Trigonometric Ratio_Cosine, Sine And Tangent

Finding trigonometric ratio

1. Mathematics Grade 12 MR. LUIS V. SALENGA (Teacher)

2. Drill Use paperless app

3. Trigonometric Ratios A RATIO is a comparison of two numbers. For example; boys to girls cats : dogs right : wrong. In Trigonometry, the comparison is between sides of a triangle ( right triangle) .

4. EXPLAIN and USE the relationship between the sine and cosine of complementary angles .

5. Trigonometric Ratios

6. What are sin, cos and tan ?

7. What are sin cos and tan ? They are the 3 ways that two sides can be selected in relation to one angle. Ɵ

8. What are sin , cos and tan ? They are the 3 ways that two sides can be selected in relation to one angle Ɵ O (opposite)

9. What are sin cos and tan ? They are the 3 ways that two sides can be selected in relation to one angle. Ɵ O (opposite)

10. What are sin, cos and tan ? They are the 3 ways that two sides can be selected in relation to one angle. Ɵ H (hypotenuse)

11. What are sin , cos and tan ? They are the 3 ways that two sides can be selected in relation to one angle Ɵ H (hypotenuse) O (opposite)

12. What are sin , cos and tan ? Ɵ A (adjacent) O (opposite) H (hypotenuse)

13. What are sin , cos and tan ? Ɵ

14. What are sin, cos and tan ? Ɵ H (hypotenuse)

15. What are sin, cos and tan ? Ɵ H (hypotenuse) O (opposite)

16. What are sin, cos and tan ? Ɵ H (hypotenuse) O (opposite) A (adjacent)

17. What are sin cos and tan ? They are the 3 ways that an angle can be found when given the lengths of two sides -1 -1 -1 Ɵ

18. What are sin , cos and tan ? They are the 3 ways that an angle can be found when given the lengths of two sides -1 -1 -1 Ɵ

19. Finding Trigo Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine , cosine , and tangent , which are abbreviated as sin, cos, and tan respectively.

20. Trig Ratios A Hypotenuse Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.

21. Trig Ratios A Opposite Hypotenuse Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.

22. Trig Ratios Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. sin A = side opposite A hypotenuse = c cos A = side adjacent to A hypotenuse = b c tan A = side opposite A side adjacent to A = a b a

23. Trig Ratios A Hypotenuse

24. Trig Ratios A Opposite Hypotenuse

25. The Trig Ratios S oh C ah T oa sin A = Side opposite A hypotenuse = O A cos A = Side adjacent to A hypotenuse = tan A = Side opposite A Side adjacent to A = A H O A O pposite H ypotenuse A djacent A

26. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside.

27. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. H O

28. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. h o a

29. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. Soh Cah Toa Soh  Sine A = o h

30. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. Soh Cah Toa Soh  Sine A = = o h 8

31. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. Soh Cah Toa Soh  Sine A = = o h 8 17

32. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. Soh Cah Toa Cah  cos A = a h = 15 17

33. Ex. 1: Finding Trig Ratios Find the sine, the cosine, and the tangent ratios for A in each triangle beside. Soh Cah Toa Toa  tan A = o a = 8 15

34. Ex. 1: Finding Trig Ratios sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent 8 17 ≈ 0.4706 15 17 ≈ 0.8824 8 15 ≈ 0.5333 Trig ratios are often expressed as decimal approximations.

35. What are sin cos and tan ? They are the 3 ways that an angle can be found when given the lengths of two sides -1 -1 -1 Ɵ

36. Let’s practice. C 2cm B 3cm A Find the measure of the angle. SOH CAH TOA Ɵ tan =   Ɵ = (   Ɵ = 33.69   Ɵ tan =   Ɵ

37. Let’s practice. Find the measure of the angle. SOH CAH TOA 33.69   C 2cm B 3cm A sin 33.69 x h =   h=   sin 33.69 =   sin =   Ɵ h= 3.61cm

38. Practice some more… Find the measure of angle A 24.19 12 A 21 SOH CAH TOA sin =   Ɵ sin =   Ɵ =   Ɵ = 29.74   Ɵ

39. Practice some more. SOH CAH TOA cos =   Ɵ cos 42 =   cos 42 x 14.5 = a a = 10.78 cm

40. Practice some more. SOH CAH TOA tan =   Ɵ tan 38 =   tan 38 x 10 = o o = 7.81 cm

41. Indirect Measurement You are measuring the height of a pine tree in Baguio City. You stand 15m from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59 °. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 15m. 15m

42. Indirect Measurement 15m SOH CAH TOA tan =   Ɵ tan =   59 tan 59 x 15 = x x = 24.96 cm The tree is about 25m tall.

43. Estimating Distance Escalators. The escalator at the Brisbane Central Rail Station rises 21m at a 30 ° angle of elevation. To find the distance d a person travels on the escalator stairs, you can write a trig ratio that involves the hypotenuse and the known leg of 21m. 30 ° d 21m

44. Estimating Distance 30 ° d 21m sin =   Ɵ SOH CAH TOA sin 30 =   sin 30 x d = 21 d =   d = 42 m A person travels 42m on the escalator stairs.

45. Group Activity: Using the inverse Trig ratios to find angles The angle that an anchor line makes with the seabed is really critical for holding the ship. This angle depends upon the type of seabed, etc. In this case, the seabed is mud and the best angle for holding this ship is between 42-50 degrees. Will the boat be safely anchored?

46. Group Activity: Using the inverse Trig ratios to find angles The angle that an anchor line makes with the seabed is really critical for holding the ship. This angle depends upon the type of seabed, etc. In this case, the seabed is mud and the best angle for holding this ship is between 42-50 degrees. Will the boat be safely anchored? SOH CAH TOA sin =   Ɵ sin =   Ɵ sin =   Ɵ =   Ɵ = 39   Ɵ No, the boat will not be safely anchored

47. Find the measurement of x (side) or (angle). Ɵ 1. 2 . 3 . 4 . x = 8.39 cm X = 48.05 cm θ = 31.59° θ = 37.69

48. Assignment: When the space shuttle is 5 miles from the runway, its glide angle is about 19 °. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth.