1. sin 2. csc 2 3. tan 1 2. Cos 8) Sin 10. sec. Honors Pre-Calculus Final Exam Review 2 nd semester. TRIGONOMETRY Solve for 0 2

Transcription

1 Honors Pre-Calculus Final Eam Review nd semester Name: TRIGONOMETRY Solve for 0 without using a calculator: 1 1. sin. csc 3. tan 1 4. cos 1) ) 3) 4) Solve for in degrees giving all solutions. 5. sin 1 6. cos 3 7. tan undefined 5) 6) 7) Give the eact value of each epression Tan 3 9. cot Sin 10. sec Cos 8) 5 9) 10) Given y sin, find 3 a. amplitude 11a.) b. period 11b.) c. graph at least two periods of the function

2 1. Given: y 4 cos3, find 1a) a. Amplitude b. Period 1b) c. Phase shift d. Vertical shift e. graph at least two periods of the function 1c) 1d) 13. Simplify: 1 sin 1 csc sin 13) sec csc 14. Prove: csc 1 tan 15. Solve: cos 1 sin for if 0. 15) 16. Solve for : cos 1 0, 0. 16) 17. Solve a right triangle ABC, if a = 6 and 0 A 30. (Give eact lengths.) 17) B b = c = 18. A boy flying a kite is standing 0 ft from a point directly under the kite. 18) If the string to the kite is 40 ft long, find the angle of elevation of the kite. 19. An airplane is at an elevation of 30,000 ft and approaches the airport 19) with an angle of descent of 5. What is the distance between the airport and the point on the ground directly below the airplane?

3 0. Given ABC with a = 30, b = 0, c = 40, find the largest angle. 0) 1. The captain of a clipper ship spots two other ships on the ocean. 1) One ship is 5 miles away while the other is 5. miles away. The angle between the two sightings is 0º. How far apart are the two observed ships? (to three decimals). In RST, 0 R 137,t = 15, and s = 1. Find r to the nearest integer. ) 3. If ABC has b = 30, C = 40 0 and A = 60 0, find a to the nearest tenth. 3) 4. Two angles of a triangle measure 9º and 51º. The longest side is 55cm. 4) Find the length of the shortest side to the nearest tenth. 5. Solve for ABC if a = 15, c = 18, and A= 3º. 5) C = B = b = OR (if two triangles) C = B = b = 6. Find the area of ABC if b = 3, c = 7, and A = 108º. 6) 7. The area of PQR is 15. If p = 5 and q= 10, find all possible measures of 7) R. 3

4 8. Find the area of a regular pentagon inscribed in a circle with 8) radius The sides of an isosceles triangle have lengths 7, 10 and 10. What are the measures of its angles? 9.) 30. At a distance of 00 meters, the angle of elevation to the top of a building is 70. Approimately how tall is the building? 30) 31. Two ships leave a port on courses that differ by 70 and each travels at 5 knots. In terms of nautical miles, how far apart are the ships after 1 hour? 31) 3. After leaving an airport, a plane flies for 1.75 hours at a speed of 00 k/h on a course of 100. The plane then flies for hours at a speed of 50 k/h on a course of 40. At this time, how far from the airport is the plane? 3.) 33. Find the eact value of the following: a. cos 75 b. sin a.) 34. Simplify the following: 33b.) a. cos 75cos15 sin75sin 15 b. sin( 30 ) sin(30 ) 34a.) 34b.) 35. Suppose angle A is acute and cos A Find: a. sin A b. cos A c. sin A 35a.) 35b. 35c.) 36. Simplify the following: a. 1 cos sin b. 1 cot cos 1 c. tan sec 1 36a.) 36b.) 4 36c.)

5 37. Evaluate the given epression: 5 1 sin 37.) Prove: 1 tan 1 cos 39. Solve the following for 0. a. cos sin b. sin 3cos 3 a.) b.) c. sin tan sin c.) POLAR COORDINATES AND COMPLEX NUMBERS 40. Convert the following into rectangular form. a., 90 6 b., 40a.) Convert the following into polar form: 40b.) a. (3,3) b. (-1, 3 ) c. (0, -) 41a.) 41b.) 41c.) 4. If z1 3 i and z 4 4i, find z 1, z, and z1z in polar form. 4.) z 1 z z 1z = 43. If z = 4cis 30, find the following in a bi form: 43a) a. z 3 b. z 5 c. z - 43b) 44. Find the cube roots of 3 i. 44.) 43c) 5

6 SEQUENCES AND SERIES 45. State whether each sequence is arithmetic, geometric or neither and find a formula for a n. a. 17, 1, 7,, b. 3, 8, 15, 4, 35, c. -81, 7, -9, 3, 45a.) 45b.) 45c.) 46 a. Find the 17 th term of 3, 6, 9, 46a.) b. Find the sum of the first 17 terms of 3, 6, 9, 46b.) 47. In a geometric sequence, a 3 4 and 4 a 6. Find a ) 48. In an arithmetic sequence, a 3 3 and a Find a ) 49. An auditorium has 30 rows of seats. There are 0 seats in the first row, 4 seats in the second row, 8 seats in the third row, and so on. Determine the seating capacity in the auditorium. 49.) Given the series a) Does it converge or diverge? 50a) b) Find the sum, if possible. 50b) 51. Find the sum of the first 8 terms of ,,,, ) 6

7 5. Find the interval of convergence and the sum in terms of of: ) 53. Epress the following series in sigma notation: ) 54. t t 1 1 k1 t k k 1 a) Write the first 6 terms 54a) b) Write an eplicit formula 54b) 55. Evaluate the following: 7 a. (4n 7) 55a.) n3 0 b. k ( k 1) 55b.) k Epress in sigma notation: ) 57. For what values of do the following converge? a) 1 + (-3) + (-3) + 57a) b) b) 7

8 Given the series a) Epress the series in sigma notation. 58a) b) Find the sum. 58b) 59. Write a recursive definition for the following sequence: 6, 10, 14, 18,, 59) 60. For what value of does the following sequence converge 60) to 5 3? Prove by Mathematical Induction. n i1 4i 1 nn 1 You may skip this one! 8

9 LIMITS, DERIVATIVES AND APPLICATIONS OF DERIVATIVES (MAX/MIN PROBLEMS) Find the following limits, if they eist. If they do not eist, write does not eist. 6. 3n n lim n 6) 63. lim 4 63) 64. lim7 3 64) 65. lim n n n n 1 65) 3n n n 66. lim 1 n 1 66) 67. lim ) 68. lim ) 69. lim ) 70. lim n sin n 70) 71. lim ) 9

10 7. lim ) 73. lim 3n n n 5n 1 73) 74. lim n n n n 4 74) 75. lim 5n 3n 1 n n 5 75) 76. lim3 n 9 76) 77. lim ) Find the derivative of the following (using the special rules/techniques) f ( ) ) f ( ) 79.) f ( ) 80.) *81. 1 f ( ) 81.) 10

11 3 8. f ( ) 3 8.) *83. 4 f ( ) 83.) f ( ) ) 3 15 *85. ) f ( 85.) ( ) 3 f 86.) 3 7 Use the difference quotient, lim h0 f ( h) h f ( ), to find the derivative. 87. f ( ) ) 88. f ( ) ) 89. f ( ) ) 11

12 Find the slope of the graph at the given point. Use the result to find an equation of the tangent line to the graph at the point. 90. f ( ) 1; (, 3) 90.) 91. f ( ) ; (, 6) 91.) 9. f ( ) 1 ; (3, ) 9.) f ( ) ; (, 6) 93.) 94. Find the equation of the tangent to the given curve: y ; when 94.) 95. If f '( ) 3 4 3, find f () (the anti-derivative) 95.) 96. Use the first and second derivative to identify the local ma and min, inflection point/s and determine the intervals where the curve is concave up and concave down. Then graph the function. (DO NOT USE A GRAPHING CALCULATOR). 3 f ( ) Local Ma 1 Local Min Pt. of Inflection Interval/s: Concave Up Concave Down

13 97. A ball is thrown upward from the top of an 80 ft building so that its height in feet above the ground after t seconds is h( t) 80 64t 16t. a. What is the instantaneous velocity at t 1 second? 97a.) b. When is the velocity = 0? 97b.) c. What is the ball s maimum height above the ground? 97c.) d. When does the ball hit the ground? 97d.) e. For what values of t is the ball falling? 97e.) Use derivatives to solve yards of fencing is used to enclose a rectangular field with a fence down the middle parallel to one of the sides. What is the maimum area which can be enclosed? 98.) 99. A cardboard poster is to have 50 square inches of printed material surrounded by a border at the top, at the bottom and 1 on each side. Find the minimum dimensions of the poster which has a minimum area. 99.) 100. An open square-base bo is to be manufactured from the least amount of material. If the bo is to have a volume of 3 cubic meters, what dimensions will minimize the amount of material used? 100.) 13