Pre-calculus: 1st Semester Review Concepts Name: Date: Period:

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1 Pre-calculus: 1st Semester Review Concepts Name: Date: Period: Problem numbers preceded by a $ are those similar to high miss questions on the Semester Final last year. You have multiple resources to study this material. Also study your lessons, OoCLs, and returned quizzes. Solve the following using what you know about right triangles. 1. In right triangle XYZ with m < Y = 90, it is also known that m < X = 4 and x 14.3 ft. What are the other measures in the triangle? m < Z = y z. The angle of depression from the top of the Eiffel Tower to a point on the Seine River is 33 o. If the point on the river is 1637 feet from the tower, approximately how tall is the Eiffel Tower? 3. From a point 400 feet from the base of a cliff the angle of elevation to the top of the cliff is some angle. It is known that the cliff is 300 feet tall. What is the angle of elevation?

2 4. A freighter travels at a speed of knots from its port on a bearing of 40 o for hours and then changes to a bearing of 130 o for 4 hours at the same speed. Determine the distance and the bearing from the port to the boat. 5. y x 3sin 4 1 Vertical Stretch/Shrink: Period: Phase Shift: Vertical Shift: y x Domain: Is the function continuous? Yes or No Range: Are there any vertical asymptotes? If so where?

3 $6. f(x) = 3 csc (x π ) + 1 Vertical Stretch/Shrink: Period: 4 Phase Shift: Vertical Shift: y x lim x π f x = 4 lim x π + 4 f x = Is the function continuous? Yes or No Are there any vertical asymptotes? If so where? Simplify 7. sec x(cos x) 8. cos x (csc x) 9. sec x tan x 10. sec x 1

4 Prove: 11. sec x tan x+cot x = sin x cot x 1 1. = cot x 1 tan x 13. sin x+cos x sec x+csc x = sin x sec x 14. csc x 1 cos x = cotx csc x $15. f(x) =.1 tan πx 4 + π 4 Vertical Stretch/Shrink: Phase Shift: y Period: Vertical Shift: x Domain: Is the function continuous? Yes or No lim f x = lim x 5 Range: Are there any vertical asymptotes? If so where? f x = x 5 +

5 $16. f x = 1 cot (x π ) Vertical Stretch/Shrink: Period: 4 Phase Shift: Vertical Shift: y x Domain: Is the function continuous? Yes or No lim x π f x = Range: Are there any vertical asymptotes? If so where? lim x π + f x = 17. The following table represents the height of a miniature replica WWI French cannonball dropped from the top of the 1,050 foot high Eiffel tower in Paris. Use regression to determine the quadratic equation representing this event and use this equation to determine: a) the cannonball height at 3.5 seconds, b) the cannonball height at 8 seconds and, c) when the cannonball is 10 feet from the ground (6 feet above a cute little Belgian tourist). d) which superhero will save the Belgian tourist and by what method. Time (seconds) Cannonball Height (feet)

6 18. Determine the domain and range of the following equations. f x ( ) 16t 1050 Domain Range g( x) 5 7 x Domain Range f( x) 6 7 3x Domain Range Prove: 19. sin x + cos x (cot x) = csc x 0. sin x (tan x) = sec x cos x 1. cos x 1 sin x = cos x sec x tan x. sin x+tan x 1+cos x = tan x 3. sec x 1 cos x = sec x+1 sin x 4. (cos x +3 sin x) + (3cos x sin x) = 13

7 Write the period, phase shift, vertical shift, and vertical stretch for the following function: Also write directions of shifts y 5sin x 4 3 Period Vertical Stretch Phase Shift Vertical Shift 6. f x = 4 cos πx Period Vertical Stretch Phase Shift Vertical Shift 7. f x = + tan π(x 3) Period Vertical Stretch Phase Shift Vertical Shift 8. Which of the parent trigonometric functions are even, meaning they have symmetry and f(-x) = f(x)? 9. Odd functions have symmetry and f(-x) =. Solve for the angle. Sketch your solution. 30. sin x = sin x 31. sin x 3cosx + 3 = 0 3. sinx cosx = sinx 33. cos x 4cosx 5 = 0

8 Use the Law of Sines or Law of Cosines to solve the following. 34. Because of the prevailing winds, a tree grew so that it was leaning 6 from the vertical. At a point 100 feet away from the tree, the angle of elevation to the top of the tree is. Find the length of the tree. 100 ft Suppose you want to fence a triangular lot. If two sides measure 84 feet and 78 feet and the angle between the two sides is 10 0, what is the length of the fence to the nearest thousandth of a foot? 36. A parallelogram has sides of 55 cm and 71 cm. Find the length of each diagonal to the nearest tenth if the largest angle measures

9 $ 37. A ship embarks from the port and travels 75 km with a compass bearing of 43. Then the ship acquires a new bearing of 160 and travels 300 km at which point the engines fail. How far from where the ship embarked did its engines fail and what course does it need to take to get back to port? Prove. 38. tan x sin x = sin x (tan x) tan x sin x = csc x + sec x 40. cot x = 1 sin x 41. cot x = cosx sin x sec x sin x 1+cot x

10 4. If one apple in a tree is 10 feet above level ground and 3 feet to the right of the trunk of the tree and a second apple is 5 feet above level ground and 8 feet to the right of the trunk of the tree, determine the equation of the straight line required to shoot an arrow through both apples. Using this equation, determine where the archer must stand relative to the trunk if the arrow is released 7 feet above level ground. Also, use trig to determine the required trajectory angle of the arrow. 43. An Angry Bird s height following launch is represented by the following equation. How many seconds will pass before an Angry Bird on a downward trajectory strikes a pig which is 9 feet off the ground? f x t t ( ) $44. Solve the following equation for y in terms of x. 7 x 7 y 9

11 45. Last Christmas your many cash gifts summed to be $1,00. You took this money and purchased Apple stock. You then sold this stock right after Thanksgiving break to use to buy Christmas presents for your teachers. The stock value had increased 45% since your purchase. Each transaction (buying and selling) required a brokerage fee of $0. How much money do you now have to spend related to this stock investment? 46. This holiday break you and Grandpa are planning to build a pea gravel drainage zone on the north side of the house. This will be a rectangular area measuring 5 feet by 35 feet with a depth of 0.75 feet. If the pea gravel comes in 3 cubic foot bags at a cost of $15/bag, how much should you budget for the pea gravel? Simplify. 47. (sin x) (tan x) (cot x) (csc x) 48. tan x sec x sin x+cos x cos x 50. sec x cos x tan x cot x

12 Simplify. 51. tan x sec x 5. sec x (cot x) 53. cos x + sin x + cot x 54. sec x tan x + cot x Solve for the angle. Sketch your solution. 55. sinx 1 = cos + 3 = tanσ + 3 = cot x + 3 = 0

13 Triangle area 59. A triangular parking lot has sides of lengths 40 feet, 350 feet, and 180 feet. What is the area? 60. The lengths of sides of a triangle are 48 meters and 6 meters and the measure of the included angle is 10 o. Find the area of the triangle. 61. An isosceles triangle has a base of 46 centimeters and a vertex angle of 64 o. Find the area of the triangle.

14 6. A triangular lot has sides of 00 meters, 180 meters, and 10 meters. What is the area of the lot? 63. The lengths of two sides of a triangle are 0 feet and 30 feet and the measure of the included angle is 60 o. Find the area. Prove. 64. (1 + csc x)(1 sin x) = cot x (cos x) 65. sec x (csc x) = csc x + sec x Solve for the angle. Sketch your solution secμ 4 = cot t 1 = 0

15 68. sin 7xcos 7x 69. sin 11 cos sin xcos x 71. cos 8 7. cos 4x 73. sin 74. sin 6 Find all the solutions to the equation in the interval [0,). 75. sin(b) = sin(b) 76. cos(b) = sin(b) Prove the identity. $77. csc(x )= csc (x) tan(x ) 78. sin(4x) = (4sin(x) cos(x))(cos (x)-1)

16 $79. A pendulum on a ginormous clock swings 50 degrees up on either side of the pendulum resting point. With a pendulum length of 150 feet (Did I mention that it was ginormous?), how far will the tip of the pendulum swing in a full cycle? Find the arclength, and don t forget to change the angle to radians.) State whether the given measurement determine zero, one, or two triangles. $ 80. A 36, a, b 7 $ 81. B 8, b 17, c 15 $ 8. C 36, a 17, c 16

17 Fall Semester Formula Page RECIPROCAL IDENTITIES: QUOTIENT IDENTITIES: sin 1 csc cos 1 sec tan 1 cot tan sin cos cot cos sin csc 1 sin sec 1 cos cot 1 tan PYTHAGOREAN IDENTITIES: 1 tan sec 1 cot csc & + = 1 DOUBLE ANGLE IDENTITIES: sin sin cos cos cos sin cos cos 1 cos 1 sin COFUNCTION IDENTITIES: ODD-EVEN IDENTITIES: sin sin cos tan cot sec sin cot tan csc cos csc sec sin cos cos tan tan csc csc sec sec cot cot Trigonometric Ratios: sin θ = y r cos θ = x r tan θ = x y cot θ = x y sec θ = r x csc θ = r y Right Triangle Trig: opp. adj. opp. sin cos tan hyp. hyp. adj. Law of Sines: a b c sin A sin B sin C or Law of Cosines: sin A sin B sin C a b c c a b ab cos C Triangle Areas: Heron s Rule: s = a b c then, Area of ABC s( s a)( s b)( s c) SAS Area: Area of 1 ABC bc sin A AAS Area: Area of ABC a sin B sin C sin A Arclength: s r if is in radians & radians deg rees 180