1 of 9 LYS Math 2412 - Final Exam Review Problems Update:11/18/ 1. Verify the identity: tan(x) sin(x) + cos(x) = sec(x) 2. Verify the identity: sin(x) - sin(x) cos2(x) = sin(x) 3. Verify the identity: tan?(x) tan?(x) + 1 = sin?(x) 4. Verify the identity: tan?(x) tan?(x) + 1 = sin? (x) 5. Verify the identity: cos(x) + sin (x) = 1 cos(x) + 1 6. Without the aid of a calculator, find the exact value of cos( Hy ) cos( 3x ) + sin( Hz ) sin(3x ). 7. Let x = tan(10) + tan(20) 1 - tan(10) tan(20) a) Write x as a single function of sine, cosine, or tangent. b) Without the aid of a calculator, find the exact value of x. 8. Given the following information i) sin( A) = +; and the terminal side of A lies in Quadrant I and ii) sin (B) = ? and the terminal side of B lies in Quadrant II, find the exact value of each of the following trigonometric functions. a) cos(A + B) b) sin(A + B) c) cot (A + B) d) sec(-A - B) 9. Given that cot(0) = 9 and the terminal side of 0 lies in Quadrant III, find the exact value of each of th following trigonometric functions. a) cos (20) b) sin(20) c) tan(20) 10. Use the appropriate half-angle formula to find the exact value of sin(75). 11. Express cos(50) + cos(70) as a product of sines and/ or cosines. 12. Verify the identity: Cos(2) - cos(3) = - tan , sin(x) + sin(y) 13. Find all solutions to 7 sin(x) + 1 = 5 sin(x). 14. Find all solutions to 2 sin(2x) + 1 = 2 on [0, 27) Created/Revised the Math DepartmentMath 2412 - Final Exam Review Problems 15. Find all solutions to tan(3x) = 1 on [0, 27). 16. Solve the equation 2sin (x) + sin(x) = 1 on [0, 27). Select the best answer choice below. D {g, 27, 5, 3x } No solution None of A through E. 17. Solve the equation 2sin?(2x) - sin(2.x) = 1 on [0, 27). 18. Solve the equation cot(0) (tan(0) + v3) = 0 on [0, 27). Select the best answer choice below. No solution None of A through F 19. Find all solutions to sin?(x) - 9 cos(x) + 9 =0 on [0, 27). Solving Triangles Use the given information to solve the triangle. Round lengths to the nearest hundredth and angles to the nearest tenth of a degree. Place your answers in the appropriate answer blanks. Sketch the triangle(s) showing the given information. If no triangle is formed from the given information, then write "No Solution". If two triangles are formed from the given information, then fill in the information for 2; otherwise, leave the information blank. 20. Given A = 30, a = 7, and c = 16, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (42) blank. 41: A = 30, B = C = , and a = 7, b = , C = 16. 42: A = 30, B = C = , and a = 7, b = , C = 16. 21. Given a = 4, b = 14, and c = 12, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (2) blank. 1: A = B = C = _, and a = 4, b = 14, c = 12. 42: A = B = C = _, and a = 4, b = 14, c = 12. 22. Given B = 45, a = 5, and c = 8, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (2) blank. 1: A= B = 45, C = and a = 5, b = , C= 8. 42: A = B = 45, C = , and a = 5, b= , C = 8. 23. Given C = 60, a = 2, and b = 5, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (42) blank. Created/Revised the Math Department 2Math 2412 - Final Exam Review Problems 1: A = B = C = 60, and a = 2, b = 5, c= 42: A = B = C = 60, and a = 2, b = 5, c = 24. Find the area of the triangle AABC having the following properties: B = 50, a = 5 ft, and c = 2 ft. 25. Given a = 400 ft and b = 300 ft and the measure of angle A is twice the measure of angle B, solve the triangle. In the case that there is only one triangle which satsifies the given information leave the information for the second trangle (2) blank. You may round the side to the nearest tenth of a foot and the angle to the nearest degree. 1: A = B = and a = 400, b = 300, c = 42: A = B = C = and a = 400, b = 300, c = 26. Find the polar coordinates of (-2V/3, 2). 27. Find the rectangular coordinates of (-3, l1x ). 28. Convert x = 14 to a polar equation that expresses r in terms of 0. 14 AT= sin(0) cos (0) 72 = 14 D r' = 3 14 sin(0) E T= COS (20) None of A through E. 29. Convert r = 16 sin(0) to a rectangular equation. A (x+8)2 + y2 = 64 B (x -8)2 + y2 = 64 Cr2 + (y+8)2 = 64 D x2 + (y - 8)2 = 64 (x+8)2 +y2 = 16 (x -8)2 + y2 = 16 Gr2 + (y+8)2 = 16 H 2 + (y - 8)2 = 16 1 None of A through 1. 30. Convert rsin(0) = 10 to a rectangular equation. 31. Convert 72 cos(20) = 1 to a rectangular equation. 32. Convert r2 = 7y to a polar equation that expresses r in terms of 0. 33. Convert rcos(0 + ) = 5 to a rectangular equation. 34. Determine whether r = 4 cos(0) is symmetric about the a) polar axis, b) the line 0 = 7, and/or c) about the pole. Place a check in the appropriate box. Yes No Symmtery Test Inconclusive a) polar axis b) 0 =5 c) pole 35. Write -1 - i in polar form (i.e., trigonometric form). 3 Created/Revised the Math Department