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Pre-Calculus 2015-2016 Name Math 112 Final Review Power Standard 7βTrigonometric Functions 1. A water wheel has a radius of 18 feet. The wheel is rotating at 15 revolutions per minutes. Find the linear speed, in feet per second, of the water. 2. A large pizza from Papa Johnβs has a radius of 9β. a) It is cut into eight equal pieces. What angle, in degrees and convert to radians, will be formed by a slice? b) Before the pizza was made, the crust was thrown into the air and spun a complete rotation every ½ second. What was the speed of the outer edge of the crust in inches per second? 3. A tower that is 98 feet tall casts a shadow of 113 feet long. Find the angle of elevation of the sun to the nearest degree. 4. The angle of elevation to the top of a pole is 62°. If the distance to the pole is 47 feet, what is the height of the pole? 5. 3 Find the exact value of cos(π ππβ1 7) without using your calculator. Show your diagram for full credit. Pre-Calculus 2015-2016 6. 4 Find the exact value of tan( cosβ1 5). 7. Find the exact value of each of the remaining trigonometric functions of π if 2 sin π = β 5 , sec π > 0. 8. Solve for x in the equation on the interval [0, 2π). Solve algebraically and give exact values for the answers. a) tan π₯ + 1 = 0 1 b) sin π₯ + 2 = 0 9. Solve cos(x) + 2 cos(x) sin(x) = 0 on the interval [0, 2Ο). Pre-Calculus 2015-2016 10. Using standard notation for a triangle, solve triangle ABC if A = 30Λ, a = 44, and b = 52. 11. Using standard notation solve the triangle if π΄ = 130Λ, πΆ = 10Λ, π = 4. 12. Two ships leave a harbor at the same time. Ship A travels on a bearing of 191° at 10 mph to location A. The other ship, Ship B, travels on a bearing of 65° at 12 mph to location B. How far apart, to the nearest mile, will the ships be after three hours? 13. A surveyor needs to find the distance between two houses that are separated by a lake. She measures angle ACB, which is found to be 15°, and then walks off the distance to each house, 80 feet and 100 feet, respectively. How far apart are the houses? Pre-Calculus 2015-2016 Power Standard 8βGraphs of Trigonometric Functions 14. Find the amplitude, period, phase shift, and vertical shift of π¦ = 6 sin(8π₯ β π) + 2. Graph the function. y 8 6 4 2 -Ο/4 -3Ο/16 -Ο/8 -Ο/16 x Ο/16 Ο/8 3Ο/16 Ο/4 -2 -4 -6 -8 π 15. Find the amplitude, period, phase shift, and vertical shift of π¦ = 2 cos(3(π₯ β )) β 2. Graph the 3 function. y 4 3 2 1 -2Ο/3 -Ο/2 -Ο/3 -Ο/6 -1 -2 -3 -4 x Ο/6 Ο/3 Ο/2 2Ο/3 Pre-Calculus 2015-2016 Power Standard 9βAnalytic Trigonometry 16. Simplify cos(π₯) sec(βπ₯). 17. Simplify csc π₯ tan2 π₯ . sec π₯ sin2 π₯+sin π₯ 1βsin(x) 18. Prove or disprove the following identity: (sec(x) β tan(x))2 = 1+sin(x). 19. Prove π ππ2 π₯πππ 2 π₯ + πππ 4 π₯ = πππ 2 π₯. 20. Prove sin π₯ + cos π₯ cot π₯ = csc π₯. Pre-Calculus 2015-2016 21. Find all of the solutions of 2sin2 (x) β cos(x) β 1 = 0. 22. Prove cos 2π’ = cos2 π’ β sin2 π’ 23. Find the exact value of each of the following under the given conditions: β24 3Ο Ο cos(x) = ,β < x < βΟ ; tan(y) = ββ6, < y < Ο 25 2 2 a) sin(x β y) = b) tan(π₯ β π¦) = 24. Find the exact value of the expression below: 12 3 Give that cos π₯ = 13, for π₯ in Quadrant IV and tan π¦ = 4 for π¦ in Quadrant I find cos(π₯ β π¦). Pre-Calculus 2015-2016 Power Standard 10βVector, Parametric, and Polar Coordinates 25. Let βββββ π΄π΅ be determined by the points π΄(2, 3) and π΅(β1, β4). What is the component form of βββββ π΄π΅? 26. Let π be a vector from initial point π1 to terminal point π2 . Write π in terms of the π and π vectors. π1 = (β8, 5) and π2 = (β3, 1) 27. Let π = 6π β 2π, and π = β3π β 8π. Find |π β π|. 28. Let π = β2π + 3π, π = 6π β 3π, and π = β4π. Find 3π β (2π β π). 29. Consider force vectors π’ and π£ acting on the same point. Find the resultant magnitude and angle ππ . βπ’β = 200 pounds, π = 29° βπ£β = 300 pounds, π = 205° Pre-Calculus 2015-2016 30. The magnitude and direction of two forces acting on an object are 60 pounds, S 40Λ E, and 70 pounds N 62Λ E, respectively. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force. 31. Find the horizontal and vertical components of the vector with magnitude 40 and direction π = 20Λ. Write the vector in terms of the vectors π and π. 32. Find the measure of the angle between the vectors π = β©β3, 4βͺ and π = β©6, 8βͺ. Round your answer to the nearest degree. Explain why the vectors are orthogonal or not orthogonal. 33. A force of 60 pounds on a rope is used to pull a box up a ramp inclined at 20Λ from the horizontal. The rope forms an angle of 43Λ with the horizontal. How much work is done pulling the box 22 feet along the ramp? Pre-Calculus 2015-2016 3π 34. Given the point (π, π) = (2, 4 ): a) Plot the point in the polar coordinate system. b) Give another representation of the point in polar form using a negative r value. c) Convert the point to rectangular coordinates (x, y). (Leave as exact values) 7π 35. Given the point (π, π) = (β3, 12 ): a) Plot the point in the polar coordinate system. b) Give another representation of the point in polar form using a negative r value. c) Convert the point to rectangular coordinates (x, y).