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1 Semester Eam Review The semester A eaminatin fr Hnrs Precalculus cnsists f tw parts. Part 1 is selected respnse n which a calculatr will NOT be allwed. Part is shrt answer n which a calculatr will be allwed. Pages with the smbl indicate that a student shuld be prepared t cmplete items like these with r withut a calculatr. The frmulas belw are prvided in the eaminatin bklet. Trignmetric Identities sin cs 1 sec 1 tan csc 1 ct sin sincs cssin cs cscs sinsin cs cs sin cs 1 1 sin sin sin cs tan tan tan 1 tan tan tan tan 1 tan 1 cs sin 1 cs cs 1 cs sin tan sin 1 cs Triangle Frmulas Law f Sines: sin A sin B sin C Law f Csines: a b c c a b abcsc Area f a Triangle: 1 sin ab C Arc Length s r ( in radians) s r ( in degrees) 360 Linear Speed v r MCPS

2 PART 1 NO CALCULATOR SECTION 1. Sketch the graph f the piece-wise functin f. Lk at the graph f the piece-wise functin belw., if 0 1, if 0 O Which f the fllwing functins is represented b the graph? A B f f,if 0,if 0,if 0,if 0 f C, if 0, if 0 f D,if 0,if 0 MCPS

3 3. Lk at the graph f the piece-wise functin belw. O Which tpe f discntinuit des the graph have at the fllwing -values? a. 3 b. 1 c Sketch the graph f the fllwing piece-wise functin. Be sure t indicate arrws, clsed circle endpints and pen circle endpints. 4, if f 3, if 3 5, if Suppse that lim f 8, and that f 4 4 is undefined. a. Write a limit statement s that there is a remvable discntinuit at 4. b. Write a limit statement s that there is a jump discntinuit at 4. MCPS

4 6. Which f the fllwing is true abut the functin f 4? 3 A The functin is cntinuus fr all real numbers. B The functin is discntinuus at 3nl. C The functin is discntinuus at 4nl. D The functin is discntinuus at 3and Let f c 7,if 5, if 5 What is the value f c that will make f cntinuus at 5? 8. Let g 4 c, if 11 5, if 11 What is the value f c that will make f cntinuus at 11? 9. Lk at the graph f a functin belw. Des the graph represent an dd functin, an even functin, r a functin that is neither dd nr even? MCPS

5 10. Lk at the graph f a functin belw. Des the graph represent an dd functin, an even functin, r a functin that is neither dd nr even? 11. Determine whether each functin belw is dd, even, r neither dd nr even. 3 a. f sin b. g 4 c. rcs 4 d. f cs sin 4 1. If f 3, which f the fllwing statements is NOT true? A The graph f f is smmetric with respect t the -ais. B f is an even functin. C The range f f is all real numbers. D As, f. MCPS

6 13. Lk at the graph f the functin belw. O a. What is the dmain f this functin? b. What is the range f this functin? Fr each functin belw, find a frmula fr f dmain f f 1. and state an restrictins n the a. f 3 b. f c. f 4 4 MCPS

7 15. Lk at the graph f f belw. O a. Sketch the graph f f. b. Sketch the graph f f. 16. True r False. a. The functin g5f represents a vertical stretch f the graph f f b a factr f 5, fllwed b a vertical translatin dwn units. b. The functin graph f f. 1 g f represents a vertical and hrizntal shrinking f the Match the transfrmatins that wuld create the graph f g frm the graph f g 3 f A Stretch the graph f f g f 3 B Stretch the graph f f hrizntall verticall f. 1 g f 3 C Shrink the graph f f hrizntall 1 3 g f D Shrink the graph f f verticall MCPS

8 Fr items 18 and 19, use the graphs f f and g belw. f g 18. Which f the fllwing represents the relatinship between f and g? A gf 3 1 B g f 3 C g f 3 D 1 g f Sketch the graph f f. MCPS

9 0. Lk at the graph f f belw. The dmain f f is 6 5. O a. Describe hw t transfrm the graph f f t g f b. Sketch the graph f g. c. What is the dmain f g? d. What is the range f g? 1 3. MCPS

10 3? 1. Which f the fllwing describes the right-end behavir f the functin f A B C D lim f lim f 0 lim f 3 lim f. Let and be related b the equatin 17. a. Determine the values f if = 0, 1, and 8. b. Is this relatin a functin? Justif ur answer. c. Determine tw functins defined implicitl b this relatin. 3. Write the definitins f the si circular functins f an angle in standard psitin, passing thrugh the pint,, with r. 4. If 4 sin with cs 0, what are the values f ther five trignmetric functins? 5 5. Fr each f the fllwing, state the quadrant in which the terminal side f lies. a. sin 0, tan 0 b. cs 0, tan 0 c. sec 0, csc 0 6. Cnvert t radian measure. Leave ur answer in terms f. a. 40 b. 165 MCPS

11 7. On the unit circle belw the crdinates f pint A are 1, 0 and the crdinates f pint B are 0.8,0.6. Find the value f the fllwing. a. sin B b. cs c. tan A 8. Determine the eact value f the fllwing. a. sin 6 b. 5 cs 4 c. 5 tan 3 d. 3 sin cs f. tan e. g. 7 tan 4 h. 4 cs 3 i. 11 sin 6 j. sin 4 k. 5 cs 6 l. 7 tan 6 m. 5 sec 4 n. 5 ct 6. 4 csc 3 p. sec 9. Sketch the graphs f the si circular functins n the interval. 30. What are the perids f the si circular functins? 31. What is the perid f tan 8? 3. What is the value f b such that csb 33. What is the value f c such that cscc has a perid f 3? has a perid f 10? MCPS

12 34. Cmplete the table fr the inverse circular functins. Dmain 1 Sin Cs Tan 1 1 Range 35. Identif the functins represented b the graphs belw. a. 3 1 O 1 3 b. c MCPS

13 36. Determine the eact (nt a decimal) value f the fllwing. a. Sin 1 1 b. Cs c. Tan 3 d. 1 3 Sin e. Cs f. Tan g. Sin 1 h. Cs 0 1 i. Tan 0 j. cs Sin k. sin Tan 1 l. tan Cs 1 1 m. sin Csc n. 1 1 tan Sin Cs cs Write a sinusidal equatin fr each f the fllwing graphs. a b. c MCPS

14 38. Determine the equatin that best describes a sine curve with amplitude 3, perid f 6, and a phase shift f t the right. 39. Fr each equatin belw, state the amplitude, perid, phase shift, vertical translatin, and an reflectins f the sinusid relative t the basic functin f sin r g cs Sketch the graph, marking the - and -aes apprpriatel. 6 a. h sin 3 5 b. h5cs 1 c. h sin Assume that angle A and angle B are supplementar. Which f the fllwing is equal t cs A? A B C D sin B sin B cs B cs B 41. Simplif each epressin belw as a single functin f a single angle. D nt evaluate. a. sin17 cs17 b. 3 3 cs cs sin sin c. sin 7 cs3 cs 7sin 3 d. tan11 tan 5 1 tan11 tan5 e. 1 sin 9 f. cs8cs5 sin 8sin 5 g. 1 cs 3 h sin cs cs sin MCPS

15 4. If 1 ct A and 0 A, determine the eact value f the fllwing. 5 a. sin A b. A cs A c. sin A d. tan A e. cs 43. Slve the fllwing equatins ver the interval a. sin 0 b. 3cs 4 5cs Slve the fllwing equatins n the interval 0. a. tan 3 0 b. sin 3sin 1 0 c. sin Cmplete the fllwing chart. Radius Angle (radians) Arc length 6 inches feet 6 10 meters 30 meters 46. Hw man triangles are pssible if a40, b80, and A 30? MCPS

16 47. Prve the fllwing identities. a. sin ct cs b. sin cs 1 sin csc c. sin 1 ct d. sin cs sec csc cs sin e. sin sin sin cs f. sin sin tan tan g. tan ct csc h. ct sec sec csc cs ct MCPS

17 PART CALCULATOR SECTION Semester Eam Review A calculatr ma be used n items 48 thrugh 65. Make sure that ur calculatr is in the apprpriate mde (radian r degree) fr each item. Unless therwise specified, answers shuld be crrect t three places after the decimal pint. 48. A wheel f radius 1 cm turns at 7 revlutins per secnd. Determine the linear speed f a pint n the circumference f the wheel in meters per secnd. 49. A mn makes a circular revlutin arund its planet in 80 hurs. The radius f the circular path is 0,000 miles. a. What is the angular speed f the planet in radians per hur? b. What is the linear speed f the planet in feet per secnd? 50. A ball n a string is swinging back and frth frm a ceiling, as shwn in the figure belw. d Let d represent the distance that the center f the ball is frm the wall at time t. Assume that the distance varies sinusidall with time. When t 0 secnds, the ball is farthest frm the wall, d 160 cm. When t 3 secnds, the ball is clsest frm the wall, d 0 cm. When t 6 secnds, the ball is farthest frm the wall, d 160 cm. a. Sketch a graph f the distance as a functin f time. b. Write a trignmetric equatin fr the distance as a functin f time. c. What is the distance f the ball frm the wall at t 5 secnds? d. What is the value f t the first time the ball is 40 cm frm the wall? Yur answer shuld be crrect t three places after the decimal pint. MCPS

18 Semester Eam Review 51. Sara is riding a Ferris wheel. Her sister Kari starts a stpwatch and recrds sme data. Let h represent Sara s height abve the grund at time t. Kari ntices that Sara is at the highest pint, 80 feet abve the grund, when t 3 secnds. When t 7 secnds, Sara is at the lwest pint, 0 feet abve the grund. Assume that the height varies sinusidall with time. a. Write a trignmetric equatin fr the height f Sara abve the grund as a functin f time. b. What will Sara s height be at t 11.5 secnds? Yur answer shuld be crrect t three places after the decimal pint. c. Determine the first tw times, t 0, when Sara s height is 70 feet. Yur answer shuld be crrect t three places after the decimal pint. 5. At Ocean Tide Dck, the first lw tide f the da ccurs at midnight, when the depth f the water is meters, and the first high tide ccurs at 6:30 a.m., with a depth f 8 meters. Assume that the depth f the water varies sinusidall with time. a. Sketch and label a graph shwing the depth f the water as a functin f the number f hurs after midnight. b. Determine a trignmetric functin that mdels the graph. c. Suppse a ship requires at least three meters f water depth is planning t dck after midnight. Determine the earliest pssible time that the ship can dck. 53. Slve the fllwing equatins fr, where a. 3cs 9 7 b. 3sin 7sin Hw man triangles ABC are pssible if A0, b40, and a 10? 55. Given ABC, where A41, B58, and c 19.7cm, determine the measure f side b. Yur answer shuld be crrect t three places after the decimal pint. 56. In ABC, a 9, b 1, c 16. What is the measure f B? Yur answer shuld be crrect t the nearest tenth f a degree. 57. Determine the remaining sides f a triangle with A58, a11.4, b 1.8. Yur answers (side and angles) shuld be crrect t the nearest tenth. MCPS

19 Semester Eam Review 58. Frm a pint 00 feet frm its base, the angle f elevatin frm the grund t the tp f a lighthuse is 55 degrees. Hw tall is the lighthuse? Yur answer shuld be crrect t three places after the decimal pint. 59. A truck is travelling dwn a muntain. A sign sas that the degree f incline is 7 degrees. After the truck has travelled 1 mile (580 feet), hw man feet in elevatin has the truck fallen? Yur answer shuld be crrect t three places after the decimal pint. 60. The wner f a garage shwn belw plans t install a trim alng the rf. The lengths required are in bld. Hw man feet f trim shuld be purchased? Yur answer shuld be crrect t three places after the decimal pint feet An airplane needs t take a detur arund a grup f thunderstrms, as shwn in the figure belw. Hw much farther des the plane have t travel due t the detur? Yur answer shuld be crrect t three places after the decimal pint miles 6. Determine the area f triangle ABC if a4, b10, and mc 30. MCPS

20 Semester Eam Review 63. A real estate appraiser wishes t find the value f the lt belw. 160 feet 6 50 feet a. Hw lng is the third side f the lt? Yur answer shuld be crrect t three decimal places. b. Find the area f the lt. Yur answer shuld be crrect t three places after the decimal pint. c. An acre is 43,560 square feet. If land is valued at $56,000 per acre, what is the value f the lt? Yur answer shuld be crrect t the nearest cent. 64. Find the area f the quadrilateral belw. Yur answer shuld be crrect t three places after the decimal pint Triangle ABC has an area f,400 with AB 80, AC 100. Determine the tw pssible measures f angle A. Yur answers shuld be crrect t the nearest tenth f a degree. MCPS

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