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  • .576 x 60=34.56’; .56 x 60=33.6’’ round to 34 ‘’: 37deg34min34sec

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  • 1. 6-1: Right-Triangle Trigonometry © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Define the six trig ratios of an acute angle •Evaluate trig ratios using triangles, a calculator and special angles •Introduce trigonometry and angle measurement
  • 2. Important Idea Trigonometry, which means triangle measurement was developed by the Greeks in the 2nd century BC. It was originally used only in astronomy, navigation and surveying. It is now used to model periodic behavior such as sound waves and planetary orbits.
  • 3. Definition Angle: the figure formed by two rays with a common endpoint
  • 4. Definition Initial Side Terminal Side Verte x Angle A is in standard position A x y
  • 5. 43 1 Definition x yThe Cartesian Plane is divided into quadrants as follows: 2
  • 6. Important Idea Angles may be measured in degrees where 1 degree (°) is 1/360 of a circle, 90° is ¼ of a circle, 180° is ½ of a circle, 270° is ¾ of a circle and 360° is a full circle. A 90° angle is also called a right angle.
  • 7. Example 90° 180° 270° 0° or 360°
  • 8. Example 90° 180° 270° 0° or 360°
  • 9. Example 90° 180° 270° 0° or 360°
  • 10. Example 90° 180° 270° 0° or 360°
  • 11. Example 90° 180° 270° 0° or 360°
  • 12. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
  • 13. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
  • 14. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
  • 15. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
  • 16. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
  • 17. Definition Fractional parts of a degree can be written in decimal form or Degree-Minute – Second (DMS) form. A minute is 1/60 of a degree. A second is 1/60 of a minute. How many seconds are in each degree?
  • 18. Example Write 29°40’20” in decimal degrees accurate to 3 decimal places. Symbol for minute Symbol for second
  • 19. Important Idea To convert from DMS to decimal, write the decimal expression 29°40’20” as: 40 20 29 60 3600 + + in your calculator.
  • 20. Try This 77.399° Write 77°23’56’’ in decimal form. 0° 90° 180° 270°
  • 21. Try This 185.751° Write 185°45’3’’ in decimal form. 0° 90° 180° 270°
  • 22. Try This 319.541° Write 319°32’28’’ in decimal degrees accurate to 3 decimal places. 0° 90° 180° 270°
  • 23. Example Write 37.576° in DMS form. Procedure: 1. Write the decimal part .576° as .576 x 60=34.56’ 2. Write the decimal part of minutes as . 56 x 60=33.6’’ 3. Round 33.6’’ to 34’’ for total 37°34’34’’
  • 24. Try This Write 185.651° in DMS form. 0° 90° 180° 270°185°39’4’’
  • 25. Try This Write 85.259° in DMS form. 0° 90° 180° 270°85°15’32’’
  • 26. Definition A B C a b c D E F d e f m A m D∠ = ∠If then a d c f = b e c f = a d b e =& & Similar Triangles
  • 27. Important Idea If 2 right triangles have equal angles, the corresponding ratios of their sides must be the same no matter the size of the triangles. This fact is the basis for trigonometry.
  • 28. Example A B C a b c D E F d e f 30m A m D∠ = ∠ = °If and 2, 4a c= = and 3d = then ?f =
  • 29. Try This A B C ab c D E F d e f 60m A m D∠ = ∠ = °If and 2, 4c b= = and 6e = then ?f = 12
  • 30. Definition The hypotenuse is the side opposite the 90° angle and is the longest side. The other 2 sides are legs.
  • 31. Definition The opposite side is the leg opposite the given angle A C
  • 32. Definition The adjacent side is the leg next to the given angle (not the hypotenuse). A C
  • 33. Important Idea Right triangles come in all sizes, shapes and orientations.
  • 34. Definition For a given acute angle in a right triangle: θ The sine of written as is the ratio θ sinθ sinθ = opposite hypotenuse (see p.416 of your text): M em orize
  • 35. Definition For a given acute angle in a right triangle: θ The cosine of written asθ cosθ cosθ = adjacent hypotenuse is the ratio: (see p.416 of your text): M em orize
  • 36. Definition For a given acute angle in a right triangle: θ The tangent of written asθ tanθ tanθ = opposite adjacent is the ratio: (see p.416 of your text): M em orize
  • 37. opposite Definition For a given acute angle in a right triangle: θ The cosecant of written asθ cscθ 1 csc sin θ θ = = hypotenuse is the ratio: (see p.416 of your text): M em orize
  • 38. adjacent Definition For a given acute angle in a right triangle: θ The secant of written asθ secθ 1 sec cos θ θ = = hypotenuse is the ratio: (see p.416 of your text): M em orize
  • 39. opposit e Definition For a given acute angle in a right triangle: θ The cotangent of written as θ cotθ 1 cot tan θ θ = = adjacent is the ratio: (see p.416 of your text): M em orize
  • 40. Example θ 13 5 12 Evaluate the 6 trig ratios of the angleθ
  • 41. Try This θ 35 4 Evaluate the 6 trig ratios of the angle θ 4 sin 5 θ = 3 cos 5 θ = 4 tan 3 θ = 5 csc 4 θ = 5 sec 3 θ = 3 cot 4 θ =
  • 42. Try This θ13 5 12 Evaluate the 6 trig ratios of the angle θ 5 sin 13 θ = 12 cos 13 θ = 5 tan 12 θ = 13 csc 5 θ = 13 sec 12 θ = 12 cot 5 θ =
  • 43. Example Using your calculator, evaluate the 6 trig ratios of 33° Be sure that mode is set to degrees
  • 44. Try This Using your calculator, evaluate the 6 trig ratios of 117.25° sin117.25 .889° = cos117.25 .458° = − tan117.25 1.942° = − csc117.25 1.125= sec117.25 2.184= − cot117.25 .515= −
  • 45. Try This Using your calculator, evaluate cos12 15'30''° cos12 15'30'' cos12.258 .977° = =
  • 46. Important Idea 1 csc sin θ θ = 1 sec cos θ θ = Since your calculator does not have a sec, csc or cot key, you must find the reciprocal of cos, sin or tan. 1 cot tan θ θ =
  • 47. Definition The special angles are: •30° •60° •45°
  • 48. Important Idea These angles are special because they have exact value trig functions.
  • 49. Consider the first two special angles in degrees... 30° 60° Long Side ShortSide Hypotenuse Analysis
  • 50. Important Idea In a 30°-60°-90° right triangle, the short side is opposite the 30° angle, the long side is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.
  • 51. LongSide Hypotenuse Short Side…orientation does not change the relationships between sides and angles 60° 30° Important Idea
  • 52. LongSide Hypotenuse Short Side …orientation does not change the relationships between sides and angles 30° 60° Important Idea
  • 53. Important Idea In a 30°,60°,90° triangle: •the short side is one-half the hypotenuse. •the long side is times the short side. 3Memoriz e
  • 54. Try This Find the length of the missing sides: 30° 60° 4 8 4 3
  • 55. Try This Find the length of the missing sides: 10 5 5 3 30°
  • 56. Try This Find the length of the missing sides: 5 5 3 3 10 3 3 60°
  • 57. Try This Find the length of the missing sides: 60° 4 8 4 3
  • 58. 30° 60° 45° 45° 45° Analysis Consider the last special angle:
  • 59. Hypotenuse…orientation does not change the relationships between sides and angles 45° 45° Important Idea
  • 60. H ypotenuse …orientation does not change the relationships between sides and angles 45° 45° Important Idea
  • 61. …and sides opposite equal angles are equal... x x and by the 2x pythagorean theorem, the hypotenuse is... 45° 45° Important Idea
  • 62. Important Idea In a 45°,45°,90° triangle: •The legs of the triangle are equal. •the hypotenuse is times the length of the leg. Memoriz e 2
  • 63. Try This Find the length of the missing sides 2 2 2 2 45°
  • 64. Try This Find the length of the missing sides 2 2 2 45° 45°
  • 65. Example Find the exact value of the 6 trig functions of 30°. This is not a calculator problem. 30°
  • 66. Example Find the exact value of the 6 trig functions of 30°. This is not a calculator problem. 30°
  • 67. Important Idea When you know the lengths of the 3 sides of a right triangle, you can evaluate any of the 6 trig functions.
  • 68. Example Find the exact value of the 6 trig functions of 45°. This is not a calculator problem. 45°
  • 69. Try This Find the exact value of the 6 trig functions of 60°. Do not use a calculator. 60°
  • 70. Solution 60° 1 2 3 3 sin60 2 ° = 1 cos60 2 ° = tan60 3° =
  • 71. Solution 60° 1 2 3 2 3 csc60 3 ° = sec60 2° = 3 cot60 3 ° =
  • 72. Lesson Close We will use the information in this lesson to solve right triangle problems in the next lesson. Right triangle problems are used in real- world applications such as indirect measurement, surveying and navigation.