Group Final Spring Is the equation a valid form of one of the Pythagorean trigonometric identities? 1 cot ß = csc., π [D] None of these 6

Transcription

1 Group Final Spring Is the equation a valid form of one of the Pythagorean trigonometric identities? 1 cot ß = csc ß. Find the exact value of the expression. sin π cos π cos π sin π Find all solutions of the equation in the interval 0, π. tan θ = secθ [A] π 5 4, π 4 [B] 5 π 7π 6, [C] π 7, π [D] None of these Use inverse functions where needed to find all solutions of the equation in the interval 0,. 1csc x cot x = 0 π [A] 9 9 arccot 4, arccot + π, π, π [B] 9 9 arccot π, arccot π,, [C] arccot, 5 arccot π, π, π π 7. Find all solutions in the interval 0,. sin x cos x = [D] arccot, 4 arccot 9 π, π, π 5π

2 Group Final Spring Use the half-angle formulas to simplify the expression. 1+ cos 4x + 6 [A] cos x + [B] cos x + [C] sec x + [D] sec x + 9. Find the exact value of cos ß using the half-angle formulas. tan ß =, 56 0 ß π [A] [B] 10 [C] 11 [D] Find the exact value of the sine, cosine, and tangent of the angle. π 1 [A] sin π = cos π π = + 1 tan = [B] sin π = cos π π = + 1 tan = [C] sin π = cos π = + 1 tan π = [D] sin π = cos π = + 1 tan π = Use inverse functions where needed to find all solutions of the equation in the interval 0,. 5 tan x 1tan x + 8 = [A] arctan, arctan,, 5 5 π π 7π 4 4 [B] 5 5 arctan π, arctan,, + π [C] 5 5 arctan, arctan,, π π 5 π 4 4 [D] arctan, arctan 5 π, π 4, 5 π 4 7π 4 π

3 Group Final Spring Identify a fundamental identity that could be used to verify the identity. x sin x cos x = 1 cos x [A] Cofunction identity [B] Even/odd identity [C] Quotient identity [D] Pythagorean identity 1. Express sinθ cos( θ ) as a sum containing only sines or cosines. 14. Use the given measures and the Law of Cosines to solve triangle ABC. a = 18, b = 16, c = 17 [A] A = 57. ; B = 10. ; C = 4. 0 [B] A = ; B = 54. ; C = [C] A = 54. ; B = ; C = [D] A = 10. ; B = 57. ; C = Find the area of the triangle having the indicated sides and angle. B = 6.5, a = 8, c = [A] 1.4 [B] 10.6 [C] 5.59 [D] None of these 16. Find all the solutions of the equation from [0, π ). (4 solutions) 1 sin( θ ) = 17. Use the given measures to find the area of triangle ABC. A = 7, a = 16., b = 1. 6 [A] 74.8 [B] [C] 54.6 [D] Use the Law of Cosines to find the third side of the triangle. C 5 A 58 7 B [A] 9 [B] 6 [C] 4 [D] None of these

4 Group Final Spring Convert the rectangular point (-5, 1) to the same point in the polar system. 0. Use Heron s Area Formula to find the area of the triangle with the given measures. Equilateral triangle, with a perimeter of 459. [A] [B] 51.6 [C] [D] Two ships leave a port at the same time. When ship A is 170 miles due east of the port, ship B is 00 miles from the port and 145 miles from ship A, in the direction shown below. What is ship B s bearing? Ship B N W E S Port Ship A [A] N W [B] N 45. E [C] N 116. W [D] N E Solve for x

5 Group Final Spring 010 5

6 Group Final Spring Evaluate the set of parametric equations for the given value of the parameter.. x = t, y = ln t, t =. Write a parametic equation for the following scenario. A banana is launced at an angle of 85 degrees with an initial velocity of 10 m/s at an initial height of 5 meters. Find the maximum height of the banana. 4. Find (a) the altitude from vertex B of the triangle to side AC, and (b) the area of the triangle. A = 1, 1,,, B = 5 11, C =

7 Group Final Spring Find the distance from 1, 4 to the line containing 1, 5 and, 8, to the nearest hundredth. 8. Solve the equation. 7 csc x 4 = 11csc x 9. A ship at sea, the Intrepid, spots two other ships, the Ranger and the Lancer, and measures the angle between them to be 4. The distance between the Intrepid and the Ranger is 190 meters. The Ranger measures an angle of 85 between the Intrepid and the Lancer. To the nearest meter, what is the distance between the Ranger and the Lancer? 40. Given a triangle with A = 41. Find the standard form of the equation of the parabola. Vertex: 0, 0 ; Focus: at (0, 4) 61, b = 18, and a = 17, find B. If there are two solutions, give both. 4. Each cable of a suspension bridge is suspended in the shape of a parabola between two towers that are 11 meters apart and whose tops are 0 meters above the roadway. The cables are 1.8 meters above the roadway midway between the towers. (a) Find an equation for the parabolic shape of the cable. (Assume the origin is on the road, midway between the towers.) (b) Find the height of the cable meters from the center of the bridge. 4. A point in rectangular coordinates is given. Convert the point to polar coordinates., [B] 4, 7 π 6 [C] 4, 7 π 6 [D] 4, π [A] 4 5 π, Identify the set of parametric equations for the given rectangular equation. y = x x [A] x = t, y = t t [B] x = x, y = x [C] x = t t, y = t [D] x = 0, y = x

8 Group Final Spring Convert the rectangular equation to polar form. x + y + = 4 [A] r = 4 cos θ [B] r = 4 cos θ [C] r = 4 sin θ [D] r = ± 46. The needle of the scale in the bulk food section of a supermarket is 0 cm long. Find the distance the tip of the needle travels when it rotates 168. [A] 58.6 cm [B] 6 cm [C] 9. cm [D] 9. cm 47. A bicycle wheel with a radius of 1 inches makes 1.1 revolutions per second. What is the speed of the bicycle? [A] in. s [B] in. s [C] 8. 6 in. s [D] in. s 48. A 95-foot long irrigation sprinkler line rotates around one end as shown. The sprinkler moves through an arc of 10 in 1.8 hours. Find the speed of the moving end of the sprinkler in feet per minute. Round your answer to the nearest tenth. s = 10 r = 95 ft 49. Use the unit circle and symmetry to help you evaluate the function(s). sec π 6 [A] [B] [C] [D] 50. Use the period of the function to select the expression that has the same value as the given expression. cos 19 π 8 [A] cos π 4 [B] cos π 8 [C] cos π 8 [D] cos 5 π 8

9 Group Final Spring Use the fundamental trigonometric identities to determine a simplified form of the expression. cos ß 1 sin ß 5. A photographer points a camera at a window in a nearby building forming an angle of 44 with the camera platform. If the camera is 50 meters from the building, how high above the platform is the window? Round to two decimal places m x 5. The point given is on the terminal side of an angle in standard position. Determine the exact value of the sine of the angle. 8, 15 [A] [B] 8 17 [C] 8 15 [D] Give the number of full cycles of the function that are found in the interval. y = sin x on the interval π, 11 π [A] 6 B] 1 [C] 5 [D] 7 55Find the amplitude and the period of the function. x y = cos 4 π [A] Amplitude = Period = π [B] Amplitude = Period = [C] Amplitude = Period = 4 [D] Amplitude = Period = 4

10 Group Final Spring cos arcsin (EXACT) A professional baseball coach is scouting players in the minor leagues to pull up to the major leagues. He must find one player each to play shortstop, first base, and centerfield. He can choose from 4 shortstops, 5 first basemen, and centerfielders. How many ways can the coach choose the players he needs? 58. How many different ways can 6 different runners finish in first, second, and third places in a race? 59. A college has 10 instructors qualified to teach a special computer lab course which requires two instructors to be present. How many different pairs of teachers could teach the class? 60. Suppose two fair dice are rolled. What is the probability that a sum of 10 or 11 turns up?

11 Group Final Spring