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Name: ________________________ Class: ___________________ Date: __________ ID: A Trig/Pre-Calc B Midterm Review - 2012-13 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Graph the function. Which choice gives the amplitude, period, phase shift, and vertical shift for the function? ÁÊÁ 1 ˜ˆ˜ y = 4sin ÁÁÁÁ 2θ + π ˜˜˜˜ − 3 ÁË 2 ˜¯ a. c. 1 4; π ; – π ; –3 4 1 −4; π ; – π ; 3 4 b. d. 1 4; π ; – π ; –3 4 1 −4; π ; – π ; –3 4 ____ 2. Which sum or difference identity would you use to verify that cos (180° − θ) = −cos θ? a. cos (α −β ) = cos α cosβ + sin α sin β b. sin (α + β ) = sin α cos β + cos α sin β c. sin (α −β ) = sin α cos β – cos α sin β d. cos (α −β ) = cos α cos β – sin α sin β 1 Name: ________________________ ____ ID: A 3. Which double-angle or half-angle identity would you use to verify that 2tanx a. tan2x = b. sin2x = 2sinx cos x 2 1 − tan x x =± c. sin d. cot 2x = 2 sin2x 2sin 2 x = cot x? 1 − cos x 1 + cos x 2sinx 1 − sin2 x Short Answer 4. Find the exact value of cos 15°. 5. Change 2.99 radians to degree measure. Round to the nearest tenth. ÊÁ 1 ˆ˜ Á ˜ 6. Graph y = sec ÁÁÁÁ θ + 3π ˜˜˜˜ + 3 . ÁË 2 ˜¯ 7. Use a graph of the sine function to find the value of θ for which sin θ = −1. 8. If sin θ = 3 5 and θ terminates in the first quadrant, find the exact value of cos 2θ. 9. Solve 5tanx = 5 3 for 0° ≤ x ≤ 180°. 10. Find the exact value of cos 105°. 11. A gear of radius 4.6 cm turns at 3 revolutions per second. What is the linear velocity of the gear in centimeters per second? 12. What basic trigonometric identity would you use to verify that cot x sin x = cos x? 13. What basic trigonometric identity would you use to verify that tan x cos x = sin x? 14. Find sec x if tan2 x = 1 2 . 15. Solve 2 − 3cos x = 5 + 3cos x for 0° ≤ x ≤ 180°. 16. Find the area of a sector with a central angle of 111° and a radius of 19.3 millimeters. Round to the nearest tenth. 17. What basic trigonometric identity would you use to verify that 18. Given sinθ = 19. Find cos x if 6 and secθ < 0, find cosθ and tanθ. 11 sin 2 x − 1 cos x = −1. 20. Find cot x if sin x cot x csc x = 2. 2 sinx + 1 sinx = 1 + csc x ? Name: ________________________ ID: A 21. For a circle of radius 5 feet, find the arc length s subtended by a central angle of 65°. 22. If α and β are the measures of two first quadrant angles and sin α = 4 5 and sin β = 5 13 , find sin (α + β ). 23. Find the amplitude, period, and phase shift of f (x ) = −4 sin (3x + 1). 24. Solve tanx = cot x for 0 ≤ x ≤ π. 25. Change 310° to radian measure in terms of π. 3 26. If sin θ = − and θ terminates in the fourth quadrant, find the exact value of tan 2θ. 5 27. Write an equation for the given function given the period, phase shift, and vertical shift. 2 2 cosecant function, period = π, phase shift = π, vertical shift = 3 7 9 28. What basic trigonometric identity would you use to verify that sin2 x + cos 2 x cos x = sec x ? 29. Solve tanx sec x − 2tanx = 0 for all real values of x. 30. Write an equation of the cosine function with amplitude 3 and period 6π. 31. A truck driver travels at 60 miles per hour. The truck tires have a diameter of 26 inches. What is the angular velocity of the wheels in revolutions per second (rps)? 32. Use a half-angle identity to find the exact value of tan 105°. 33. Write an equation of the sine function with the given amplitude, period, phase shift, and vertical shift. 2 2 amplitude: 4, period = π , phase shift = – π , vertical shift = 1 3 9 34. Find cos x if sin x cot x = 4. 35. Find csc x if sin x + cot x cos x = 36. Solve cos θ ≥ 3 2 3. for 0 ≤ θ ≤ 2π. ÊÁ 1 ˆ˜ ˜ˆ˜ ÁÊÁ 37. Find the value of tanÁÁÁ sin -1 ÁÁÁÁ − ˜˜˜˜ ˜˜˜ . Á Ë 2 ¯ ˜¯ Ë → 38. Identify the ordered pair that represents the vector from A ÊÁË −3, 1ˆ˜¯ to B ÊÁË 2, − 2 ˆ˜¯ and the magnitude of AB . ä = 2, − 1, − 7 and ä ä − 2v ä. 39. Given u v = −2, 3, − 3 find an ordered triple that represents 3u 3 Name: ________________________ ID: A 40. State the amplitude, period, phase shift, and vertical shift for the function. Then graph the function. ÊÁ 3 ˆ˜˜ Á y = −2 cos ÁÁÁÁ 3θ − π ˜˜˜˜ + 3 ÁË 2 ˜¯ ÊÁ 1 ˆ˜ ÁÁ ˜ Á 41. Graph y = tan ÁÁ θ − 3π ˜˜˜˜ + 3. ÁË 2 ˜¯ 42. Verify that cos 2 x = sec 2 x − tan 2 x − sin2 x is an identity. 4