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# Hprec6 4

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### Hprec6 4

1. 1. 6-4: Trigonometric Functions © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Define the Trigonometric functions in terms of the unit circle. •Define the Trigonometric functions in the coordinate plane.
2. 2. Important Idea Trig ratios depend only the angle and not on a point on the terminal side of the angle. θ (3,4) (6,8)
3. 3. Example Findsin ,cosθ θ & when the terminal side of the angle passes through (3,4) tanθ θ (3,4) (6,8)
4. 4. Try This Findsin ,cosθ θ & when the terminal side of the angle passes through (6,8) tanθ θ (3,4) (6,8)
5. 5. Solution θ (6,8) 2 2 2 6 8r = + 10r = 8 6 10 8 4 sin 10 5 θ = = ⇒ 6 3 cos 10 5 θ = = 8 4 tan 6 3 θ = =
6. 6. Important Idea θ ( , )x y r x y opp cosθ = x r = hyp sinθ = y r = hyp adj tanθ = opp adj y x = See p. 444. of your text
7. 7. Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12) θ Try This θ (5,-12)
8. 8. Solution θ 5 -12 13 12 sin 13 θ = − 5 cos 13 θ = 12 tan 5 θ = − (5,-12)
9. 9. Important Idea Trig ratios may be positive or negative
10. 10. Find sin, cos & tan of the angle whose terminal side passes through the point (-5,-5) θ Try This θ (-5,-5)
11. 11. Solution θ (-5,-5) -5 -5 5 2 5 2 sin 25 2 θ = − = − 5 2 cos 25 2 θ = − = − 5 tan 1 5 θ − = = −
12. 12. Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12) θ Try This θ (5,-12)
13. 13. Solution θ 5 -12 13 12 sin 13 θ = − 5 cos 13 θ = 12 tan 5 θ = − (5,-12)
14. 14. Example Find ,sint cost & when the terminal side of an angle passes through the given point on the unit circle. tant 1 3 , 10 10   − ÷   1 10 3 10 −1
15. 15. Important Idea cos x t r = sin y t r = tan y t x = In the unit circle, r=1, therefore 1 y y= = 1 x x= = sint y= cost x= and
16. 16. Try This sintFind , cost when the terminal side of an angle passes through tant& on the unit circle. 3 4 , 5 5    ÷  
17. 17. Solution 4 sin 5 t = 3 cos 5 t = 3 35tan 4 4 5 t = =
18. 18. Definition Coterminal Angles: Angles that have the same terminal side. x y y x
19. 19. Important Idea To find coterminal angles, simply add or subtract either 360° or 2 radians to the given angle or any angle that is already coterminal to the given angle. π
20. 20. Example Find an angle coterminal with 420°. Find sin420° and cos420° 1. Find smallest positive coterminal angle. 3. Apply definition of sin and cos. Procedure: 2. Draw picture of coterminal angle.
21. 21. Example Find an angle coterminal 1. Find smallest positive coterminal angle. 3. Apply definition of sin and cos. Procedure: 2. Draw picture of coterminal angle. 7 4 π −with Find the sin and cos.
22. 22. Important Idea The trig ratios of a given angle and all its coterminal angles are the same.
23. 23. Try This Find an angle that is coterminal with 780°. Find sin780°and cos780°. 3 sin780 sin60 2 ° = ° = 1 cos780 cos60 2 ° = ° =
24. 24. Try This Find an angle that is coterminal with . Find and . sin( 10 ) sin0 0π− = = cos( 10 ) cos0 1π− = = 10π− sin( 10 )π− cos( 10 )π− Hint: use the unit circle to find the trig ratio.
25. 25. Important Idea In addition to finding trig ratios of angles ( ), we can also find trig ratios of real numbers in radians (t). Radians may be in terms of θ sin 4 π   ÷   cos( 2.56)− tan 3 π   ÷   π or just a number, for example:
26. 26. Important Idea There are times when we must be satisfied with approximate values of trig ratios. At other times, we can find and prefer exact values.
27. 27. Example cos( 2.56)− Find the approximate value: Since the degree symbol (°) is not used, this must be radians. mode
28. 28. Try This Use your calculator in radian mode to approximate the sin, cos and tan. Round to 4 decimal places. Use the signs of the functions to identify the quadrant of the terminal side. -18 7 8 π 2 5 π − 35.6π
29. 29. Definition sint is the sin of a number t where t is in radians. sint = opposite hypotenuse y r = where 2 2 r x y= + See page 445 of your text.
30. 30. Definition cost is the cos of a number t where t is in radians. cost = adjacent hypotenuse x r = where 2 2 r x y= + See page 445 of your text.
31. 31. Definition tant is the tan of a number t where t is in radians. tant = opposite adjacent y x = See page 445 of your text.
32. 32. Important Idea cos costθ = = The definitions of the trig ratios are the same for angles and radians, for example: sin sintθ = = hyp opp y r = hyp adj x r =
33. 33. Example Find the exact value: cos45° 45° cos 4 π   ÷   10 10 4 π 10 10
34. 34. Example Find the exact value: sin30° 1 sin 6 π   ÷   3 1 6 π 3 30°
35. 35. Definition Reference Angle: the angle between a given angle and the nearest x axis. (Note: x axis; not y axis). Reference angles are always positive.
36. 36. Important Idea How you find the reference angle depends on which quadrant contains the given angle. The ability to quickly and accurately find a reference angle is going to be important in future lessons.
37. 37. Example Find the reference angle if the given angle is 20°. In quad. 1, the given angle & the ref. angle are the same. x y 20°
38. 38. Example Find the reference angle if the given angle is . x y 9 π 9 π In quad. 1, the given angle & the ref. angle are the same.
39. 39. Example Find the reference angle if the given angle is 120°. For given angles in quad. 2, the ref. angle is 180° less the given angle. ? 120° x y
40. 40. Example Find the reference angle if the given angle is . ? x y 2 3 π 2 3 π For given angles in quad. 2, the ref. angle is less the given angle. π
41. 41. Example Find the reference angle if the given angle is . x y 7 6 π 7 6 π For given angles in quad. 3, the ref. angle is the given angle less π
42. 42. Try This Find the reference angle if the given angle is 7 4 π For given angles in quad. 4, the ref. angle is less the given angle. 2π 7 4 π 4 π
43. 43. Try This Find the reference angle if the given angle is x y 4 π − Hint: Don’t forget the definition. 4 π
44. 44. Important Idea The trig ratio of a given angle is the same as the trig ratio of its reference angle except, possibly, for the sign.
45. 45. Example Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator. 135°
46. 46. Procedure 1.Sketch the given angle. 2.Find and sketch the reference angle. Label the sides using special angle facts. 3.Find sin, cos and tan using definition. 4.Add the correct sign.
47. 47. Example Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator. 7 6 π
48. 48. Try This Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator. 60°
49. 49. 2 Solution 60° 3 1 3 sin60 2 ° = 1 cos60 2 ° = tan60 3° =
50. 50. Important Idea x or y can be positive or negative depending on the quadrant but the hypotenuse ( r ) is always positive.
51. 51. Try This Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator. 11 6 π
52. 52. Solution 11 6 π -1 3 2 11 1 sin 6 2 π  = − ÷   11 3 cos 6 2 π  = ÷   11 1 3 tan 6 33 π  = − = − ÷  
53. 53. Try This Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator. 4 3 π
54. 54. Solution 4 3 π -1 23− 4 3 sin 3 2 π  = − ÷   4 1 cos 3 2 π  = − ÷   4 tan 3 3 π  = ÷  
55. 55. The unit circle is a circle with radius of 1. We use the unit circle to find trig functions of quadrantal angles. -1 1 -1 1 1 Definition
56. 56. The unit circle -1 1 -1 1 1 Definition (1,0) (0,1) (-1,0) (0,-1) x y
57. 57. Definition -1 1 -1 1 (1,0) (0,1) (-1,0) (0,-1) For the quadrantal angles: The x values are the terminal sides for the cos function.
58. 58. Definition -1 1 -1 1 (1,0) (0,1) (-1,0) (0,-1) For the quadrantal angles: The y values are the terminal sides for the sin function.
59. 59. Definition -1 1 -1 1 (1,0) (0,1) (-1,0) (0,-1) For the quadrantal angles : The tan function is the y divided by the x
60. 60. -1 1 -1 1 Find the values of the 6 trig functions of the quadrantal angle in standard position: Example sinθ cosθ tanθ cscθ secθ cotθ 0° (1,0) (0,1) (-1,0) (0,-1)
61. 61. -1 1 -1 1Find the values of the 6 trig functions of the quadrantal angle in standard position: Example θ sinθ cosθ tanθ cscθ secθ cotθ90° (1,0) (0,1) (-1,0) (0,-1)
62. 62. -1 1 -1 1 Find the values of the six trig functions of the given angle in standard position. 2 π Example θ sinθ cosθ tanθ cscθ secθ cotθ
63. 63. -1 1 -1 1 Find the values of the six trig functions of the given angle in standard position. 2π Example sinθ cosθ tanθ cscθ secθ cotθ
64. 64. -1 1 -1 1 Find the values of the six trig functions of the given angle in standard position. 3π Try This sinθ cosθ tanθ cscθ secθ cotθ
65. 65. -1 1 -1 1Find the values of the 6 trig functions of the quadrantal angle in standard position: Example sinθ cosθ tanθ cscθ secθ cotθ540° (1,0) (0,1) (-1,0) (0,-1)
66. 66. -1 1 -1 1Find the values of the 6 trig functions of the quadrantal angle in standard position: Example sinθ cosθ tanθ cscθ secθ cotθ270° (1,0) (0,1) (-1,0) (0,-1)
67. 67. -1 1 -1 1 Find the values of the six trig functions of the given angle in standard position. 7 2 π Try This sinθ cosθ tanθ cscθ secθ cotθ
68. 68. -1 1 -1 1Find the values of the 6 trig functions of the quadrantal angle in standard position: Try This sinθ cosθ tanθ cscθ secθ cotθ360° (1,0) (0,1) (-1,0) (0,-1)
69. 69. Important Ideas •Trig functions of quadrantal angles have exact values. •Trig functions of all other angles have approximate values. •Trig functions of special angles have exact values.
70. 70. Example Use a calculator to approximate cos 710° to 4 decimal places. Don’t forget to check “Mode”.
71. 71. Example Use a calculator to approximate sin(72°30’30”) to 4 decimal places.
72. 72. Example Use a calculator to approximate csc 15° to 4 decimal places.
73. 73. Lesson Close How do you evaluate the trig ratios of quadrantal angles?