Advanced Math Final Exam Review Name: Bornoty May June 2013 Use the following schedule to complete the final exam review. Homework will e checked in every day. Late work will NOT e accepted. Homework answers will e provided at the eginning of each class period for you to check your work from the previous night. FINAL EXAM SCHEDULE: Thursday, May 30 th Friday, May 31 st Monday, June 3 rd Tuesday, June 4 th All exams egin promptly at 8:00 a.m. 6 th & 7 th hour exams [90 min each] 4 th & 5 th hour exams [90 min each] 2 nd & 3 rd hour exams [90 min each] 1 st hour exam [90 min] Day Date Assignment Completed Tuesday May 21, 2013 Trig Part 1 #1-37 all Wednesday May 22, 2013 Trig Part 2 #1-33 all Thursday May 23, 2013 Trig Part 3 #1-20 all Friday May 24, 2013 Conics #1-8, 11-17 all Tuesday May 28, 2013 Polars #1-14 all Wednesday May 29, 2013 Vectors #1-43 odds Thursday May 30, 2013 Matrices #1-18, 21, 22 Friday May 31, 2013 STUDY!! 1
Advanced Math Name Final Exam Review Packet 2013 Date Hour Trig Part 1 - SOH CAH TOA, Special Rt Triangles, & Unit Circle Fill in the ratios for each trig function using the letters for opposite (O), adjacent(a) and hypotenuse(h) Find each value. No decimals reduce fractions. 1. 2. 3. 4. D 5. 6. 7. 8. 4 5 9. 10. 11. 12. F 3 E Using the 30-60-90 and 45-45-90 rules, find each length or angle measure. Express all lengths in simplest radical form. 13. AB = BC = 14. PM = MN = 15. AC = AB = 16. ST = TV = A P A S 6 15 10 C B N M C 18 B T V Fill in the missing sides of the special right triangles and answer the following questions. Exact answers only 17. 18. 19. 20. Use a calculator to find each value. Round to the nearest 100 th. Use SOH CAH TOA to solve for the following parts of the right triangles. C is a right angle. 21. c = 20, a = 15, find A 22. a = 30, = 12, find A A 23. A = 35, c = 9, find A 24. a = 6, = 12, find B A c c c c C a B C a B C a B C a B 2
Use SOH CAH TOA to solve each story prolem 25. Katlyn leans a 16 foot ladder against the wall. If the ladder makes an angle of with the ground, how far up the wall does the ladder reach? 26. Tom leans a 20-ft ladder against a wall. The ase of the ladder is 4 feet from the wall. What angle does the ladder make with the ground? Find the reference angle for each. 27. 28. 29. 30. 31. Find the value of n to the nearest degree. A. B. C. D. E. None of the aove 32. A school flagpole casts a 16-foot shadow on the lawn. A teacher stood at the shadow's edge and measured the angle of elevation to the top of the pole at. How tall is the pole? A. 12 ft B. 5.6 ft C. 17.8 ft D. 14.4 ft E. None of the aove 33. Find the value of j rounded to the nearest tenth. A. 663.4 B. 197.1 C. 49.3 D. 13.2 E. None of the aove 34. Change to degrees. A. B. C. D. E. None of the aove 35. Determine the exact value of. A. B. C. D. E. None of the aove 36. Find the exact value of. A. B. C. 37. Point is located on a unit circle. Find and. D. E. None of the aove A. B. C. D. 3
Trig Part 2 - More Unit Circle, Graphs, Identities, and Formulas Convert from degrees to radians Convert from radians to degrees 1. 2. 3. 4. Complete the following. 90 Family Sin 90 Cos 90 Tan 90 Sin 270 Cos 270 Tan 270 Csc 90 Sec 90 Cot 90 Csc 270 Sec 270 Cot 270 Sin 180 Cos 180 Tan 180 Sin 360 Cos 360 Tan 360 Csc 180 Sec 180 Cot 180 Csc 360 Sec 360 Cot 360 Give the exact value for each prolem. 0 360 5. 6. 7. 8. Draw a triangle for each prolem and find the exact value no decimals!! 9. Find, if and 10. Find if and 11. Find, if and the angle is in the third quadrant 12. Find, if and Tell which quadrant each angle lies in. (I, II, III, or IV) 13. is positive and is negative. 14. is positive and is negative. 15. is positive and is negative. 4
Give the amplitude, period and phase shifts (up/down/left/right) of each graph you do not need to graph these. 16. 17. 18. Amplitude = Period = Amplitude = Period = Amplitude = Period = h = left or right Reflection? Y/N h = left or right Reflection? Y/N h = left or right Reflection? Y/N k = up or down k = up or down k = up or down Directions: Graph one period of the following trigonometric equations. 19. 20. 19. 20. 21. 22. 21. 22. 23. 24. 23. 24. 25. 25. 5
Use your formula sheet to remind yourself of the trig identities. Simplify. Show all work. 26. 27. Use your formula sheet to remind yourself of the sum and difference formulas. Use the sum and difference formulas to find the EXACT value of each trig. function. Rationalize when necessary! 28. 29. 30. 31. 32. 33. 6
Trig Part 3 - Laws, Inverses, Solving, and Areas of Triangles Law of Sines: Law of Cosines: Determine whether to use the Law of Sines of the Cosines to solve the given triangles. Circle one. You do NOT need to solve! 1. 2. Sines Sines Cosines Cosines Use the Law of Sines and/or the Law of Cosines to solve each triangle. Round all decimals to the nearest 100 th. 3. 4. 5. 6. 7
7. An isosceles triangle has a ase of 22 cm and a vertex angle of. Find the perimeter of the triangle. 8. In, A = 32, a =8, and = 7. Find C. A. B. C. D. E. None of the aove 9. Find the length of side a to the nearest tenth. A. 11.8 B. 8.7 C. 6.1 D. 10.9 E. None of the aove 10. Solve (Recall restricted domain!) A. B. C. D. E. None of the aove 11. Solve A. B. C. D. E. None of the aove 8
Solve the following given that. The numer of solutions you should find is given in parentheses. 12. (5) 13. (4) 14. (1) 15. (2) 16. (4) 17. (4) Use your formula sheet to find the area of the given triangle. Draw the triangle and then decide if you can find the area, or if you have to use the Law of Sines or the Law of Cosines to find more information. Lael your triangle as you find additional information. 18. 19. 20. 9
CONICS Paraola: or distance from vertex to focus or focus to directrix. Given the equation, find the information. Given the information, find the equation. 1. 2. 3. Vertex (1, -1) and focus 4. Vertex (3, 0) and Directrix is Vertex Axis Focus Directrix Vertex Axis Focus Directrix Equation in standard form: Equation in standard form: Circle: center = radius = Given the equation, find the information. Given the information, find the equation. 5. 6. Center =, Radius = 5 Equation in standard form: Center Radius 10
Ellipse: or center =, igger numer, Given the equation, find the information. Given the information, find the equation. 7. 8. Center (-2, 1) with major axis length = 8 and parallel to the x-axis. Minor axis length = 6. Center (, ) a = = c = Foci (, ) (, ) Vertices (, ) (, ) Co-Vertices (, ) (, ) Equation in correct form Major Axis Minor Axis Hyperola: or center =, FIRST numer, Given the equation, find the information Given the information, find the equation 9. 10. Center and. The transverse (major) axis is vertical. Center (, ) a = = c = Foci (, ) (, ) Vertices (, ) (, ) Trans.Axis Conj. Axis Slopes of Asymptotes 11
Identify which conic the equation descries. Rememer to change the form if necessary. 11. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 12. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 13. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 14. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 15. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 16. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 17. A. Line B. Paraola C. Circle D. Ellipse E. Hyperola 12
Polar Coordinates Plot the following point on the polar coordinate system. Be sure to lael each point y its letter. 1. 5. 7. 2. 3. 4. 6. 8. Convert the following rectangular coordinates into TWO PAIRS of polar coordinates where is in degrees. Use the equations for converting or the special right triangles. 9. 10. 11. Convert the following polar coordinates into rectangular coordinates. Use the equations for converting or the special right triangles. EXACT answers only! 12. 13. 14. 13
Vectors Draw the vector from to. Then find the vector in component form with the initial point and terminal point. Finally, draw the vector in component form. 1. and 2. and 1.) 2.) 3. and 4. and 3.) 4.) 5. and 6. and 5.) 6.) 14
Find the magnitude and direction for the given vector. Draw the vector to help decide the direction! Estimate the magnitude to the nearest tenth. Estimate the direction to the nearest tenth of a degree. 7. 8. 7.) Magnitude 7.) Direction 8.) Magnitude 8.) Direction 9. 10. 9.) Magnitude [these two vectors will go off the graph a little when you graph them that s OK just estimate!] 9.) Direction 10.) Magnitude 10.) Direction 15
Find the magnitude and direction for the given vector. Draw the vector to help decide the direction! Estimate the magnitude to the nearest tenth. Estimate the direction to the nearest tenth of a degree. 11. 12. 11.) Magnitude 11.) Direction 12.) Magnitude 12.) Direction 13. 14. 13.) Magnitude 13.) Direction 14.) Magnitude 14.) Direction 16
Given, and. Find the following. 15. 16. 17. 15.) 16.) 17.) 18. 19. 20. 18.) 19.) 20.) 21. 22. 23. 21.) 22.) 23.) 24. 25. 26. 24.) 25.) 26.) 17
Find the unit vector in the direction of y dividing each component of y the magnitude of. 27. 28. 29. 27.) 28.) 29.) Given and, find the following: 30. 31. 32. 30.) 31.) 32.) 33. 34. 35. 33.) 34. ) 35. ) 36. 37. 38. 36.) 37.) 38.) 18
Draw vectors and on the same coordinate plane. Then, find the dot product of and, and. Finally, find the measure of the angle etween the two vectors. 39. and Angle etween the two vectors = 40. and Angle etween the two vectors = 41. and Angle etween the two vectors = 19
Draw vectors and on the same coordinate plane. Then, find the dot product of and, and. Finally, find the measure of the angle etween the two vectors. 42. and Angle etween the two vectors = 43. and Angle etween the two vectors = 44. and Angle etween the two vectors = 20
Systems and Matrices Solve each system y hand. State which method you used: Sustitution or Elimination. 1. 2. 3. 4. Method = Method = Method = Method = State the dimensions of each matrix. 5. 6. 7. 8. For the following questions, use the matrices elow. For the following questions, use the matrices elow. 9. 10. 11. 12. 21
Use the following matrices for the multiplication prolems: 13. 14. 15. 16. What are the dimensions of the product matrix when the following two matrices are multiplied:? 17. Find the inverse of matrix using your calculator. 18. Find the inverse of matrix using your calculator. 19. Use either formula to find the determinant of the 3 x 3 matrix. DO THIS BY HAND!!!!!!!!! a a a 1 2 3 1 2 3 c1 c2 a1 c 3 2 3 c c 2 3 a 2 1 3 c c 1 3 a 3 1 2 c c 1 2 20. Use your calculator to find the determinant of the 3 x 3 matrix. Use the inverse matrix and matrix multiplication to solve the following systems. Start y filling in the Coefficient, Variale and Constant matrices. Rememer to write your answer as an ordered pair or ordered triple. 21. 22. AX=B AX=B = 22