Review for the final exam (Math 127)

Transcription

1 . Evaluate 3 tan tan 4 3 (b) (c) cs cs sec cs 4 4 (d) cs tan 3 Review fr the final eam (Math 7). If sec, and 7 36, find cs, sin, tan, ct, csc tan (b) If, evaluate cs, sin 7 36 (c) Write the csc in terms f ct fr in the Quadrant III. (d) Find the values f all the trignmetric functins f t frm the given infrmatin: tan t = -3/4; cs t < 3. Determine whether the functin is even, dd, r neither f sin cs (b) f cs (c ) f sin cs 4. Find the eact value f the epressin tan cs 4 cs cs ; (b) sec sin ; (c) sin sin ; (d) sin sin 3 (e) cs cs ; (f) tan tan ; (g) 5 sin sin Epress cs tan and sin tan as an algebraic epressin in 4 4 (b ) epress sintan tan y in terms f and y nly (c) Write the epressin sin cs as an algebraic epressin in (d) The dmain and range fr the functin cs are : ;

2 The dmain and range fr the functin sin are : The dmain and range fr the functin tan are : The dmain and range fr the functin ct are : (e) Find the eact value f each epressin sincs tan 6. Find the length f the arc that subtends an angle f measure 6 n a circle f diameter 8cm; (b) Find the area f a sectr in part 7. (a ) The area f the triangle ABC is A 5 3. If BAC 3 find AC BC? (b) Given triangle ABC with sides length a=6, b=6, c=4. Find the area f the triangle. (c) Find the area f the shaded regin in the figure. (d) An issceles triangle has an area f cm, and the angle between tw equal sides is 5. What is the length f the tw equal sides? 6 8. Simplify (b) Simplify (c ) Simplify (d) Simplify (e) Simplify (f) Prve ct csc sin sin sin cs cs 3 sin t tan t cs t ct tan ct csc csc ct cs cs 4ct csc cs cs

3 tan cs (g) Prve sec cs sin cs sin (h) Prve tan ct (i) Prve tan ct sin 9. Evaluate sin75 ; (b) cs75 ; (c) cs ; (d) sin ; (e) sin(.5 ) ; (f) cs(67.5 ) 5 5. Given that sin, is in Quadrant I, and cs y, y is in the Quadrant III, calculate 3 5 y y y cs / cs y ; (b) cs ; (c ) sin ; (d) sin ; (e ) cs ; (f). Given that csc 3, 9 8, find sin, cs, tan. Find the value cs 55 cs 55 sin 55 cs95 cs95 sin95 sin 55 sin95 3. Slve the given equatin sec sin (b) 3 tan3 (c) cs cs,, (d) 4sin 4cs,, (e) cs 5sin 4,,

4 4. Find the sum f the series: (b) k k k (c ) k 5. The 8 th term f arithmetic sequence is 6, and the 8 th term is 56. Find the sum f the first terms. Find a +a + +a. 6. A persn gets a jb with a salary f $, a year. He prmised a $45 raise each subsequent year. Find his ttal earning fr -year perid. 7. Write z cs isin in rectangular frm. 6 6 (b) Write z 3 i and z i in plar frm, and then find the prduct z z, and the z qutients and z z. 8. Find the indicated pwer using De Mivre s Therem i 9. Find the square rts f 4 3 4i. Graph rts in the cmple plane.. Find the angle between vectrs (b) Let be an angle between vectrs w 3 i 3 j and 3, 3 v w i 3 j and v 5,. What is cs? What issin?. A cnstant frce 3, 3, t the pint 3,4. Find the distance D. Find the wrk dne if the distance is measured in feet, and F mves an bject alng a straight line frm the pint the frce is measured in punds. Find the csine f an angle between the frce F and the distance D.. Frm a pint C n the grund level, the angle f elevatin t the tp f a tree is 3. Frm a pint D, which is clser t the tree, the angle f elevatin is measured t be 45. Find the distance between pints C and D if the height f the tree is 4 m. 3. A - ft ladder leans against a building. The angle between the grund and the ladder is 6. Hw high des the ladder reach n the building?

5 (b) Frm the tp f a -ft lighthuse, the angle f depressin t a ship in the cean is 3. Hw far is the ship frm the base f the lighthuse? (c) 6-ft guy wire is attached t the tp f a cmmunicatins twer. If the wire makes an angle f 6 with the grund, hw tall is the cmmunicatins twer? (d) An airplane is flying at an elevatin f 5 ft. directly abve a straight highway. Tw mtrists are driving cars n the highway n ppsite sides f the plane and the angle f depressin t ne car is 3 and t the ther car is 6. Hw far apart are the tw cars? (e) An airplane is flying at an elevatin f 9 feet, directly abve a straight highway. Tw mtrists are driving cars n the highway, bth n ne side f the plane. If the angle f depressin t ne car is 6 and t the ther is 45, hw far apart are the cars? 4. The law f csines: Tw bats leave the same prt at the same time. One travels at a speed 4 mi/h in the directin N 48 E and the ther travels at a speed f mi/h in a directin S E. Hw far apart are the tw bats after 3/ hur? (b) Tw bats leave the same prt at the same time. One travels at a speed mi/h in the directin S W and the ther travels at a speed f mi/h in a directin S 7 W. Hw far apart are the tw bats after ne hur? (c) A pilt flies in a straight path fr hurs. She then makes a curse crrectin, heading 45 t the right f her riginal curse, and flies hur in the new directin. If she maintains a cnstant speed f mile per hur, hw far is she frm her starting psitin? (d) Tw straight rads diverge at an angle 45. Tw cars leave the intersectin at 4: P.M., ne travelling at 4 mi/h, and the ther at 6 mi/h. Hw far apart are the cars at 4:3 P.M.? (e) Use either the Law f sines r the Law f csines t determine pssibilities fr angle b 3, a, c 3. (f) Triangle ABC has a=7, b=8 and c=. Slve fr cs A. 5. The law f sines: A satellite rbiting the Earth passes directly verhead at bservatin statins in Pheni and Ls Angeles, 34 mi apart. At an instant when the satellite is between these tw statins, its angle f elevatin bserved at Ls Angeles is 6. If the distance frm the satellite t Pheni is knwn t be 7 6 miles, find the angle f elevatin bserved at Pheni. (b) T find the distance acrss a river, a surveyr chses pints A and B, which are 8 ft apart n ne side f the river. She then chses a reference pint C n the ppsite side f the river and finds that BAC 6 and ABC 75. Find the distance frm B t C? (c ) Assume yu have a triangle ABC with a=5, b=5, and angle ppsite the side a, is A=3. Find B (the angle ppsite the side b). Hw many slutins d yu have?

6 (d) A fire is sighted frm statins A and B. The bearing f the fire frm statin A is S 6 E, and the bearing frm the statin B is S 3 W. Statin B is 4 miles due east f statin A. Hw far is the fire frm statin A? (e) Use the Law f sines t slve fr the missing values f the triangle which satisfy the given cnditins belw. (Determine if there are,, r triangles): a 5 3, b 5, B 3. (f) Use the Law f sines t slve fr the missing values f the triangle which satisfy the given cnditins belw. (Determine if there are,, r triangles): a, b 4, B I. A straight river flws east at a speed f mi/h. A bater starts at the suth shre f the river and heads in the directin which is 6 t the suth shre. The mtrbat has a speed mi/h relative t the water. Epress the velcity f the river as a vectr in cmpnent frm; (b) Epress the velcity f the mtrbat relative t the water as a vectr in cmpnent frm; (c) Find the true velcity f the mtrbat; (d) Find the true speed and directin f the mtrbat. II. A migrating salmn heads in the directin N45 E, swimming at 5 mi/h relative t the water. The prevailing cean currents flw due east at 3 mi/h. Find the true velcity f the fish as a vectr. III. A straight river flws east at a speed f mph. A bater starts at the suth shre f the river and heads in a directin 6 frm the shre (see figure in bk). The mtrbat has a speed f mph relative t the water. Epress the velcity f the river as a vectr in cmpnent frm. (b) Epress the velcity f the mtrbat relative t the water as a vectr in cmpnent frm. (c) Find the true velcity f the mtrbat (d) Find the true speed and directin f the mtrbat. 7. Find the amplitude, perid, and phase shift f the functin, and graph ne cmplete perid (c ) (e) y 5cs3 ; (b) y sin y 3cs 3 ; (d) y 3sin 3 y 4cs ; (f) y 4sin Find the perid and phase shift, identify the asympttes and graph

7 y 5sec 3 ; (b) y tan 3 (c) y 5csc 3 ; (d) y ct ; 3 (e) y sec ; (f) y tan 9. Sketch a graph f the plar equatin (d) r cs ; (b) r sin ; (c) r cs3 r cs ; (e) r sin 3. A pair f parametric equatins is given. Find a rectangular-crdinate equatin fr the curve by eliminating the parameter. Sketch the graph. sec t, y tan t, t (b) cs t, y cst, (c) t, y t, t 4 (d) 3cs t, y 3sin t, t (e) 3sin t, y 3cs t, t 3. Cmplete the square t determine whether the equatin represents an ellipse, a parabla, a hyperbla r a degenerate cnic. Fr ellipse: find center, fci, vertices, and length f the majr and minr aes; Fr parabla: find the verte, fcus, and directri; Fr hyperbla: find the center, fci, vertices, and asympttes. Sketch the graph. 6 y 9 (b) y y (c ) 6 9y (d) 6 4y 3. Find the radius and the center f the circle given by 8 y y 5