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.

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1 Unit Circle Class Work Find the exact value of the given expression. 1. cos π 3. sin 7π 3. sec π 3. tan 5π 6 5. cot 15π 6. csc 9π 7. Given the terminal point ( 3, 10 ) find tanθ Given the terminal point ( 5, ) find cotθ 9. Knowing cosx= and the terminal point is in the fourth quadrant find sinx Knowing cotx= and the terminal point is in the third quadrant find secx. 5 Pre-Calc Trig ~1~ NJCTL.org

2 Unit Circle Home Work Find the exact value of the given expression. 11. cos 5π 3 1. sin 3π 13. sec π 3 1. tan 7π cot 13π 16. csc 11π 17. Given the terminal point ( 7, 5 5 ) find cotθ 18. Given the terminal point ( 9, 7 ) find tanθ Knowing sinx= 7 and the terminal point is in the second quadrant find secx Knowing cscx= and the terminal point is in the third quadrant find cotx. 5 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. 1. y = cos ( (x + π )) + 1. y = 3 cos(x π) 3 3. y = sin ( (x + π )) + 3. y = 1 cos(3x π) y = cos(x π) + 3 Pre-Calc Trig ~3~ 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 ( 1 (x π )) + 7. y = cos(x 3π) y = sin ( 1 (x + π )) y = 1 cos(6x π) y = 3 cos(x 3π) Pre-Calc Trig ~~ NJCTL.org

5 Law of Sines Class Work Solve triangle ABC. 31. A = 70, B = 30, c = 3. B = 65, C = 50, a = b = 6, A = 5, B = 5 3. c = 8, B = 60, C = c = 1, b = 6, C = b = 1, a = 15, B = A = 35, a = 6, b = 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 35 when the plane is directly between them. How far is the plane from JFK? What is the plane s elevation? 39. 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? Pre-Calc Trig ~5~ NJCTL.org

6 Law of Sines Home Work Solve triangle ABC. 0. A = 60, B = 0, c = 5 1. B = 75, C = 50, a = 1. b = 6, A = 35, B = 5 3. c = 8, B = 50, C = 0. c = 1, b = 8, C = b = 1, a = 16, B = A = 0, a = 5, b = 1 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? 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 30 toward the tree, the angle is 60. How far is she from the bird? Pre-Calc Trig ~6~ NJCTL.org

7 Law of Cosines Class Work Solve triangle ABC. 9. a = 3, b =, c = a = 5, b = 6, c = a = 7, b = 6, c = 5. A = 100, b =, c = B = 60, a = 5, c = 9 5. C = 0, a = 10, b = A ship at sea noticed two lighthouses that according to the charts are 1 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 300 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? 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? Pre-Calc Trig ~7~ NJCTL.org

8 Law of Cosines Home Work Solve triangle ABC. 57. a =, b = 5, c = a =, b = 10, c = a = 11, b = 8, c = A = 85, b = 3, c = B = 70, a = 6, c = 1 6. C = 5, a = 1, b = A ship at sea noticed two lighthouses that according to the charts are 1 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 35 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? 6. A student takes his dogs for a walk. He lets them off their leash in a field where Edison runs at 10 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? Pre-Calc Trig ~8~ NJCTL.org

9 Pythagorean Identities Class Work Simplify the expression 65. csc x tan x 66. cot x sec x sin x 67. sin x (csc x sin x) 68. (1 + cot x)(1 cos x) tan x sec x 70. (sin x cos x) 71. cot x 1 sin x 7. cosx secx+tanx 73. sin x tan x + cos x Verify the Identity 7. (1 sin x)(1 + sin x) = cos x 75. tan x cot x sec x = cos x 76. (1 cos x)(1 + tan x) = tan x sec x+tan x + 1 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 ) sin x cos x + cos x 1 sin x 80. cos x cos y sin x+sin y + sin x sin y cos x+cos y sin x csc x 8. 1+sec x 1+tan x 83. sin x + cos x tan x cot x 8. tan x 1+tan x 85. cos x + sin x sec x csc x sec x + cos x 1+tan x cot x Verify the Identity 87. cos x sin x = 1 sin x 88. tan x cos x csc x = cot x = sin x + cos x 90. cos x csc x = 1 csc x cot x Pre-Calc Trig ~10~ NJCTL.org

11 Angle Sum/Difference Identity Class Work Use Angle Sum/Difference Identity to find the exact value of the expression. 91. sin cos tan sin π cos 19π tan π 1 Verify the Identity. 97. sin (x + π 3 ) + sin (x π 3 ) = sin x 98. cos (x + π ) cos (x π ) = cos x tan (x π tan x 1 ) = tan x sin(x+y) sin(x y) cos(x+y)+cos(x y) = tan y Pre-Calc Trig ~11~ NJCTL.org

12 Angle Sum/Difference Identity Home Work Use Angle Sum/Difference Identity to find the exact value of the expression sin cos tan sin 11π cos 17π tan 7π 1 Verify the Identity sin (x + π 3 ) + sin (x π 3 ) = sin x 108. cos (x + 3π ) cos (x 3π ) = cos x tan (x + 5π tan x+1 ) = 1 tan x 110. cos ( 5π 6 + x) cos (5π 6 x) = 3 sin x Pre-Calc Trig ~1~ NJCTL.org

13 Double Angle Identity Class Work Find the exact value of the expression cosθ = 1, find cos θ if θ is in the first quadrant. 11. cosθ = 1, find sin θ if θ is in the fourth quadrant sinθ = 3, find tan θ if θ is in the third quadrant sinθ = 3, 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. 9 Verify the Identity sin 3x = 3 sin x sin 3 x 118. tan 3x = 3 tan x tan3 x 1 3tan x sin x sin x = cos x cos x 10. csc x = csc x cos x Pre-Calc Trig ~13~ NJCTL.org

14 Double Angle Identity Home Work Find the exact value of the expression. 11. cosθ = 3, find cos θ if θ is in the first quadrant. 1. cosθ = 3, find sin θ if θ is in the fourth quadrant. 13. 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. 9 Verify the Identity. 17. sec x = sec x sec x sin x sin x = sec x cscx cos 10x = cos 5x Pre-Calc Trig ~1~ NJCTL.org

15 Half Angle Identity Class Work Find the exact value of the expression. 1 cos 6x cos ( x ) sin ( x ) 13. sin tan 67.5 Verify the Identity. 13. sec x = ± tanx tan x+sin x Half Angle Identity Home Work Find the exact value of the expression. 1+cos x cos ( x ) sin (x ) 137. cos tan 15 Verify the Identity tan x = csc x cot x Pre-Calc Trig ~15~ NJCTL.org

16 Power Reducing Identity Class Work Simplify the expression. 10. cos x 11. sin 8 x 1. sin x cos x 13. Find sin θ if cos θ = 3 and θ is in the first quadrant Find cos θ if tan θ = 3 and θ is in the third quadrant. 5 Pre-Calc Trig ~16~ NJCTL.org

17 Power Reducing Identity Home Work Simplify the expression. 15. sin x cos x 16. sin x cos x 17. sin x cos x 18. Find sin θ if cos θ = 3 and θ is in the fourth quadrant Find cos θ if sin θ = 7 and θ is in the third quadrant. Pre-Calc Trig ~17~ NJCTL.org

18 Sum to Product Identity Class Work Find the exact value of the expression sin 75 + sin cos 75 cos cos 75 + cos 15 Verify the Identity. sin x+ sin5x 153. = tan3x cos x+cos5x 15. sin x + sin y x y = cot cos x cos y 155. cos x+cos 3x sin 3x sin x = cot x Sum to Product Identity Home Work Find the exact value of the expression sin sin cos 105 cos cos cos 15 Verify the Identity. cosx+cosx 159. = cot3x 160. sin x+sin 5x+sin 3x = tan 3x sin x+sinx cos x+cos 5x+cos 3x 161. cos 87 + cos 33 = sin 63 Pre-Calc Trig ~18~ NJCTL.org

19 Product to Sum Identity Class Work Find the exact value of the expression. 16. cos 75 cos sin 37.5 sin sin 5.5 cos cos 6x sin x Product to Sum Identity Home Work Find the exact value of the expression cos 37.5 cos sin 5 sin cos 195 sin sin 8x cos x Pre-Calc Trig ~19~ NJCTL.org

20 Inverse Trig Functions Class Work Evaluate the expression sin (cos ) 170. cos (tan ) 171. tan (sin 1 3 ) 17. sin (tan ) 173. cos (sin 1 6 ) 17. tan 11 (cos 1 3 ) sin 1 (sin π ) 176. sin 1 (sin 3π ) 177. cos 1 (cos π 3 ) 178. cos 1 (cos π 3 ) Inverse Trig Functions Home Work Evaluate the expression sin (cos ) 180. cos 13 (tan 1 7 ) tan (sin 1 1 ) 18. sin (tan ) 183. cos (sin 1 9 ) 18. tan 11 (cos 1 ) sin 1 (sin π 6 ) 186. sin 1 (sin 5π 6 ) 187. cos 1 (cos π 3 ) 188. cos 1 (cos π 3 ) 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 sin x = tan x = sec x = sin x + 3 = 7 sin x 193. csc x = 19. 3sec x = 195. sin x cos x sin x = (sin x + 1) = cos x 197. sin x + cos x = sin x + cos x = cos x + cos x = Pre-Calc Trig ~1~ NJCTL.org

22 Trig Equations Home Work Find the value(s) of x such that 0 x < π, if they exist. 00. cos x = sin x = 1 0. csc x = sin x 3 = sin x 0. sec x = 05. 3csc x = 06. cos x cos x sin x = (sin x 1) = cos x 08. sin x = tan x 09. tan x sin x = sin x sin x = 0 Pre-Calc Trig ~~ NJCTL.org

23 1. Given the terminal point of ( a. b. π c. -1 d. 1. Knowing sec x = 5 a. b. c. d. π , Trigonometry Unit Review Multiple Choice ) find tan θ. and the terminal point is in the second quadrant find cot θ. 3. What is the phase shift of y = 5 cos(6x π) + 3? 3 a. b. 1 π π 3 1 c. 3 d. π. The difference between the maximum of y = cos ( (x + π )) + 1 and y = 3 cos(x π) is 3 a. 1 b. c. 3 d Given ABC, with A = 35, a = 5, & c = 7, find B. a b c d. both a and b 6. Given ABC, with A = 50, a = 6, & c = 8, find B. a b. 0 c d. no solution 7. Given ABC, with A = 50, b = 6, & c = 8, find B. a b c d (sec x + tan x)(sec x tan x) = a. 1 + sec x tan x b. 1 sec x tan x c. 1 d. 1 sin x cos x Pre-Calc Trig ~3~ NJCTL.org

24 9. Find the exact value of sin π 1 a. b. c d. 10. On the interval [0, π), sin x = 0, thus x = a. 0 b. π 3π c. d. all of the above 11. Find the exact value of cos 105 a. 3 b. 3 c. + 3 d sin x = a. b. c. d (3 cos x + cos x) (3 + cos x + cos x) (3 + cos x + cos x) (3 cos x + cos x) 13. Rewrite cos 6x sin x as a sum or difference. a. b. c. 1 cos 10x 1 cosx 1 cos 10x + 1 cosx 1 1 sin 10x sinx d. sin 10x 1 sinx 1. On the interval [0, π), sin 5x + sin 3x = 0 a. b. c. π kπ kπ, where k Integers, where k {0,1,,6} d. no solution on the interval given 15. sin 1 (sin π 3 ) = a. π 3 b. π 3 c. both a and b d. Undefined Pre-Calc Trig ~~ NJCTL.org

25 16. On the interval [0, π), solve sin x + 3 cos x = 3 I. 0 II. π III. 5π 3 3 a. I only b. II and III c. I and III d. I, II, and III Extended Response 1. The range of a projectile launched at initial velocity v 0 and angle θ, is r = 1 16 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. b. A golf ball is hit 00 yards, if the initial velocity 00 ft/sec, what was the angle it was hit? c. If the golfer struck the ball at 5, how far would the ball traveled?. 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= 350, how wide is the lake? 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=300, how wide is the river? (Remember distance is measured along perpendiculars.) c. The surveyor measured the angle to the top of a hill at the center of the park to be 3. She moved 00 closer and the angle to the top of the hill was 3. How tall was the hill? Pre-Calc Trig ~5~ NJCTL.org

26 3. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by M = 19.6 sin ( πd + 1.6) where d is the day, d=1 is January first. a. What is the period of the function? b. What is the average daily production for the year? c. Using the graph of M(d), what months during the year is production over 5500 gallons a day?. 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 + 1 = sin x cos x + cos x + 1 = sin x cos x + cos x + 1 = 1 cos x cos x = 0 cos x = 0 π, 3π Pre-Calc Trig ~6~ NJCTL.org