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CHAPTER 8: TRIGONOMETRY # 0 1 2 3 4 5 6 7 8 9 Title Right Triangles + Magic = Trigonometry (DUE ON 2/5) – NO EXCEPTIONS! Video: What is SOH CAH TOA? SOH CAH TOA: The Set Up Solving for a missing side Solving for a missing angle Connecting the dots of SOH CAH TOA Video: Drawing and Word Problems Drawing and Word Problems Review CHAPTER TEST Complete? February (2016) 3 4 Chp 7 Test 8 #3 Solving a Missing Side/ #4 Solving a Missing Angle HW: Video #6 9 #5Connecting the dots of SOH CAH TOA #7 Drawing and Word Problems 5 #2 SOH CAH TOA: The Set Up HW: Right Triangles+Magic=Trig #3 Solving a Missing Side Video #1 10 11 #7 Drawing and #8 Chapter Review Word Problems TEST 12 Name _________________________________ CC Geometry Video: What is SOH CAH TOA? Chp. 8 Wksht #1 SOH CAH TOA The Sine Ratio (sin) Opposite sin Hypotenuse The Cosine Ratio (cos) Adjacent cos Hypotenuse The Tangent Ratio (tan) Opposite tan Adjacent These ratios ONLY work in ______________ triangles! Finding the opposite, adjacent and hypotenuse: A Angle A Opposite = Angle C Opposite = Adjacent = Adjacent = B C The Opposite and Adjacent sides ALWAYS depend on the _________________ ____________. The hypotenuse is always located ______________ from the right angle. Examples: A 60o 5 1 3 B 30o C 4 2 √𝟑 sin A= sin B= sin 30= sin 60= cos A= cos B= cos 30= cos 60= tan A= tan B= tan 30= tan 60= Solving for a missing side: First, you must choose...... sin cos tan Now, actually solve them...... Solving for a missing angle..... sin cos tan sin cos tan Name _________________________________ CC Geometry The Sine Ratio (sin) Opposite sin Hypotenuse SOH CAH TOA: The Set Up Chp. 8 Wksht #2 The Cosine Ratio (cos) Adjacent cos Hypotenuse The Tangent Ratio (tan) Opposite tan Adjacent 1. Match the following. a) _______ Opposite Leg to A C b) _______ Sine Ratio of C c) _______ Opposite Angle to AB B A d) _______ The Hypotenuse 1. A e) _______ Adjacent Leg to A BC AC 8. AB AC 9. BC AB 2. B f) _______ Tangent Ratio of C BC g) _______ Reference angle if is the Cosine Ratio. AC h) _______ Adjacent Leg to C 3. C 4. AB i) ________ Cosine Ratio of A 5. BC j) ________ The Longest Side k) _______ Reference angle if 7. BC is the Sine Ratio. AC 10. 6. AC AB BC 2. Label the sides of the triangle using the reference angle -- (O) for Opposite, (A) for Adjacent and (H) for Hypotenuse. After you have labeled the triangle, then choose which trigonometric ratio that you would use to solve for the missing info. a) b) c) C 12 cm x 29 cm B 30° C A A B x d) 13° A 21 cm 25 cm B θ θ C 34 cm C B A SIN COS TAN e) SIN COS TAN f) C B x 66° SIN COS TAN g) A 18 cm 67° 12 cm 8 cm B B x x C A TAN SIN COS TAN SIN COS TAN 63° 34 cm B COS TAN 35° C A SIN COS h) C x SIN 25 cm SIN COS A TAN Name _________________________________ CC Geometry Solving for a missing side Chp. 8 Wksht #3 Name _________________________________ CC Geometry Solving for a missing angle Chp. 8 Wksht #4 Name _________________________________ CC Geometry Connecting the dots of SOH CAH TOA Chp. 8 Wksht #5 1. A teacher asks (while looking at the trigonometry table), “What does the 0.6691 mean?” How would you respond to that? Degrees 42 Sine 0.6691 Cosine 0.7431 Tangent 0.9004 2. Thomas sees two triangles on the board in geometry class. From those he makes two claims: CD EF . How could he know this without having any CB EG numbers on the sides? F B #1) that C 24° E 24° D G CD 0.9135, how could he know CB this without any numbers on the sides? What did he type into his calculator to get this value? #2) that after clicking a couple of buttons on his calculator he states that 3. Sarah, the student who sits next to you in geometry class, notices that the values for sine and cosine are only from 0 to 1 for the angles 0 to 90. She leans over and asks, “Why can’t sine and cosine be greater than 1?” How would you respond to her? 4. A student who did very well in Algebra 1 looked at this trigonometry problem, and said “What a minute, Tangent is the same as slope!!” Why would she says this? How is tangent the same as slope? 14 cm θ 38 cm 5. Jessy claims that AB is the opposite side. Is he correct? Explain. C o A θ B 6. What does similarity have to do with Trigonometry? This stuff is IMPORTANT! 7. When looking closely at the trigonometry table a student notices that certain sine values are the same as certain cosine values (partial table shown). What do the angles that have the same value have in common? Degree 10 9 8 7 6 5 4 3 2 1 0 Sine 0.1736 0.1564 0.1392 0.1219 0.1045 0.0872 0.0698 0.0523 0.0349 0.0175 0.0000 Degree 80 81 82 83 84 85 86 87 88 89 90 Cosine 0.1736 0.1564 0.1392 0.1219 0.1045 0.0872 0.0698 0.0523 0.0349 0.0175 0.0000 8. Why does the sin Ɵ = cos (90 – Ɵ)? 9. Solve the following. a) sin 42 = cos _______ d) cos 0 = sin _______ b) cos 12 = sin _______ c) sin 45 = cos _______ e) cos 65 = sin _______ f) sin 78.5 = cos _______ b) sin (2x – 17) = cos (x – 4) 3 1 c) sin ( x ) = cos ( x ) 4 4 10. Solve for the unknown. a) sin (x – 5) = cos (35) Name _________________________________ CC Geometry THE ANGLE OF ELEVATION THE ANGLE OF DEPRESSION Video: Drawing and Word Problems Chp. 8 Wksht #6 Solve the given word problem. 1. A 3.4 guy wire is attached to a tree 3 feet from the ground. What is the angle that is formed between the wire and the ground (to the nearest degree)? 2. A 15 feet ladder leans against a wall at 52. How far from the wall is the foot of the ladder (to the nearest foot)? 3. The sun’s ray strike the ground at 55, 21 m from the base of the tree. What is the height of the tree (to the nearest meter)? 4. A little boy flies his kite. The string formes an angle of elevation of 37 and from where he stands to directly under the kite is 45 ft. How long is the kite string (to the nearest ft)? 5. A 10 feet ladder reaches a window that is 8 feet up from the ground. What is the angle that is formed between the ladder and the wall (to the nearest degree)? 6. An airplane spots the floating debris at an angle of depression of 15. If the plane is at an altitude of 3,000 feet, what is the horizontal distance before they fly over it (to the nearest hundred feet)? Name _________________________________ CC Geometry Drawing and Word Problems Chp. 8 Wksht #7 1. Circle (or Draw) the side or angle that is represented by the description. a) The Leaning Ladder Height on the wall that the ladder reaches. b) The Leaning Ladder The distance from the foot of the ladder to the wall. e) Flying a Kite c) The Leaning Ladder The angle the ladder forms with the wall. d) The Shadow The length of his shadow. f) Flying a Kite The length of the string. The height of the kite. What are some of the assumptions that are made about the kite example so that it works easily as a trigonometry question? g) The Support Guy Wire h) The Support Guy Wire The distance from the base of the tree to where the guy wire is fastened to the ground. The angle between the antenna and the guy wire. i) The Support Guy Wire The height of where the guy wire is fastened to the antenna. j) The Support Guy Wire The angle formed between the wire and the ground. What are some of the assumptions that are made about the guy wire example so that it works easily as a trigonometry question? 2. Create the diagram for the following descriptions. Label the diagram completely including putting the x for the unknown missing value. a) A young boy lets out 30 ft of string on his kite. If the angle of elevation from the boy to his kite is 27°, how high is the kite? DIAGRAM b) A 20 ft ladder leans against a wall so that it can reach a window 18 ft off the ground. What is the angle formed at the foot of the ladder? DIAGRAM c) To support a young tree, Jack attaches a guy wire from the ground to the tree. The wire is attached to the tree 4 ft above the ground. If the angle formed between the wire and the tree is 70°, what is the length of the wire? DIAGRAM d) A helicopter is directly over a landing pad. If Billy is 110 ft from the landing pad, and looks up to see the helicopter at 65° to see it. How high is the helicopter? DIAGRAM 3. Solve the following problems. (All answers to 2 decimals places, unless otherwise instructed.) a) A tree casts a shadow 21 m long. The angle of elevation of the sun is 55. What is the height of the tree? b) You are flying a kite and have let out 30 ft of string but it got caught in a 8 ft tree. What is the angle of elevation to the location of the kite? c) A 15 m pole is leaning against a wall. The foot of the pole is 10 m from the wall. Find the angle that the pole makes with the ground. d) A lighthouse operator sights a sailboat at an angle of depression of 12. If the sailboat is 80 m away, how tall is the lighthouse? 4. a) Using the drawbridge diagram, determine the distance from one side to the other. (exact answer) 45° 40 ft 45° 40 ft b) Now that you know the distance from side to side, determine how high the drawbridge would be if the angle of elevation was 60.(exact answer) 60° 60° 40 ft c) How far apart would the drawbridge be if the angle of elevation of the drawbridge was 20? x 20° 20° For each problem, first draw the diagram and then solve for the requested information. (All answers to 2 decimals places, unless otherwise instructed.) 5. An airplane is flying at an altitude of 6000 m over the ocean directly toward an island. When the angle of depression of the coastline from the airplane is 14, how much farther does the airplane have to fly before it crosses the coast? 6. A loading ramp is 25 m long with a height of 10 m. What is the horizontal distance of the ramp and what is the angle of incline that the ramp forms with the ground? 7. A telephone pole casts a shadow 18 m long when the sun’s rays strike the ground at an angle of 70. How tall is the pole? For each problem, first complete the diagram and then solve for the requested information. 8. From an apartment window 24 m above the ground, the angle of depression of the base of a nearby building is 38 and the angle of elevation of the top is 63. Find the height of the nearby building (to the nearest meter). 24 m 9. A flagpole is at the top of a building. 400 ft from the base of the building, the angle of elevation of the top of the pole is 22 and the angle of elevation of the bottom of the pole is 20. Determine the length of the flagpole (to the nearest foot). 22° 20° 400 ft 10. From a lighthouse 1000 ft above sea level, the angle of depression to a boat (A) is 29. A little bit later the boat has moved closer to the shore (B) and the angle of depression measures 44. How far (to the nearest foot) has the boat moved in that time? 29° 44° 1000 ft A B 11. Jack and Jill are on either side of the church and 50 m apart. Jack sees the top of the steeple at 40 and Jill sees the top of the steeple at 32. How high is the steeple? h 32° 40° 50 m 12. Jack and Jill are 20 m apart. Jack sees the top of the building at 30 and Jill sees the top of the building at 40. What is the height of building? h 30° 20 m 40° x Name _________________________________ CC Geometry Chapter Review Chp. 8 Wksht #8 Vocabulary: Sine Cosine Tangent 1. Which of the following is equal to cos 35 A) sin 35 B) cos 55 C) sin 55 D) cos 145 1. _________ 2. If cos Ɵ = sin ß then which of the following must be true? A) Ɵ + ß = 180 B) Ɵ - ß = 90 C) ß = 90Ɵ D) ß - Ɵ = 90 2. _________ 3. The angle of depression from the girl to the car is: A) 1 B) 2 C) 3 D) 4 3. _________ 4. Julie has a large red apple in her hand that is 4 ft off the ground. A blue bird sees the apple at an angle of depression of 55. If Julie is 15 ft from the tree, how tall is the tree (round to the nearest foot)?? A) 16 ft B) 17 ft C) 21 ft 4. _________ D) 25 ft 5. A ladder reaches a window 12 ft above the ground and the foot of the ladder is 4.8 ft from the wall. How long is the ladder? A) 14 ft B) 13 ft C) 12 ft 5. _________ D) 11 ft 6. A lighthouse operator sights a sailboat at an angle of depression of 25. If the lighthouse is 40 ft tall, how far is the boat from the base of the lighthouse? A) 95 ft B) 86 ft C) 44 ft 6. _________ d) 19 ft 7. Solve for the unknown. a) sin (22)= cos (x) b) sin (x + 18) = cos (45) c) sin (2x – 15) = cos (x – 12) 8. A guy wire is attached to a tree 3.5 ft above the ground to stabilize it. If the guy wire forms an angle with the tree of 50, what is the length of the guy wire? (2 decimal places) 9. A 15 ft ladder is leaning against a wall. The foot of the ladder is 4 ft from the wall. Find the angle that the pole makes with the ground. (2 decimal places) 10. A man stands between two trees and he is 70 ft from the tall tree and 50 ft from the shorter tree. If he sees the taller tree at an angle of 38 and the smaller at 45 , what is the difference in the heights of the two trees (to the nearest foot) ? 11. Two bird watchers position themselves at point A (the beach) and point C (the shed) which are 111 ft apart. The both spot the rare Blue Breasted Turk at point B using their binoculars. If they see the Blue Breasted Turk at 94 and 46 respectively, how far is the bird from the bird watcher in the shed (point C) (round to the nearest ft)? B 46° A 12. Some marine biologists are studying rare red belly salmon in the north portion of the lake. They have gathered some of the measurements of the area but still need to determine the width (from A to B) of the north portion of the lake. Determine the width of the north lake (from A to B) to the nearest foot. 94° 111 ft B 178 m 53° C 589 ft A C