Trigonometric Ratios

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. Trigonometric Ratios • MM2G2b:Explain the relationship between the trigonometric ratios of complementary angles. • MM2G2c:Solve application problems using the trigonometric ratios.

3. Warm Up Put 4 30-60-90 triangles with the following sides listed and have students determine the missing lengths.

4. Or Warm Up Put 4 3-4-5 triangles on the board and ask: Are these triangles right triangles? How do you know? Are they similar? How do you know?

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. Ratio of Opposite / Adjacent • NOTE: Greek letters are often used to designate the angle when using trigonometry. Greek letters that are popular are: α, β, θ, and φ • NOTE: If the angles are designated by a capital letter, the sides opposite that angle are often designated by the same letter in lower case.

7. Ratio of Opposite / Adjacent • Calculate the ratio of opposite/adjacent • In a right triangle, this ratio is called “Tangent” • Please do not confuse this tangent with the tangent of a curve. Same word, different meaning.

8. Ratio of Opposite / Adjacent • Align the triangles with the angle of interest at the origin of the Cartesian Coordinate System. • Opposite can be thought of as “rise” • Adjacent can be thought of as “run” • Look at the ratio of opposite divided by adjacent • Opposite divided by adjacent can be thought of as slope

9. The following slides have been come from the following sources: www.mccd.edu/faculty/bruleym/.../trigonometric%20ratios http://ux.brookdalecc.edu/fac/cos/lschmelz/Math%20151/ www.scarsdaleschools.k12.ny.us /202120915213753693/lib/…/trig.ppt Emily FreemanMcEachern High School

10. Trigonometric Ratios in Right Triangles M. Bruley

11. Trigonometric Ratios are based on the Concept of Similar Triangles!

12. 1 45 º 2 1 1 45 º 2 45 º All 45º- 45º- 90º Triangles are Similar!

13. 30º 30º 2 60º 60º 1 30º 60º All 30º- 60º- 90ºTriangles are Similar! 4 2 1 ½

14. All 30º- 60º- 90ºTriangles are Similar! 10 60º 2 60º 5 1 30º 30º 1 60º 30º

15. adjacent opposite 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 We’ll label them a, b, and c and the angles  and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.  hypotenuse c First let’s look at the three basic functions. b leg SINE COSINE  TANGENT leg a They are abbreviated using their first 3 letters

16. The Trigonometric Functions SINE COSINE TANGENT

17. SINE Prounounced “sign”

18. COSINE Prounounced “co-sign”

19. TANGENT Prounounced “tan-gent”

20. Greek Letter q Pronounced “theta” Represents an unknown angle

21. Greek Letter α Pronounced “alpha” Represents an unknown angle

22. Greek Letter β Pronounced “Beta” Represents an unknown angle

23. hypotenuse hypotenuse opposite opposite adjacent adjacent

24. We need a way to remember all of these ratios…

25. Question ! 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 ?

26. 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.

27. An example of an acronym for SohCahToa. Seven old horses Crawled a hill To our attic..

28. Some Old Hippie Came A Hoppin’ Through Our Old Hippie Apartment

29. Sin SOHCAHTOA Opp Hyp Cos Adj Hyp Tan Opp Adj Old Hippie

30. Other ways to remember SOH CAH TOA Some Of Her Children Are Having Trouble Over Algebra. Some Out-Houses Can Actually Have Totally Odorless Aromas. She Offered Her Cat A Heaping Teaspoon Of Acid. Soaring Over Haiti, Courageous Amelia Hit The Ocean And ... Tom's Old Aunt Sat On Her Chair And Hollered. -- (from Ann Azevedo)

31. Other ways to remember SOH CAH TOA Stamp Out Homework Carefully, As Having Teachers Omit Assignments. Some Old Horse Caught Another Horse Taking Oats Away. Some Old Hippie Caught Another Hippie Tripping On Apples. School! Oh How Can Anyone Have Trouble Over Academics.

32. opposite adjacent We could ask for the trig functions of the angle by using the definitions. You MUST get them memorized. Here is a mnemonic to help you.  c The sacred Jedi word: b SOHCAHTOA SOHCAHTOA  INE a ANGENT OSINE PPOSITE DJACENT PPOSITE DJACENT YPOTENUSE YPOTENUSE

33. opposite adjacent It is important to note WHICH angle you are talking about when you find the value of the trig function.  Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so hypotenuse 5 c b 4 opposite  a 3 Let's choose: sin  = Use a mnemonic and figure out which sides of the triangle you need for tangent. Use a mnemonic and figure out which sides of the triangle you need for sine. tan  =

34. Oh, I'm acute! So am I! 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. This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.  5 4  3

35. Go to book and do problems

36. Pg 159 # 3 Pg 159 # 11 Pg 160 # 20a Pg 166 # 3 Pg 167 # 25

37. Trigonometry Ratios Tangent A = Sine A = Cosine A = Soh Cah Toa A

38. The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side. hypotenuse 1.9 cm opposite adjacent 14º 7.7 cm 0.25 Tangent 14º 0.25

39. Tangent A = 3.2 cm 24º 7.2 cm 0.45 Tangent 24º 0.45

40. Tangent A = 5.5 cm 46º 5.3 cm 1.04 Tangent 46º 1.04

41. Tangent A = 1.76 6.7 cm Tangent 60.5º 1.76 60.5º 3.8 cm

42. Tangent A = As an acute angle of a triangle approaches 90º, its tangent becomes infinitely large very large Tan 89.9º = 573 Tan 89.99º = 5,730 etc. very small

43. 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 = Cosine A = Sine 89º = .9998 A Sine 89.9º = .999998

44. Sin α = 7.9 cm 3.2 cm 24º 0.41 0.41 Sin 24º

45. Cosine β = 7.9 cm 46º 5.5 cm Cos 46º 0.70 0.70

46. 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? After 60 sec., at 240 mph, the plane has traveled 4 miles 4 x 18º

47. SohCahToa Soh Sine A = Sine 18 = 0.3090 = 1 x = 1.236 miles or 6,526 feet 4 x opposite hypotenuse 18º

48. 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? Tan 21 = 0.3839 = 1 x = 5.49 miles = 29,000 feet x 21º 14.3

49. 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? 0.3249x = 150 Tan 18 = 0.3249 0.3249 0.3249 = X = 461.7 ft 1 150 18º x

50. 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? Tan x = Tan x = 0.5 Tan-1(0.5) = 26.6º 60 x 120