Name: Date: 1. Find the exact value of the given trigonometric function of the angle θ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) Find: sinθ c a θ a a = 8 7 2 2 2 2 1 2 7 2. Solve for y. y 60 y = 17 3 y = 17 2 3 5 3 y = 17 y = 17 3 y = 17 2 3. Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set to the correct angle mode.) sin ( 112 ) 0.7771 0.9638 0.7598 0.6293 0.8098 Page 1
4. Use a graphing utility to graph the function below. Be sure to include at least two full periods. Page 2
5. Determine the quadrant in which the angle lies. 150 Quadrant IV Quadrant III Quadrant I Quadrant II 6. Determine two coterminal angles (one positive and one negative) for the given angle. Give your answer in degrees. 7. Rewrite the given angle in radian measure as a multiple of π. (Do not use a calculator.) 270 π 3π 3π 5π 2 2π 4 2 4 8. Rewrite the given angle in degree measure. (Do not use a calculator.) 5π 18 696 318 378 333 50 9. Evaluate the trigonometric function using its period as an aid. 11π cos 3 1 2 1 2 3 2 3 2 2 3 3 Page 3
10. Find the exact value of the given trigonometric function of the angle θ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) Find: secθ c a θ a a = 8 6 2 2 2 2 1 2 6 11. State the quadrant in which θ lies. cot( θ ) < 0 and sec( θ ) < 0 Quadrant IV Quadrant I Quadrant III Quadrant I or Quadrant III Quadrant II 12. Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set to the correct angle mode.) csc ( 230 ) 1.0024 0.0698 1.4492 14.3356 14.3007 13. Which of the following is equivalent to the expression below? Page 4
14. If 4 3 csc x = and cos x 0 3 <, evaluate the function below. 15. Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. 16. Verify the identity shown below. Page 5
17. Solve the following equation. 18. Solve the following equation. Page 6
19. Find all solutions of the following equation in the interval [ ) 0, 2π. 20. Determine the area of a triangle having the following measurements. Round your answer to two decimal places. A= 119, b= 6, and c = 12 (d1) sq. units 17.54 sq. units 26.96 sq. units 21.43 sq. units 15.59 sq. units 21. Given C = 128, a = 13.9, and c = 9.3, use the Law of Sines to solve the triangle (if possible) for the value of b. If two solutions exist, find both. Round answer to two decimal places. b = 4.78 b = 2.34 and 6.66 b = 8.03 b = 1.41 and 6.79 not possible 22. Given C = 130, a = 9.93, and c = 18, use the Law of Sines to solve the triangle (if possible) for the value of b. If two solutions exist, find both. Round answer to two decimal places. b = 3.39 b = 11.89 and 4.03 b = 0.58 and 2.73 not possible b = 10.10 Page 7
23. Given a = 6, b = 7, and c = 8, use the Law of Cosines to solve the triangle for the value of B. Round answer to two decimal places. C b a Figure not drawn to scale A c B 60.33 88.26 80.44 39.98 51.75 24. Given a = 5, b = 10, and c = 6, use the Law of Cosines to solve the triangle for the value of B. Round answer to two decimal places. 60.33 97.90 52.41 80.44 29.69 25. A vertical pole 30 feet tall stands on a hillside that makes an angle of 19 with the horizontal. Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 51 feet downhill from the base of the pole. Round answer to two decimal places. 70.16 feet 70.19 feet 58.49 feet 62.39 feet 49.54 feet Page 8
26. Sketch the graph of the function. Page 9
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27. Identify the graph of the function. Page 11
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28. Sketch the graph of the function. Page 13
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29. 3 1 4 = Rewrite the exponential equation 64 1 log 3 2 9 = log3125 = 3 1 log2 3 8 = in logarithmic form. log5 25 = 2 log16 4 = 2 30. Rewrite the logarithmic equation 1 log = in exponential form. = 1/ = 1 = 1 = 1 = 31. Identify the x-intercept of the function y=+ log x. 1 3 6 The function has no x-intercept. 32. Write the logarithmic equation ln 5 = 1.609... in exponential form. 1.099... e = 3 6 2.303 10 = 1.792... 6 10 = 1.792... 3 e = 1.099... 1.099... 2.303e = 3 33. Evaluate the function f( x) = 1.568lnx at x= 28.481. Round to 3 decimal places. (You may use your calculator.) 4.748 3.291 3.291 8.239 undefined 34. Solve the equation log(1 x) = log(1) for x using the One-to-One Property. 101 99 101 1 The equation has no solution. 35. Rewrite the logarithm log6 25 in terms of the natural logarithm. ln13 ln 3 ln 3 ln13 ln3 ln13 ln13 log3 e ln 25 36. Evaluate the logarithm 1/3 log 1.975 using the change of base formula. Round to 3 decimal places. 0.005 0.006 219.173 0.005 0.002 Page 15
37. Expand the expression as a sum, difference, and/or constant multiple of logarithms. 38. Condense the expression log5x + log56 3 log(6 x ) log6 3x to the logarithm of a single term. log3 7 x log x 3 7 log 6 ( x + 4 ) 39. x 1 = 8 Solve 2 for x. 1 1 3 4 no solution 40. 2 Solve ln x ln11 = 0 for x. 9 3, 3 9 e 3/2 e no solution 41. Approximate the solution to ln 5x = 3.2. Round to 3 decimal places. 1.896 4.907 4.809 0.446 316.979 42. Find the standard form of the parabola with the given characteristic and vertex at the origin. focus: (0, 9) x 2 = 36y x 2 = 9y x 2 = 9y y 2 = 36x y 2 = 9x Page 16
43. Find the standard form of the parabola with the given characteristics. directrix: x = 4 vertex: ( 5, 1) (x 5) 2 = 36(y 1) (y 1) 2 = 36(x 5) (x + 5) 2 = 36(y + 1) (y + 1) 2 = 36(x + 5) (x 1) 2 = 36(y 5) 44. Match the graph with its equation. 45. Find the standard form of the equation of the ellipse with the given characteristics. vertices: ( 4, 3), ( 4, 15) minor axis of length 4 ( ) 2 ( ) 2 x+ 5 y+ 9 2 2 ( x 9 ) ( y 5) + = 1 + = 1 36 81 81 36 2 2 ( x 5 ) ( y 9 2 2 ) ( x+ 9 ) ( y+ 5) + = 1 + = 1 36 81 81 36 2 2 ( x 5 ) ( y 9) + = 1 81 36 Page 17
46. Find the standard form or the equation of the ellipse with vertices (±6, 0) and eccentricity 2 2 2 2 2 2 2 2 x y x y x y x y = 1 + = 1 + = 1 + = 1 4 25 4 25 25 4 25 21 2 2 x y + = 1 21 25 2 e = 5. Page 18
Answer Key 1. C 2. A 3. D 4. A 5. B 6. Answers may vary. One possible response is given below. 7. D 8. E 9. A 10. B 11. C 12. D 13. A 14. C 15. A 16. 17. B 18. C 19. A 20. B 21. E 22. C 23. D 24. C 25. A 26. A 27. B 28. B 29. A 30. C 31. B 32. A 33. C 34. B 35. A 36. D 37. D Page 19
38. B 39. C 40. B 41. B 42. A 43. E 44. D 45. B 46. D Page 20