1. Review – Right Triangle Trigonometry

2. Objectives Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle measures in right triangles.

3. History Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be used to make indirect measurements like those using similar triangles.

4. History Early mathematicians discovered trig by measuring the ratios of the sides of different right triangles. They noticed that when the ratio of the shorter leg to the longer leg was close to a specific number, then the angle opposite the shorter leg was close to a specific number.

5. Trigonometric Ratios The word trigonometry originates from two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements. A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent.

6. Trigonometric Ratios Only Apply to Right Triangles

7. In right triangles : The segment across from the right angle ( ) is labeled the hypotenuse “Hyp.”. The “angle of perspective” determines how to label the sides. Segment opposite from the Angle of Perspective( ) is labeled “Opp.” Segment adjacent to (next to) the Angle of Perspective ( ) is labeled “Adj.”. * The angle of Perspective is never the right angle. Hyp. Angle of Perspective Opp. Adj.

8. Labeling sides depends on the Angle of Perspective Angle of Perspective Hyp. Opp. Adj. Ifis the Angle of Perspective then …… * ”Opp.” means segment opposite from Angle of Perspective “Adj.” means segment adjacent from Angle of Perspective

9. If the Angle of Perspective is then Opp Hyp Adj then Opp Adj Hyp

10. The 3 Trigonometric Ratios The 3 ratios are Sine, Cosine and Tangent

11. Trigonometric Ratios To help you remember these trigonometric relationships, you can use the mnemonic device, SOH-CAH-TOA, where the first letter of each word of the trigonometric ratios is represented in the correct order. A C B b c a Sin A = Opposite side SOH Hypotenuse Cos A = Adjacent side CAH Hypotenuse Tan A = Opposite side TOA Adjacent side

12. Trigonometric Ratios hypotenuse side adjacent  side opposite 

13. SohCahToa

14. Chief SohCahToa The Amazing Legend of…

15. Chief SohCahToa Once upon a time there was a wise old Native American Chief named Chief SohCahToa. He was named that due to an chance encounter with his coffee table in the middle of the night. He woke up hungry, got up and headed to the kitchen to get a snack. He did not turn on the light and in the darkness, stubbed his big toe on his coffee table….

16. Solving a right triangle Every right triangle has one right angle, two acute angles, one hypotenuse and two legs. To solve a right triangle, means to determine the measures of all six (6) parts. You can solve a right triangle if one of the following two situations exist: – One side length and one acute angle measure (use trigonometric ratios to find other side). – Two side lengths (use inverse trigonometric ratios to find an angle).

17. Using a Calculator – Trigonometric Ratios Use a calculator to approximate the sine, cosine, and the tangent of 74 . Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

18. Sample keystrokes Sample keystroke sequences Sample calculator displayRounded Approximation 74 0.9612626950.9613 0.2756373550.2756 3.4874144443.4874 sin ENTER 74 COS ENTER 74 TAN ENTER

19. Using a Calculator – Trigonometric Ratios Using a calculator to find the following trigonometric ratios. Sin 37 ˚ Sin 63 ˚ Cos 82 ˚ Cos 16 ˚ Tan 29 ˚ Tan 55 ˚.6018.8910.1392.9613.5543 1.4281

20. Using a Calculator – Trigonometric Ratios Given one acute angle and one side of a right triangle, the trigonometric ratios can be used to find another side of the triangle. Example 1: Trig. ratio used to find side adj. to 42 ˚ angle. Multiple both sides by 15 to solve for x. Use calculator to find the length of the adj. side.

21. Using a Calculator – Trigonometric Ratios Example 2: Trig. ratio used to find hyp. of a right triangle. Multiple both sides by x. Multiple both sides by sin66 ˚ to solve for x. Use calculator to find the length of the hyp.

22. Using a Calculator – Trigonometric Ratios Practice finding a side of a right triangle. Solve for x. Sin 32 = x/8 Sin 54 = 21/x Cos 81 = x/8.8 Tan 60 = 25/x 4.24 25.96 1.38 14.43

23. Inverse Trigonometry As we learned earlier, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. Once you know the sine, cosine, or tangent (trig. ratio) of an acute angle, you can use a calculator to find the measure of the angle. To find an angle measurement in a right triangle given any two sides, use the inverse of the trig. ratio.

24. Using a Calculator – Inverse Trigonometric Ratios Given two sides of a right triangle, the inverse trigonometric ratios can be used to find the measure of an acute angle of the triangle. In general, for an acute angle A: – If sin A = x, then sin -1 x = m  A – If cos A = y, then cos -1 y = m  A – If tan A = z, then tan -1 z = m  A The expression sin -1 x is read as “the inverse sine of x.” On your calculator, this means you will be punching the 2 nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle.

25. Using a Calculator – Inverse Trigonometric Ratios “sin -1 x ” is read “the angle whose sine is x ” or “inverse sine of x ”. arcsin x is the same thing as sin -1 x. “inverse sin” is the inverse operation of “sin”.

26. Example Given the trig. Ratio, solve for the angle. KEYSTROKES: [COS] 13 1946.82644889 2nd( ÷)ENTER Answer: So, the measure of  P is approximately 46.8°.

27. Using a Calculator – Inverse Trigonometric Ratios Given a trigonometric ratio in a right triangle, use inverse trig. ratios to solve for an acute angle. Example: Trig. ratio for ∠ A. To solve for ∠ A, take the sin -1 of both sides of the equation. Inverse operations, sin and sin -1, cancel out. Use calculator to solve for ∠ A.

28. Using a Calculator – Inverse Trigonometric Ratios Practice finding an acute angle of a right triangle. Solve for the indicated angle. Sin B = 3.5/8 Cos D = 12/14 Tan A = 17/12 ∠ B = 25.94 ˚ ∠ D = 31.0 ˚ ∠ A = 54.78 ˚

29. To use Trigonometric Ratios to find lengths, given an interior angle and one side of a RAT Finding the length of an unknown side of a right angled triangle: The appropriate ratio to use is Tangent, i.e. TOA Tan 26 0 = a/7 a/7 = Tan 26 0 a = 7 x Tan 26 0 a = 7 x 0.4877 a =3.41cm (adjacent) (opposite) 26 o 7cm a Calculate the length of y. The appropriate ratio is sine, SOH Sine 34 0 = y/5 y/5 = Sine 34 0 y = 5 x Sine 34 0 y = 5 x 0.559 y =2.80m (opposite) (hypotenuse) y 5m 34 o

30. Your Turn: Calculate the length of side x The appropriate ratio to use is Cosine, i.e. CAH Cos 67 0 = x/10.6 0.39 = x/10.6 x =10.6 X 0.39 x =4.14m Calculate the length of y The appropriate ratio is Sine, i.e. SOH Sine 42 0 = y/6.2 y/6.2 = Sine 42 0 y = 6.2 X Sine 42 0 y = 6.2 X 0.669 y =4.19m 10.6m 67 0 x (Adjacent) Hypotenuse y (Opposite) 6.2m Hypotenuse 42 0

31. To use Inverse Trigonometric Ratios to find an interior angle, given two sides of a RAT To find angle a The appropriate ratio to use is Tangent, i.e. TOA Tan a = 32/25 Tan a = 1.28 a = Tan -1 1.28 a = 52.0 0 To find the size of angle y 32cm (Opposite) 25cm Adjacent a The appropriate ratio is sin, i.e. SOH Sine y = 30/50 Sine y = 0.6 y = Sine -1 0.6 y = 36.9 0 30cm ( Opposite ) 50cm Hypotenuse y0y0

32. Your Turn: Find angle b The appropriate ratio to use is Sine, i.e. SOH Sin b = 6/12 Sin b = 0.5 b = Sin -1 0.5 b = 30 0 Find angle y The appropriate ratio is Cosine, i.e. CAH Cos y = 12.4/19.7 Cos y = 0.639 y = Cos -1 0.639 y = 50.28 0 6cm Opposite Hypotenuse 12cm b 19.7m Hypotenuse 12.4m (Adjacent) y

33. Solving Trigonometric Equations There are only three possibilities for the placement of the variable ‘x”. Sin = Sin = 0.48 X = Sin (0.48) X = 28.6854 Sin 25 = x = (12) (0.4226) x = 5.04 cm 0.4226 = Sin 25 = 0.4226 = x = x = 28.4 cm

34. EXAMPLES

35. Example 1a A. Express sin L as a fraction and as a decimal to the nearest hundredth. Answer:

36. Example 1b B. Express cos L as a fraction and as a decimal to the nearest hundredth. Answer:

37. Example 1c C. Express tan L as a fraction and as a decimal to the nearest hundredth. Answer:

38. Example 1d D. Express sin N as a fraction and as a decimal to the nearest hundredth. Answer:

39. Example 1e E. Express cos N as a fraction and as a decimal to the nearest hundredth. Answer:

40. Example 1f F. Express tan N as a fraction and as a decimal to the nearest hundredth. Answer:

41. Your Turn: A. Find sin A. A. B. C. D.

42. Your Turn: B. Find cos A. A. B. C. D.

43. Your Turn: C. Find tan A. A. B. C. D.

44. Your Turn: D. Find sin B. A. B. C. D.

45. Your Turn: E. Find cos B. A. B. C. D.

46. Your turn: F. Find tan B. A. B. C. D.

47. Example 2 Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth. Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse. The side adjacent to the 60° angle has a measure of x.

48. Example 2 Definition of cosine ratio Substitution Simplify.

49. Your Turn: Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D.

50. Example 3 Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

51. Example 3 Step 1Find m  A by using a tangent ratio. 29.7448813≈m  AUse a calculator. So, the measure of  A is about 30 . Definition of inverse tangent

52. Example 3 Step 2Find m  B using complementary angles. m  B≈60Subtract 30 from each side. So, the measure of  B is about 60 . 30 + m  B≈90m  A ≈ 30 m  A + m  B=90Definition of complementary angles

53. Example 3 Step 3Find AB by using the Pythagorean Theorem. (AC) 2 + (BC) 2 =(AB) 2 Pythagorean Theorem 7 2 + 4 2 =(AB) 2 Substitution 65=(AB) 2 Simplify. Take the positive square root of each side. 8.06≈ ABUse a calculator.

54. Example 3 Answer: m  A ≈ 30, m  B ≈ 60, AB ≈ 8.06 So, the measure of AB is about 8.06.

55. Your Turn: A.44.1° B.48.3° C.55.4° D.57.2° Use a calculator to find the measure of  D to the nearest tenth.

56. Your Turn: A.m  A = 36°, m  B = 54°, AB = 13.6 B.m  A = 54°, m  B = 36°, AB = 13.6 C.m  A = 36°, m  B = 54°, AB = 16.3 D.m  A = 54°, m  B = 36°, AB = 16.3 Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

57. APPLICATION PROBLEMS

58. Example 4 You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet 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 45 feet.

59. Solution tan 59 ° = opposite adjacent tan 59 ° = h 45 45 tan 59° = h 45 (1.6643) ≈ h 74.9 ≈ h Write the ratio Substitute values Multiply each side by 45 Use a calculator or table to find tan 59 ° Simplify  The tree is about 75 feet tall.

60. Example 5 Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes. A. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.

61. Solution: You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle. Glide  = x ° 59 miles 15.7 miles tan x ° = opp. adj. tan x ° = 15.7 59 tan 15.7/59 ≈ 14.9 Use correct ratio Substitute values  When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9 °.

62. B. Solution 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. Glide  = 19 ° 5 miles h tan 19 ° = opp. adj. tan 19 ° = h 5 Use correct ratio Substitute values 5 tan 19 ° = h 5 5 Isolate h by multiplying by 5. 1.7 ≈ hApproximate using calculator  The shuttle’s altitude is about 1.7 miles.

63. A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75˚, how far is the bottom of the ladder from the base of the building? ladder building 20 x 75˚ Using the 75˚ angle as a reference, we know hypotenuse and adjacent side. Use cos 75˚ = 20 (cos 75˚) = x 20 (.2588) = x x ≈ 5.2 About 5 ft. cos Your Turn:

64. When the sun is 62˚ above the horizon, a building casts a shadow 18m long. How tall is the building? x 18 shadow 62˚ Using the 62˚ angle as a reference, we know opposite and adjacent side. Use tan 62˚ = 18 (tan 62˚) = x 18 (1.8807) = x x ≈ 33.9 About 34 m tan Your Turn:

65. A kite is flying at an angle of elevation of about 55˚. Ignoring the sag in the string, find the height of the kite if 85m of string have been let out. string 85 x 55˚ kite Using the 55˚ angle as a reference, we know hypotenuse and opposite side. Use sin 55˚ = 85 (sin 55˚) = x 85 (.8192) = x x ≈ 69.6 About 70 m sin Your Turn: