Pre-Calc Trig ~1~ NJCTL.org. Unit Circle Class Work Find the exact value of the given expression. 7. Given the terminal point ( 3, 2 10.

Transcription

1 Unit Circle Class Work Find the exact value of the given expression.. cos π. tan 5π 6. sin 7π 5. cot 5π. sec π 6. csc 9π 7. Given the terminal point (, 0 ) find tanθ 7 tan θ = Given the terminal point ( 5 cot θ = 5, ) find cotθ 9. Knowing cosx= and the terminal point is in the fourth quadrant find sinx. = 5 0. Knowing cotx= and the terminal point is in the third quadrant find secx. 5 cot x = 0 Pre-Calc Trig ~~ NJCTL.org

2 Unit Circle Home Work Find the exact value of the given expression.. cos 5π. tan 7π 6. sin π 5. cot π. sec π 6. csc π 7. Given the terminal point ( 7 cot θ = 7, 5 5 ) find cotθ 8. Given the terminal point ( tan θ = 7 8 9, 7 ) find tanθ 9 9. Knowing sinx= 7 and the terminal point is in the second quadrant find secx. 8 sec x = Knowing cscx= and the terminal point is in the third quadrant find cotx. 5 cot x = Pre-Calc Trig ~~ NJCTL.org

3 Graphing Class Work State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and then check it with a graphing calculator.. y = cos ( (x + π )) +. y = cos(x π) A: P: π PS: π VS: A: P: π PS: π VS: -. y = sin ( (x + π )) +. y = cos(x π) 6 A: P: π PS: π 6 VS: A: P: π PS: π VS: - 5. y = cos(x π) + A: P: π PS: π VS: Pre-Calc Trig ~~ NJCTL.org

4 Graphing Home Work State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and then check it with a graphing calculator. 6. y = cos ( (x π )) + 7. y = cos(x π) A: P: π PS: π VS: A: P: π PS: π VS: - 8. y = sin ( (x + π )) + 9. y = cos(6x π) A: P: 8π PS: π VS: A: P: π PS: π VS: 0. y = cos(x π) A: P: π PS: π VS: - Pre-Calc Trig ~~ NJCTL.org

5 Law of Sines Class Work Solve triangle ABC.. A = 70, B = 0, c =. B = 65, C = 50, a = C = 80 o a =. 8 b =. 0 A = 65 o b = c = 0.. b = 6, A = 5, B = 5. c = 8, B = 60, C = 0 C = 0 o a =. 59 c = A = 80 o a =. 6 b = c =, b = 6, C = b =, a = 5, B = 0 A = 8 o B = 8 o a =. 65 A = 5. 5 o C = o c = 8. 6 or A = 6. 5 o C =. 5 o c = A = 5, a = 6, b = No Solution 8. An airplane is on the radar at both Newark Liberty International and JFK airports that are 0 miles apart. The angle of elevation from Newark to the plane is and from JFK is 5 when the plane is directly between them. How far is the plane from JFK? What is the plane s elevation?. 7 miles from JFK 9. miles in elevation 9. A mathematician walking in the woods noticed that the angle the angle of elevation to a bird at the top of a tree is 50, after walking 0 toward the tree, the angle is 55. How far is she from the bird? feet from the bird Pre-Calc Trig ~5~ NJCTL.org

6 Law of Sines Home Work Solve triangle ABC. 0. A = 60, B = 0, c = 5. B = 75, C = 50, a = C = 80 o a =. b =. 6 A = 55 o b = 6. 5 b =. 09. b = 6, A = 5, B = 5. c = 8, B = 50, C = 0 C = 00 o a =. 87 c = 8. 6 A = 90 o a =. 5 b = c =, b = 8, C = b =, a = 6, B = 50 A = o B = 7. o a =. 9 No Solution 6. A = 0, a = 5, b = No Solution 7. An airplane is on the radar at both Newark Liberty International and JFK airports that are 0 miles apart. The angle of elevation from Newark to the plane is 5 and from JFK is 5 when the plane is directly between them. How far is the plane from JFK? What is the plane s elevation? miles from JFK. miles in elevation 8. A mathematician walking in the woods noticed that the angle the angle of elevation to a bird at the top of a tree is 5, after walking 0 toward the tree, the angle is 60. How far is she from the bird? feet from the bird Pre-Calc Trig ~6~ NJCTL.org

7 Law of Cosines Class Work Solve triangle ABC. 9. a =, b =, c = a = 5, b = 6, c = 7 A = 6. o B = 6. o C = 7. o A =. o B = 57. o C = o 5. a = 7, b = 6, c = 5. A = 00, b =, c = 5 A = 86. o B = o C =. 8 o a = 6. 9 B =. 7 o C = 5. o 5. B = 60, a = 5, c = 9 5. C = 0, a = 0, b = A =. 7 o b = 7. 8 C = 86. o A = 56. o B = 8. 7 o c = A ship at sea noticed two lighthouses that according to the charts are mile apart. The light at lighthouse A is 00 above sea level and the navigator on the ship measures the angle of elevation to be, how far is the ship from lighthouse A? The light at lighthouse B is 00 above sea level and the navigator on the ship measures the angle of elevation to be 5, how far is the ship from lighthouse B? How far is the ship from shore? The ship is ft from lighthouse A The ship is 9. 0 ft from lighthouse B The ship is 66. ft from shore 56. A student takes his dogs for a walk. He lets them off their leash in a field where Edison runs at 7 m/s and Einstein runs at 6 m/s. The student determines the angle between the dogs is 0, how far are the dogs from each other in 8 seconds? The dogs are 9. 7 m apart Pre-Calc Trig ~7~ NJCTL.org

8 Law of Cosines Home Work Solve triangle ABC. 57. a =, b = 5, c = a =, b = 0, c = A =. o B = 0. 7 o C = 5. o A =. o B = 5. o C =. 6 o 59. a =, b = 8, c = A = 85, b =, c = 7 A = 0. 6 o B = 5. o C =. o a = 7. 7 B =. 9 o C = 7. o 6. B = 70, a = 6, c = 6. C = 5, a =, b = 9 A = 9. 6 o b =. C = 80. o A = o B = 68. o c = A ship at sea noticed two lighthouses that according to the charts are mile apart. The light at lighthouse A is 75 above sea level and the navigator on the ship measures the angle of elevation to be, how far is the ship from lighthouse A? The light at lighthouse B is 5 above sea level and the navigator on the ship measures the angle of elevation to be 8, how far is the ship from lighthouse B? How far is the ship from shore? The ship is 9. 7 ft from lighthouse A The ship is. 5 ft from lighthouse B The ship is ft from shore 6. A student takes his dogs for a walk. He lets them off their leash in a field where Edison runs at 0 m/s and Einstein runs at 8 m/s. The student determines the angle between the dogs is 5, how far are the dogs from each other in 5 seconds? The dogs are. 8 m apart Pre-Calc Trig ~8~ NJCTL.org

9 Pythagorean Identities Class Work Simplify the expression 65. csc x tan x 66. cot x sec x sec x 67. (csc x ) 68. ( + cot x)( cos x) cos x 69. tan x sec x cos x 70. ( cos x) 7. cot x sin x 7. cosx secx+tanx csc x 7. tan x + cos x sec x Verify the Identity 7. ( )( + ) = cos x 75. sin x cos x tan x cot x sec x = cos x sec x cos x 76. ( cos x)( + tan x) = tan x = sec x sec x+tan x sec x tan x (sin x)(sec x) (sin x) ( ) cos x (sec x tan x)+(sec x+tan x) sec x tan x sec x tan x sec x Pre-Calc Trig ~9~ NJCTL.org

10 Pythagorean Identities Home Work Simplify the expression 78. (tan x + cot x ) 79. sec x + csc x cos x + cos x sec x 80. cos x cos y +sin y + sin y cos x+cos y 8. csc x 0 cos x cot x 8. +sec x +tan x cos x + 8. sin x + cos x tan x cot x 8. tan x +tan x 85. cos x + sec x csc x sin x 86. +sec x + cos x +tan x cot x Verify the Identity 87. cos x sin x = sin x 88. tan x cos x csc x = ( sin x) sin x sin x ( cos x ) (cos x) ( ) 89. +cot x = + cos x 90. cos x csc x = csc x cot x ( + cos x + cos x x ) (sin ) + cos x cos x cos x cos x cos x Pre-Calc Trig ~0~ NJCTL.org

11 Angle Sum/Difference Identity Class Work Use Angle Sum/Difference Identity to find the exact value of the expression. 9. sin cos tan sin π cos 9π tan π + Verify the Identity. 97. sin (x + π ) + sin (x π ) = 98. cos (x + π ) cos (x π ) = cos x cos π + cos x sin π + cos π cos x sin π + (cos x cos π sin π ) (cos x cos π + sin π ) (cos x ) (cos x + cos x ) cos x ( cos x) cos x 99. tan (x π tan x ) = tan x+ tan x tan π +tan x tan π tan x +tan x() 00. sin(x+y) sin(x y) cos(x+y)+cos(x y) = tan y ( cos y+cos x sin y) ( cos y cos x sin y) (cos x cos y sin y)+(cos x cos y+ sin y) cos x sin y cos x cos y tan y Pre-Calc Trig ~~ NJCTL.org

12 Angle Sum/Difference Identity Home Work Use Angle Sum/Difference Identity to find the exact value of the expression. 0. sin cos tan sin π cos 7π tan 7π + Verify the Identity. 07. sin (x + π ) + sin (x π ) = 08. cos (x + π ) cos (x π ) = cos x ( cos π + cos x sin π ) + ( cos π cos x sin π ) (cos x cos π sin π ) (cos x cos π + sin π ) + (cos x ) (cos x + cos x ) cos x ( cos x) cos x 09. tan (x + 5π tan x+ ) = tan x tan x+tan 5π tan x tan 5π tan x+ tan x() 0. cos ( 5π 6 + x) cos (5π 6 x) = (cos 5π cos x sin 5π ) (cos 5π cos x + sin 5π ) ( cos x ) ( cos x + ) cos x Pre-Calc Trig ~~ NJCTL.org ( )

13 Double Angle Identity Class Work Find the exact value of the expression.. cosθ =, find cos θ if θ is in the first quadrant cosθ =, find sin θ if θ is in the fourth quadrant sinθ =, find tan θ if θ is in the third quadrant sinθ =, find cos θ if θ is in the fourth quadrant tanθ = 5, find sin θ if θ is in the second quadrant cotθ = 5, find tan θ if θ is in the third quadrant Verify the Identity. 7. = sin x 8. tan x = tan x tan x tan x sin(x + x) tan(x + x) cos x + cos x ( cos x) cos x + ( sin x) cos x + sin x ( sin x) + sin x sin x tan x+tan x tan x tan x tan x tan x tan x = cos x cos x 0. csc x = csc x cos x tan x+tan x tan x tan x tan x tan +tan x x tan x tan tan x x tan x tan x tan x cos x ( cos x) cos x cos x cos x cos x cos x csc x cos x Pre-Calc Trig ~~ NJCTL.org

14 Double Angle Identity Home Work Find the exact value of the expression.. cosθ =, find cos θ if θ is in the first quadrant. 8. cosθ =, find sin θ if θ is in the fourth quadrant sinθ = 5, find tan θ if θ is in the third quadrant sinθ = 5, find cos θ if θ is in the fourth quadrant tanθ =, find sin θ if θ is in the second quadrant cotθ =, find tan θ if θ is in the third quadrant Verify the Identity. 7. sec x = sec x sec x cos x cos x cos x cos x cos x 8. + = + sec x cscx + cos x + csc x sec x + sec x sec x 9. + cos 0x = cos 5x + ( cos 5x ) cos 5x Pre-Calc Trig ~~ NJCTL.org

15 Half Angle Identity Class Work Find the exact value of the expression. cos 6x 0.. cos ( x ) sin ( x ) cos x. sin.5. tan 67.5 or + Verify the Identity.. sec x = ± tanx tan x+ = x sin cos x + cos x cos x + cos x cos x cos x cos x (+cos x) +cos x +cos x cos x sec x Half Angle Identity Home Work Find the exact value of the expression. +cos x cos ( x ) sin (x ) cos x 7. cos.5 8. tan or Verify the Identity. 9. tan x = csc x cot x = cos x cos x csc x cot x Pre-Calc Trig ~5~ NJCTL.org

16 Power Reducing Identity Class Work Simplify the expression. 0. cos x. sin 8 x + cos x + 5 cos x 8 8 cos x + cos x 8 8 cos x cos x + cos 8x 8 8. sin x cos x cos x cos x + cos x cos x Find sin θ if cos θ = and θ is in the first quadrant Find cos θ if tan θ = and θ is in the third quadrant. 5 7( 5 ) Pre-Calc Trig ~6~ NJCTL.org

17 Power Reducing Identity Home Work Simplify the expression. 5. sin x cos x 6. sin x cos x 8 8 cos x 8 cos x + cos 8x 8 7. sin x cos x + cos x cos x cos x cos x Find sin θ if cos θ = and θ is in the fourth quadrant Find cos θ if sin θ = 7 and θ is in the third quadrant. (7 ) Pre-Calc Trig ~7~ NJCTL.org

18 Sum to Product Identity Class Work Find the exact value of the expression. 50. sin 75 + sin 5 5. cos 75 cos 5 5. cos 75 + cos Verify the Identity. + sin5x 5. = tanx cos x+cos5x 5. + sin y x y = cot cos x cos y 55. cos x+cos x = cot x sin 6x x cos cos 6x x cos cos x tan x Sum to Product Identity Home Work Find the exact value of the expression. +y cosx y +y sinx y cosx y y cot x y cos x x cos x cos cos x cot x 56. sin 05 + sin cos 05 cos cos 05 + cos Verify the Identity. cosx+cosx 59. = cotx 60. +sin 5x+ = tan x +sinx cos x+cos 5x+cos x cos 6x cosx sin 6x cosx cos x cot x 6. cos 87 + cos = sin 6 cos 0 cos 5 cos 60 cos 7 sin(90 7) sin 6x x cos + cos 6x x cos + cos x cos( x) + cos x cos( x) + cos x ( cos( x) + ) cos x ( cos( x) + ) cos x tan x sin 6 Pre-Calc Trig ~8~ NJCTL.org

19 Product to Sum Identity Class Work Find the exact value of the expression. 6. cos 75 cos 5 6. sin 7.5 sin sin 5.5 cos cos 6x + 5 sin 0x 5 Product to Sum Identity Home Work Find the exact value of the expression. 66. cos 7.5 cos sin 5 sin cos 95 sin sin 8x cos x sin 0x + sin 6x Pre-Calc Trig ~9~ NJCTL.org

20 Inverse Trig Functions Class Work Evaluate the expression. 70. sin (cos 5 ) 70. cos (tan 6 5 ) tan (sin ) 7. sin (tan 7 ) cos (sin 6 ) 7. tan (cos ) sin (sin π ) 76. sin (sin π ) π undefined 77. cos (cos π ) 78. cos (cos π ) π undefined Inverse Trig Functions Home Work Evaluate the expression. 79. sin (cos ) 80. cos (tan 7 ) tan (sin ) 8. sin (tan 5 ) cos (sin 9 ) 8. tan (cos ) sin (sin π 6 ) 86. sin (sin 5π 6 ) π 6 undefined 87. cos (cos π ) 88. cos (cos π ) π undefined Pre-Calc Trig ~0~ NJCTL.org

21 Trig Equations Class Work Find the value(s) of x such that 0 x < π, if they exist. 89. = 90. tan x = x = π x = π 6, 5π 6, 7π 6, π 6 9. sec x = 0 9. sin x + = 7 x = π, π, 5π, 7π x = π 6, 5π 6 9. csc x = 9. sec x = x = π 6, 5π 6, 7π 6, π 6 x = π 6, 5π 6, 7π 6, π sin x cos x = ( + ) = cos x x = 0, π, π, 5π x = π cos x = cos x = 0 x = π, π, 7π 6, π 6 x = π, π, 5π 99. cos x + cos x = x = 0 Pre-Calc Trig ~~ NJCTL.org

22 Trig Equations Home Work Find the value(s) of x such that 0 x < π, if they exist. 00. cos x = 0. sin x = x = π x = π, π, 5π, 7π 0. csc x = 0 0. sin x = x = π, π, 5π, 7π x = π 0. sec x = 05. csc x = x = π, π, π, 5π x = π, π, π, 5π 06. cos x cos x = ( ) = cos x x = π, π, π, 5π x = 7π, π, π = tan x 09. tan x = 0 x = 0, π, π, π x = 0, π, π, π 0. = 0 x = 0, π, π, 5π Pre-Calc Trig ~~ NJCTL.org

23 . Given the terminal point of ( a. b. π c. - d.. Knowing sec x = 5 a. b. c. d. π 5 5, Trigonometry Unit Review Multiple Choice ) find tan θ. and the terminal point is in the second quadrant find cot θ.. What is the phase shift of y = 5 cos(6x π) +? a. b. π π c. d. π. The difference between the maximum of y = cos ( (x + π )) + and y = cos(x π) is a. b. A c. d Given ABC, with A = 5, a = 5, & c = 7, find B. a. 8.8 b. 5.8 C c d. both a and b 6. Given ABC, with A = 50, a = 6, & c = 8, find B. a..0 b. 0 D c d. no solution 7. Given ABC, with A = 50, b = 6, & c = 8, find B. a b..56 C c d (sec x + tan x)(sec x tan x) = a. + sec x tan x b. sec x tan x c. d. cos x C C B D Pre-Calc Trig ~~ NJCTL.org

24 9. Find the exact value of sin π a. b. c d. 0. On the interval [0, π), = 0, thus x = a. 0 b. π π c. d. all of the above. Find the exact value of cos 05 a. b. c. + d. +. sin x = a. b. c. d ( cos x + cos x) ( + cos x + cos x) ( + cos x + cos x) ( cos x + cos x). Rewrite cos 6x as a sum or difference. a. b. c. cos 0x cosx cos 0x + cosx sin 0x sinx d. sin 0x sinx. On the interval [0, π), sin 5x + = 0 a. b. c. π kπ kπ, where k Integers, where k {0,,,6} d. no solution on the interval given 5. sin (sin π ) = a. π b. π c. both a and b d. Undefined A D B D D B C Pre-Calc Trig ~~ NJCTL.org

25 6. On the interval [0, π), solve sin x + cos x = I. 0 II. π III. 5π a. I only b. II and III c. I and III d. I, II, and III D Extended Response. The range of a projectile launched at initial velocity v 0 and angle θ, is r = 6 v 0 sin θ cos θ, where r is the horizontal distance, in feet, the projectile will travel. a. Rewrite the formula using double angle formula. r = v 0 sin θ b. A golf ball is hit 00 yards, if the initial velocity 00 ft/sec, what was the angle it was hit? θ =. o c. If the golfer struck the ball at 5, how far would the ball traveled? r = 50 feet. A state park hires a surveyor to map out the park. a. A and B are on opposite sides of the lake, if the surveyor stands at point C and measures angle ACB= 50 and CA= 00 and CB= 50, how wide is the lake? 0. feet b. At a river the surveyor picks two spots, X and Y, on the same bank of the river and a tree, C, on opposite bank. Angle X= 60 and angle Y= 50 and XY=00, how wide is the river? (Remember distance is measured along perpendiculars.). 8 feet c. The surveyor measured the angle to the top of a hill at the center of the park to be. She moved 00 closer and the angle to the top of the hill was. How tall was the hill? feet Pre-Calc Trig ~5~ NJCTL.org

26 . The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by M = 9.6 sin ( πd +.6) where d is the day, d= is January first. a. What is the period of the function? 65 b. What is the average daily production for the year? 566 gallons c. Using the graph of M(d), what months during the year is production over 5500 gallons a day? February thru May. A student was asked to solve the following equation over the interval [0, π). During his calculations he might have made an error. Identify the error and correct his work so that he gets the right answer. cos x + = cos x + cos x + = sin x cos x + cos x + = cos x cos x = 0 cos x = 0 π, π Error is on line Line should read cos x + cos x = 0 The rest of the problem is cos x (cos x + ) = 0 cos x = 0 cos x + = 0 cos x = 0 cos x = x = π, π, π Pre-Calc Trig ~6~ NJCTL.org