mph 9
Trignmetry, 8th ed; Lial, Hrnsby, Schneider Chapter 8: Cmplex Numbers, Plar Equatins, and Parametric Equatins 7. 1 i 8. x = ± i 1 9. a) i b) 1 1 40. a) 10 + i b) +16i c) + i 41. a) i b) i 4. i ; 4. + i 44. 5 ( cs5 + i sin5 ) 45. a) 0 + 0i b) 0.8580 + 18.5414 i 46. i 1 47. i 7 7 48. a) i b) 18 + 18i 49. a) cis 100, cis 0, cis 40 b) ( cs 110 + i sin 110 ), ( cs 0 i sin 0 ) ( cs 50 + i sin 50 ) +, 50. a) cs.5 + i sin.5, cs 11.5 + i sin 11.5, cs 0.5 + i sin 0.5, cs 9.5 + i sin 9.5 b) cis 45, cis 15, cis 5, cis 15 51. a) (, 495 ), (, 15 ), (, ) π 4π b),,,,, 5. (1, 10 ), (-1, 0 ) 5. a) 6 r = b) r = r r = csθ sinθ 54. a. F b. A c. D d. G e. H f. B g. C h. E 10
Trignmetry, 8th ed; Lial, Hrnsby, Schneider 55a. 55b. 56a. 56b. 57. HA: y = 1 key pint : (4, 0) x y 4 1 4 0 5-1 6-58. a. x = b. x = 11 c. x = 65 59. a. $811.1 b. $55.0 60. 10% 61. a. x = 4 b. x = 4 c. 1 x = 11
Trignmetry, 8th ed; Lial, Hrnsby, Schneider y + 4 6. VA: x = ; expnential frm f the equatin: x = x y 17-6 9 5-5 -1-4 1-7 - 6. a. 1 lg b y + lg b x 4 lg bz b. 64. 1.5 x lg a yz 65. 7. 10 66..9069 67. a. 7 b. ln 1.0986 c. ln = ln = ln 9.197 68. a. 50 b. abut decibels 69. a. 6 b. abut 5,000,000 I 0 70. a..764 b. 8.5476 c..5767 d. 7.046 71. a. 1 b..69 c. 4.5 d. Ø 7. 5.1 years 7. 6.% 1