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MAC 1114 – Trigonometry – Final Exam Review Find the least positive coterminal angle. 750 34 2. 15 13 3. 3 1. Convert from radians to degrees or degrees to radians without a calculator. 115 5. 240 7 6. 6 4. 7. 9 Find the exact value of each expression. 8 3 9. csc120 10. arccos0 8. cos 2 2 11. cos sin 1 3 12. tan 2 3 13. csc 1 1 3 14. cot 1 Solve. 15. Find the length of the arc on a circle of radius 9 ft and central angle 60 . 16. Find the central angle in radians of a circle of radius 14.5 centimeters that intercepts an arc of length 25 centimeters. 17. A lawn roller with a 10-inch radius makes 1.2 revolutions per second. Find the angular velocity of the roller in radians per second. Also, find the linear velocity of the tractor pulling the roller. 18. Let 5 and evaluate the six trigonometric functions without using a calculator. 3 19. Let be an angle in standard position whose terminal side passes through the point 1, 3 . Find the exact values of all six trigonometric functions for . 1 and assume that is in quadrant 4. Determine the exact value of csc . 3 21. Graph the function f x 2 cos 6 x . What is the amplitude, period, and phase shift? 2 20. Let cos What is the range of f x ? 22. Graph the function f x 1 sin x . What is the amplitude, period, and phase shift? 2 3 23. A 15-foot ladder leans against a wall at an angle of 60 degrees. How high up the wall does the ladder rest? How far is the base of the ladder from the wall? 24. At some distance from a tower, the angle of elevation to the top of the tower is 36.5⁰. Sixty feet farther away from that point (in the same direction), the angle of elevation is 21.6⁰. How tall is the tower? 3 2 3 with and cos with 2 . Determine the exact 7 2 5 2 value of cos ,sin ,cos ,sin , tan , tan 2 ,cos ,sin . 2 2 25. Given sin Verify/Prove Each Identity 26. 1 1 2sec2 x 1 sin x 1 sin x 27. tan 4 tan 2 sec2 tan 2 28. sin 3 x cos3 x 1 1 sin 2 x sin x cos x 2 Find all real numbers that satisfy the equation. 29. 3cot 2 x 1 0 Find all values of x in [0, 2 ) that satisfy the equation. 30. 2sin x 3cos x 1 31. 2sin 2 x cos x 1 0 Solve each triangle. If no triangle exists explain why. If two different triangles are possible, find both. 32. 102.3 , 28.7 , b 27.4 33. a 12, b 31, 20.5 34. a 18, b 20, 76 35. a 8, b 19, c 14 Find the area of the triangle. 36. a 8, b 19, c 14 37. 102.3 , 28.7 , b 27.4 Let v=<-2,5> and w=<3,4>. 38. 39. 40. 41. 42. 43. v-2w |v+w| Write v as a linear combination of the unit vectors i and j. Find the magnitude and direction angle for v. Find the dot product of v and w. Determine the angle between v and w. 1 2 44. Show that the vectors ,8 and 10, 5 are perpendicular. 8 45. An airplane is flying with airspeed 186 mph is at a compass heading of 136⁰. If the wind speed at that elevation is 18mph directly out of the west, then what is the groundspeed and actual compass heading of the plane? Complex Numbers 46. Write z 2 2 3i in trigonometric form. 47. Let z1 24 cos300 i sin 300 and z 2 8 cos75 i sin 75 and find answer in standard form, a bi. 48. Determine 2 2 3i 9 Polar Coordinates 49. Convert the polar coordinates 3 3, to rectangular coordinates. 6 50. Convert the rectangular coordinates 3, 4 to polar coordinates. z1 . Write final z2 51. Convert the polar equation 5r sec into rectangular form. What does the graph look like? Parametric Equations 52. Sketch the curve given by x t 2 , y t 3 for 1 t 2 . 53. Using the parametric equations from (#50) eliminate the parameter and state an equivalent rectangular equation. State the domain. 54. The center field fence in a baseball park is 10 feet high and 400 feet from home plate. The baseball is hit 3 feet from above the ground. It leaves the bat at an angle of with a horizontal speed of 146 feet per second. a. Write a set of parametric equations for the path of the baseball. b. Use your calculator to graph to sketch the path of the baseball for = 15 and then for 23 . c. Of the two angles in part b, which one will be a home run (ball goes over the fence)?