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2 trigonometric ratios conglomerate keep

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2 trigonometric ratios conglomerate keep

1. 1. Trigonometric Ratios  MM2G2. Students will define and apply sine, cosine, and tangent ratios to right triangles.  MM2G2a: Discover the relationship of the trigonometric ratios for similar triangles.
2. 2. Trigonometric Ratios  MM2G2b: Explain the relationship between the trigonometric ratios of complementary angles.  MM2G2c: Solve application problems using the trigonometric ratios.
3. 3. The following slides have been come from the following sources: www.mccd.edu/faculty/bruleym/.../trigonome http://ux.brookdalecc.edu/fac/cos/lsch melz/Math%20151/ www.scarsdaleschools.k12.ny.us / 202120915213753693/lib/…/trig.ppt Emily Freeman McEachern High School
4. 4. Warm Up Put 4 30-60-90 triangles with the following sides listed and have students determine the missing lengths. 30 S 5 2 7√3 √2 90 H 10 4 14√3 2√2 60 L 5√3 2√3 21 √6
5. 5. Trigonometric Ratios  Talk about adjacent and opposite sides: have the kids line up on the wall and pass something from one to another adjacent and opposite in the room.  Make a string triangle and talk about adjacent and opposite some more
6. 6. Trigonometric Ratios  Determine the ratios of all the triangles on the board and realize there are only 3 (6?) different ratios.  Talk about what it means for shapes to be similar.  Make more similar right triangles on dot paper, measure the sides, and calculate the ratios.
7. 7. Trigonometric Ratios  Try to have the students measure the angles of the triangles they made on dot paper.  Do a Geosketch of all possible triangles and show the ratios are the same for similar triangles  Finally: name the ratios
8. 8. Warm Up  Pick up a sheet of dot paper, a ruler, and protractor from the front desk.  Draw two triangles, one with sides 3 & 4, and the other with sides 12 & 5  Calculate the hypotenuse  Calculate sine, cosine, and tangent for the acute angles.  Measure the acute angles to the nearest degree.  Show how to find sine, cosine, & tangent of angles in the calculator
9. 9. Yesterday  We learned the sine, cosine, and tangent of the same angle of similar triangles are the same  Another way of saying this is: The sine, cosine, tangent of congruent angles are the same
10. 10. Trigonometric Ratios in Right Triangles M. Bruley
11. 11. Trigonometric Ratios are based on the Concept of Similar Triangles!
12. 12. All 45º- 45º- 90º Triangles are Similar! 45 º 2 2 22 45 º 1 1 2 45 º 1 2 1 2 1
13. 13. All 30º- 60º- 90º Triangles are Similar! 1 60º 30º ½ 2 3 32 60º 30º 2 4 2 60º 30º 1 3
14. 14. All 30º- 60º- 90º Triangles are Similar! 10 60º 30º 5 35 2 60º 30º 1 3 1 60º 30º 2 1 2 3
15. 15. hypotenuse leg leg In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse a b c We’ll label them a, b, and c and the angles α and β. Trigonometric functions are defined by taking the ratios of sides of a right triangle. β α First let’s look at the three basic functions. SINE COSINE TANGENT They are abbreviated using their first 3 letters c a == hypotenuse opposite sinα opposite c b == hypotenuse adjacent cosα adjacent b a == adjacent opposite tanα
16. 16. The Trigonometric Functions SINE COSINE TANGENT
17. 17. SINE Prounounced “sign”
18. 18. Prounounced “co-sign” COSINE
19. 19. Prounounced “tan-gent” TANGENT
20. 20. Pronounced “theta” Greek Letter θ Represents an unknown angle
21. 21. Pronounced “alpha” Greek Letter α Represents an unknown angle
22. 22. Pronounced “Beta” Greek Letter β Represents an unknown angle
23. 23. θ opposite hypotenuse Sin Opp Hyp = adjacent Cos Adj Hyp = Tan Opp Adj = hypotenuse opposite adjacent
24. 24. We could ask for the trig functions of the angle β by using the definitions. a b c You MUST get them memorized. Here is a mnemonic to help you. β α The sacred Jedi word: SOHCAHTOA c b == hypotenuse opposite sin β adjacent cos hypotenuse a c β = = opposite tan adjacent b a β = = opposite adjacent SOHCAHTOA
25. 25. It is important to note WHICH angle you are talking about when you find the value of the trig function. a b c α Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so 222 cba =+ Let's choose: 222 543 =+3 4 5 sin α = Use a mnemonic and figure out which sides of the triangle you need for sine. h o 5 3 = opposite hypotenuse tan β = a o 3 4 = opposite adjacent Use a mnemonic and figure out which sides of the triangle you need for tangent. β
26. 26. You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle. α This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle. 3 4 5 β Oh, I'm acute! So am I!
27. 27. We need a way to remember all of these ratios…
28. 28. What is SohCahToa? Is it in a tree, is it in a car, is it in the sky or is it from the deep blue sea ?
29. 29. This is an example of a sentence using the word SohCahToa. I kicked a chair in the middle of the night and my first thought was I need to SohCahToa.
30. 30. An example of an acronym for SohCahToa. Seven old horses Crawled a hill To our attic..
31. 31. Old Hippie Some Old Hippie Came A Hoppin’ Through Our Apartmen
32. 32. SOHCAHTOA Old Hippie Sin Opp Hyp Cos Adj Hyp Tan Opp Adj
33. 33. Other ways to remember SOH CAH TOA 1.Some Of Her Children Are Having Trouble Over Algebra. 2.Some Out-Houses Can Actually Have Totally Odorless Aromas. 3.She Offered Her Cat A Heaping Teaspoon Of Acid. 4.Soaring Over Haiti, Courageous Amelia Hit The Ocean And ... 5.Tom's Old Aunt Sat On Her Chair And Hollered. -- (from Ann Azevedo)
34. 34. Other ways to remember SOH CAH TOA 1.Stamp Out Homework Carefully, As Having Teachers Omit Assignments. 2.Some Old Horse Caught Another Horse Taking Oats Away. 3.Some Old Hippie Caught Another Hippie Tripping On Apples. 4.School! Oh How Can Anyone Have Trouble Over Academics.
35. 35. A Trigonometry Ratios Tangent A = opposite adjacent Sine A = opposite hypotenuse Cosine A = adjacent hypotenuse Soh Cah Toa
36. 36. 14º 24º 60.5º 46º 82º
37. 37. 1.9 cm 7.7 cm 14º 1.9 7.7 ≈0.25 Tangent 14º ≈0.25 The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side. opposite adjacent hypotenuse
38. 38. 3.2 cm 7.2 cm 24º 3.2 7.2 ≈0.45 Tangent 24º ≈0.45 Tangent A = opposite adjacent
39. 39. 5.5 cm 5.3 cm 46º 5.5 5.3 ≈1.04 Tangent 46º ≈1.04 Tangent A = opposite adjacent
40. 40. 6.7 cm 3.8 cm 60.5º 6.7 3.8 ≈1.76 Tangent 60.5º ≈1.76 Tangent A = opposite adjacent
41. 41. As an acute angle of a triangle approaches 90º, its tangent becomes infinitely large Tan 89.9º = 573 Tan 89.99º = 5,730 Tangent A = opposite adjacent etc. very large very small
42. 42. Since the sine and cosine functions always have the hypotenuse as the denominator, and since the hypotenuse is the longest side, these two functions will always be less than 1. Sine A = opposite hypotenuse Cosine A = adjacent hypotenuse A Sine 89º = .9998 Sine 89.9º = .999998
43. 43. 3.2 cm 7.9 cm 24º 9.7 2.3 ≈0.41 Sin 24º ≈0.41 Sin α = hypotenuse opposite
44. 44. 5.5 cm 7.9 cm 46º 9.7 5.5 ≈0.70 Cos 46º ≈0.70 Cosine β = hypotenuse adjacent
45. 45. A plane takes off from an airport an an angle of 18º and a speed of 240 mph. Continuing at this speed and angle, what is the altitude of the plane after 1 minute? 18º x After 60 sec., at 240 mph, the plane has traveled 4 miles 4
46. 46. 18º x 4opposite hypotenuse SohCahToa Sine A = opposite hypotenuse Sine 18 = x 4 0.3090 = x 4 x = 1.236 miles or 6,526 feet 1 Soh
47. 47. An explorer is standing 14.3 miles from the base of Mount Everest below its highest peak. His angle of elevation to the peak is 21º. What is the number of feet from the base of Mount Everest to its peak? 21º x Tan 21 = x 14.3 0.3839 = x 14.3 x = 5.49 miles = 29,000 feet 1
48. 48. A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is 150 feet high. What is the number of feet from the swimmer to the shore? 18º 150 Tan 18 = x 150 x 0.3249 = 150 x 0.3249x = 150 0.3249 0.3249 X = 461.7 ft1
49. 49. A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly below the dragon. At what angle does the archer need to aim his arrow to slay the dragon? x 60 120 Tan x = 60 120 Tan x = 0.5 Tan-1 (0.5) = 26.6º
50. 50. Solving a Problem with the Tangent Ratio 60º 53 ft h = ? We know the angle and theWe know the angle and the side adjacent to 60º. We want toside adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use the tangent ratio:tangent ratio: ft92353 531 3 53 60tan ≈= = == h h h adj opp 1 2 3 Why?
51. 51. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? 50 71.5 ° ? tan 71.5° tan 71.5° 50 y = y = 50 (tan 71.5°) y = 50 (2.98868) 149.4y ft≈ Ex. = Opp Hyp
52. 52. A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? 200 x Ex. 5 60° cos 60° x (cos 60°) = 200 x X = 400 yards
53. 53. Trigonometric Functions on a Rectangular Coordinate System x y θθ Pick a point on the terminal ray and drop a perpendicular to the x-axis. r y x The adjacent side is x The opposite side is y The hypotenuse is labeled r This is called a REFERENCE TRIANGLE. y x x y x r r x y r r y == == == θθ θθ θθ cottan seccos cscsin
54. 54. Trigonometric Ratios may be found by: 45 º 1 1 2 Using ratios of special trianglesUsing ratios of special triangles 145tan 2 1 45cos 2 1 45sin = = = For angles other than 45º, 30º, 60º you will need to use aFor angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)calculator. (Set it in Degree Mode for now.)