1. Angular Measurements

2. Angular Measurements There are two systems of angular measurements

3. Angular Measurements There are two systems of angular measurements. I. The degree system II. The radian system

4. Angular Measurements There are two systems of angular measurements. I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o.

5. Angular Measurements There are two systems of angular measurements. I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'.

6. Angular Measurements There are two systems of angular measurements. I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1".

7. Angular Measurements There are two systems of angular measurements. I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1". The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering.

8. Angular Measurements There are two systems of angular measurements. I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1". The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering. In mathematics, the radian system is used more often because it's relationship with the geometry of circles.

9. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1.

10. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. r = 1

11. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Its given by the equation x2 + y2 = 1. r = 1

12. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle. r = 1

13. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Arc length as angle measurement for  Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.  r = 1

14. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Arc length as angle measurement for  Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.  r = 1 Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.

15. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Arc length as angle measurement for  Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.  r = 1 Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle. The following formulas convert the measurements between Degree and Radian systems

16. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Arc length as angle measurement for  Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.  r = 1 Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle. The following formulas convert the measurements between Degree and Radian systems 180o = π rad

17. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Arc length as angle measurement for  Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.  r = 1 Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle. The following formulas convert the measurements between Degree and Radian systems 180o = π rad 1o = rad π 180

18. Radian Measurements The unitcircle is the circle centered at (0, 0) with radius 1. Arc length as angle measurement for  Its given by the equation x2 + y2 = 1. The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.  r = 1 Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle. The following formulas convert the measurements between Degree and Radian systems 180o = π rad 1o = rad = 1 rad  57o π 180o π 180

19. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position.

20. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.

21. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is +  is –

22. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is +  is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), (x , y) (1,0) 

23. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is +  is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y) (1,0)  y x

24. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is +  is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y) y=sin() (1,0)  x=cos() y x

25. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is +  is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y) y=sin() (1,0)  x=cos() y x Note: tan() = slope of the dial

26. Definition of Trigonometric Functions When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is +  is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y) tan() y=sin() (1,0)  x=cos() y x Note: tan() = slope of the dial

27. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4

28. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. π/4

29. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. π/4 π/4 π/4

30. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, π/4 π/4 π/4

31. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 π/4 π/4 π/4

32. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414)

33. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414) For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. π/6

34. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414) For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. π/6 π/6 c c = 2a π/3 π/3

35. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414) For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. π/6 π/6 c c = 2a π/3 π/3

36. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414) For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 π/6 π/6 c c = 2a π/3 π/3

37. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414) For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 a2 + b2 = 4a2 b2 = 3a2 π/6 π/6 c c = 2a π/3 π/3

38. Two Important Right Triangles There are two important classes of right triangles, the rt-triangles, and the rt-triangles. π/6 π/4 A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = c π/4 π/4 π/4 (21.414) For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 a2 + b2 = 4a2 b2 = 3a2 b = 3a2 or b = a3 π/6 π/6 c c = 2a π/3 π/3

39. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .

40. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 . π/4 1 2/2 π/4 2/2

41. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 . π/4 π/3 1 1 2/2 1/2 π/6 π/4 3/2 2/2

42. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 . π/4 π/3 1 1 2/2 1/2 π/6 π/4 3/2 2/2 Important Trigonometric Values

43. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 . π/4 π/3 1 1 2/2 1/2 π/6 π/4 3/2 2/2 Important Trigonometric Values The trig-values of angles depend on the positions of the angle.

44. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 . π/4 π/3 1 1 2/2 1/2 π/6 π/4 3/2 2/2 Important Trigonometric Values The trig-values of angles depend on the positions of the angle. Following are the important positions of the dials for which we may find the exact trig-values.

45. Two Important Right Triangles From the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 . π/4 π/3 1 1 2/2 1/2 π/6 π/4 3/2 2/2 Important Trigonometric Values The trig-values of angles depend on the positions of the angle. Following are the important positions of the dials for which we may find the exact trig-values. Note that  and  + 2nπhave the same position where n is an integer.

46. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.

47. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. 0, ±2π, ±4π..

48. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π..

49. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. Kπ 2

50. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2

51. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2..

52. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4

53. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4

54. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ 4

55. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 5π/4, -3π/4..

56. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4..

57. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. Angles with measurements of (reduced) orrad. Kπ Kπ 6 3

58. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. Angles with measurements of (reduced) orrad. π/6, -11π/6.. Kπ Kπ 6 3

59. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. π/3, -5π/3.. Angles with measurements of (reduced) orrad. π/6, -11π/6.. Kπ Kπ 6 3

60. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. π/3, -5π/3.. 2π/3,.. Angles with measurements of (reduced) orrad. 5π/6,.. π/6, -11π/6.. Kπ Kπ 6 3

61. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. π/3, -5π/3.. 2π/3,.. Angles with measurements of (reduced) orrad. 5π/6,.. π/6, -11π/6.. Kπ Kπ 7π/6,.. 6 3 4π/3,..

62. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. π/3, -5π/3.. 2π/3,.. Angles with measurements of (reduced) orrad. 5π/6,.. π/6, -11π/6.. Kπ Kπ 7π/6,.. 6 3 11π/6,.. 4π/3,.. 5π/3,..

63. Important Trigonometric Values Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles. ±π, ±3π.. 0, ±2π, ±4π.. Angles with measurements of rad correspond to the y-axial angles. π/2, 5π/2.. Kπ 2 -π/2, 3π/2.. π/4, -7π/4.. 3π/4, -5π/4.. Angles with measurements of rad are diagonals. Kπ  Frank Ma 2006 4 7π/4, -π/4.. 5π/4, -3π/4.. π/3, -5π/3.. 2π/3,.. Angles with measurements of (reduced) orrad. 5π/6,.. π/6, -11π/6.. Kπ Kπ 7π/6,.. 6 3 11π/6,.. 4π/3,.. 5π/3,..

64. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π

65. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π -3π (-1, 0)

66. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0

67. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0 b.  = 5π/4

68. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0 b.  = 5π/4 5π/4

69. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0 b.  = 5π/4 Place the π/4-rt-triangle as shown, 5π/4

70. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0 b.  = 5π/4 Place the π/4-rt-triangle as shown, 1 5π/4

71. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0 b.  = 5π/4 Place the π/4-rt-triangle as shown, we get the coordinate = (-2/2, -2/2). 1 5π/4 (-2/2, -2/2)

72. Important Trigonometric Values Example A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values. a.  = -3π sin(-3π) = 0 cos(-3π) = -1 -3π (-1, 0) tan(-3π) = 0 b.  = 5π/4 Place the π/4-rt-triangle as shown, we get the coordinate = (-2/2, -2/2). sin(5π/4) = -2/2 cos(5π/4) = -2/2 1 5π/4 (-2/2, -2/2) tan(5π/4) = 1

73. Important Trigonometric Values c.  = -11π/6

74. Important Trigonometric Values c.  = -11π/6 -11π/6 1

75. Important Trigonometric Values c.  = -11π/6 Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2). -11π/6 (3/2, ½) 1

76. Important Trigonometric Values c.  = -11π/6 Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2). -11π/6 (3/2, ½) 1 sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3

77. Important Trigonometric Values c.  = -11π/6 Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2). -11π/6 (3/2, ½) 1 sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3 SOHCAHTOA

78. Important Trigonometric Values c.  = -11π/6 Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2). -11π/6 (3/2, ½) 1 sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3 SOHCAHTOA Given arighttriangle and one of the small angles, say A, the adjacent and the opposite of the angle A are as shown:

79. SOHCAHTOA If the angle A is placed in the standard position, hypotenuse opposite A adjacent

80. SOHCAHTOA If the angle A is placed in the standard position, then the trig-values of A are: opposite O Sin(A) = = hypotenuse H hypotenuse opposite adjacent A A Cos(A) = = adjacent hypotenuse H opposite O = Tan(A) = adjacent A

81. SOHCAHTOA If the angle A is placed in the standard position, then the trig-values of A are: opposite O Sin(A) = = hypotenuse H hypotenuse opposite adjacent A A Cos(A) = = adjacent hypotenuse H opposite O = Tan(A) = adjacent A One checks easily that these trig-values are the same as the ones defined via the unit circle.

82. SOHCAHTOA If the angle A is placed in the standard position, then the trig-values of A are: opposite O Sin(A) = = hypotenuse H hypotenuse opposite adjacent A A Cos(A) = = adjacent hypotenuse H opposite O = Tan(A) = adjacent A One checks easily that these trig-values are the same as the ones defined via the unit circle. Hence we use "SOHCAHTOA" to remember the definition of trig-functions for positive angles that are smaller than 90o.

83. SOHCAHTOA Example B: Given the rt-triangle, find tan(A) and sin(B) a

84. SOHCAHTOA Example B: Given the rt-triangle, find tan(A) and sin(B) a To find a first,

85. SOHCAHTOA Example B: Given the rt-triangle, find tan(A) and sin(B) a To find a first, we've a2 + 82 = 112

86. SOHCAHTOA Example B: Given the rt-triangle, find tan(A) and sin(B) a To find a first, we've a2 + 82 = 112 a2 = 121 – 64 = 57 a = 57

87. SOHCAHTOA Example B: Given the rt-triangle, find tan(A) and sin(B) a To find a first, we've a2 + 82 = 112 a2 = 121 – 64 = 57 a = 57 Opp 57 Hence tan(A) = = Adj 8 8 Opp = sin(B) = 11 Hyp