4.2 angles and trigonometric functions

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4.2 angles and trigonometric functions

  1. 1. Angular Measurements<br />
  2. 2. Angular Measurements<br />There are two systems of angular measurements<br />
  3. 3. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />
  4. 4. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. <br />
  5. 5. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. <br />One degree is divided into 60 minutes, each minute is denoted as 1'. <br />
  6. 6. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. <br />One degree is divided into 60 minutes, each minute is denoted as 1'. <br />One minute is divided into 60 seconds, each second is denoted as 1". <br />
  7. 7. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. <br />One degree is divided into 60 minutes, each minute is denoted as 1'. <br />One minute is divided into 60 seconds, each second is denoted as 1". <br />The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering. <br />
  8. 8. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. <br />One degree is divided into 60 minutes, each minute is denoted as 1'. <br />One minute is divided into 60 seconds, each second is denoted as 1". <br />The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering. <br />In mathematics, the radian system is used more often because it's relationship with the geometry of circles. <br />
  9. 9. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />
  10. 10. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />r = 1<br />
  11. 11. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Its given by the equation <br />x2 + y2 = 1.<br />r = 1<br />
  12. 12. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br />r = 1<br />
  13. 13. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Arc length as angle<br />measurement for <br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br /><br />r = 1<br />
  14. 14. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Arc length as angle<br />measurement for <br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br /><br />r = 1<br />Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.<br />
  15. 15. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Arc length as angle<br />measurement for <br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br /><br />r = 1<br />Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.<br />The following formulas convert the measurements between Degree and Radian systems<br />
  16. 16. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Arc length as angle<br />measurement for <br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br /><br />r = 1<br />Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.<br />The following formulas convert the measurements between Degree and Radian systems<br />180o = π rad <br />
  17. 17. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Arc length as angle<br />measurement for <br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br /><br />r = 1<br />Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.<br />The following formulas convert the measurements between Degree and Radian systems<br />180o = π rad 1o = rad <br />π<br />180<br />
  18. 18. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />Arc length as angle<br />measurement for <br />Its given by the equation <br />x2 + y2 = 1.<br />The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.<br /><br />r = 1<br />Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.<br />The following formulas convert the measurements between Degree and Radian systems<br />180o = π rad 1o = rad = 1 rad  57o<br />π<br />180o<br />π<br />180<br />
  19. 19. Definition of Trigonometric Functions<br />When an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. <br />
  20. 20. Definition of Trigonometric Functions<br />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. <br />
  21. 21. Definition of Trigonometric Functions<br />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. <br /> is +<br /> is – <br />
  22. 22. Definition of Trigonometric Functions<br />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. <br /> is +<br /> is – <br />Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), <br />(x , y)<br />(1,0)<br /><br />
  23. 23. Definition of Trigonometric Functions<br />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. <br /> is +<br /> is – <br />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: <br />cos() = x,sin() = y, tan() = <br />(x , y)<br />(1,0)<br /><br />y<br />x<br />
  24. 24. Definition of Trigonometric Functions<br />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. <br /> is +<br /> is – <br />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: <br />cos() = x,sin() = y, tan() = <br />(x , y)<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />y<br />x<br />
  25. 25. Definition of Trigonometric Functions<br />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. <br /> is +<br /> is – <br />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: <br />cos() = x,sin() = y, tan() = <br />(x , y)<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />y<br />x<br />Note: tan() = slope of the dial<br />
  26. 26. Definition of Trigonometric Functions<br />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. <br /> is +<br /> is – <br />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: <br />cos() = x,sin() = y, tan() = <br />(x , y)<br />tan()<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />y<br />x<br />Note: tan() = slope of the dial<br />
  27. 27. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />
  28. 28. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. <br />π/4<br />
  29. 29. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. <br />π/4<br />π/4<br />π/4<br />
  30. 30. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br />π/4<br />π/4<br />π/4<br />
  31. 31. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br />π/4<br />π/4<br />π/4<br />
  32. 32. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />
  33. 33. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. <br />π/6<br />
  34. 34. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. <br />π/6<br />π/6<br />c <br />c = 2a<br />π/3<br />π/3<br />
  35. 35. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />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. <br />π/6<br />π/6<br />c <br />c = 2a<br />π/3<br />π/3<br />
  36. 36. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />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. <br />So a2 + b2 = (2a)2<br />π/6<br />π/6<br />c <br />c = 2a<br />π/3<br />π/3<br />
  37. 37. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />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. <br />So a2 + b2 = (2a)2<br /> a2 + b2 = 4a2<br /> b2 = 3a2<br />π/6<br />π/6<br />c <br />c = 2a<br />π/3<br />π/3<br />
  38. 38. Two Important Right Triangles<br />There are two important classes of right triangles, the rt-triangles, and the rt-triangles.<br />π/6<br />π/4<br />A rt-triangle is an isosceles triangle, <br />so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, <br /> 2a2 = c2 2a2 = c2<br /> or a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />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. <br />So a2 + b2 = (2a)2<br /> a2 + b2 = 4a2<br /> b2 = 3a2<br /> b = 3a2 or b = a3<br />π/6<br />π/6<br />c <br />c = 2a<br />π/3<br />π/3<br />
  39. 39. Two Important Right Triangles<br />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 .<br />
  40. 40. Two Important Right Triangles<br />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 .<br />π/4<br />1<br />2/2<br />π/4<br />2/2<br />
  41. 41. Two Important Right Triangles<br />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 .<br />π/4<br />π/3<br />1<br />1<br />2/2<br />1/2<br />π/6<br />π/4<br />3/2<br />2/2<br />
  42. 42. Two Important Right Triangles<br />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 .<br />π/4<br />π/3<br />1<br />1<br />2/2<br />1/2<br />π/6<br />π/4<br />3/2<br />2/2<br />Important Trigonometric Values<br />
  43. 43. Two Important Right Triangles<br />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 .<br />π/4<br />π/3<br />1<br />1<br />2/2<br />1/2<br />π/6<br />π/4<br />3/2<br />2/2<br />Important Trigonometric Values<br />The trig-values of angles depend on the positions of the angle. <br />
  44. 44. Two Important Right Triangles<br />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 .<br />π/4<br />π/3<br />1<br />1<br />2/2<br />1/2<br />π/6<br />π/4<br />3/2<br />2/2<br />Important Trigonometric Values<br />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. <br />
  45. 45. Two Important Right Triangles<br />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 .<br />π/4<br />π/3<br />1<br />1<br />2/2<br />1/2<br />π/6<br />π/4<br />3/2<br />2/2<br />Important Trigonometric Values<br />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. <br />Note that  and  + 2nπhave the same position where n is an integer.<br />
  46. 46. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />
  47. 47. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />0, ±2π, ±4π..<br />
  48. 48. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />
  49. 49. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />Kπ<br />2<br />
  50. 50. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />
  51. 51. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />
  52. 52. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />
  53. 53. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />
  54. 54. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br />4<br />
  55. 55. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />5π/4, -3π/4..<br />
  56. 56. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />
  57. 57. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />Angles with measurements of <br />(reduced) orrad.<br />Kπ<br />Kπ<br />6<br />3<br />
  58. 58. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />Angles with measurements of <br />(reduced) orrad.<br />π/6, -11π/6..<br />Kπ<br />Kπ<br />6<br />3<br />
  59. 59. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />π/3, -5π/3..<br />Angles with measurements of <br />(reduced) orrad.<br />π/6, -11π/6..<br />Kπ<br />Kπ<br />6<br />3<br />
  60. 60. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />π/3, -5π/3..<br />2π/3,..<br />Angles with measurements of <br />(reduced) orrad.<br />5π/6,..<br />π/6, -11π/6..<br />Kπ<br />Kπ<br />6<br />3<br />
  61. 61. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />π/3, -5π/3..<br />2π/3,..<br />Angles with measurements of <br />(reduced) orrad.<br />5π/6,..<br />π/6, -11π/6..<br />Kπ<br />Kπ<br />7π/6,..<br />6<br />3<br />4π/3,..<br />
  62. 62. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />π/3, -5π/3..<br />2π/3,..<br />Angles with measurements of <br />(reduced) orrad.<br />5π/6,..<br />π/6, -11π/6..<br />Kπ<br />Kπ<br />7π/6,..<br />6<br />3<br />11π/6,..<br />4π/3,..<br />5π/3,..<br />
  63. 63. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />±π, ±3π..<br />0, ±2π, ±4π..<br />Angles with measurements of <br />rad correspond<br />to the y-axial angles.<br />π/2, 5π/2..<br />Kπ<br />2<br />-π/2, 3π/2..<br />π/4, -7π/4..<br />3π/4, -5π/4..<br />Angles with measurements of <br />rad are diagonals.<br />Kπ<br /> Frank Ma<br />2006<br />4<br />7π/4, -π/4..<br />5π/4, -3π/4..<br />π/3, -5π/3..<br />2π/3,..<br />Angles with measurements of <br />(reduced) orrad.<br />5π/6,..<br />π/6, -11π/6..<br />Kπ<br />Kπ<br />7π/6,..<br />6<br />3<br />11π/6,..<br />4π/3,..<br />5π/3,..<br />
  64. 64. Important Trigonometric Values<br />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.<br />a.  = -3π<br />
  65. 65. Important Trigonometric Values<br />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.<br />a.  = -3π<br />-3π<br />(-1, 0) <br />
  66. 66. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />
  67. 67. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />b.  = 5π/4<br />
  68. 68. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />b.  = 5π/4<br />5π/4<br />
  69. 69. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />b.  = 5π/4<br />Place the π/4-rt-triangle as shown,<br />5π/4<br />
  70. 70. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />b.  = 5π/4<br />Place the π/4-rt-triangle as shown,<br />1<br />5π/4<br />
  71. 71. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />b.  = 5π/4<br />Place the π/4-rt-triangle as shown,<br />we get the coordinate = (-2/2, -2/2).<br />1<br />5π/4<br />(-2/2, -2/2) <br />
  72. 72. Important Trigonometric Values<br />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.<br />a.  = -3π<br />sin(-3π) = 0 <br />cos(-3π) = -1 <br />-3π<br />(-1, 0) <br />tan(-3π) = 0 <br />b.  = 5π/4<br />Place the π/4-rt-triangle as shown,<br />we get the coordinate = (-2/2, -2/2).<br />sin(5π/4) = -2/2 <br />cos(5π/4) = -2/2 <br />1<br />5π/4<br />(-2/2, -2/2) <br />tan(5π/4) = 1 <br />
  73. 73. Important Trigonometric Values<br />c.  = -11π/6<br />
  74. 74. Important Trigonometric Values<br />c.  = -11π/6<br />-11π/6<br />1<br />
  75. 75. Important Trigonometric Values<br />c.  = -11π/6<br />Place the π/6-rt-triangle as shown, <br />we get the coordinate = (3/2, 1/2).<br />-11π/6<br />(3/2, ½) <br />1<br />
  76. 76. Important Trigonometric Values<br />c.  = -11π/6<br />Place the π/6-rt-triangle as shown, <br />we get the coordinate = (3/2, 1/2).<br />-11π/6<br />(3/2, ½) <br />1<br />sin(-11π/6) = 1/2 <br />cos(-11π/6) = 3/2 <br />tan(-11π/6) = 1/3 =3/3 <br />
  77. 77. Important Trigonometric Values<br />c.  = -11π/6<br />Place the π/6-rt-triangle as shown, <br />we get the coordinate = (3/2, 1/2).<br />-11π/6<br />(3/2, ½) <br />1<br />sin(-11π/6) = 1/2 <br />cos(-11π/6) = 3/2 <br />tan(-11π/6) = 1/3 =3/3 <br />SOHCAHTOA<br />
  78. 78. Important Trigonometric Values<br />c.  = -11π/6<br />Place the π/6-rt-triangle as shown, <br />we get the coordinate = (3/2, 1/2).<br />-11π/6<br />(3/2, ½) <br />1<br />sin(-11π/6) = 1/2 <br />cos(-11π/6) = 3/2 <br />tan(-11π/6) = 1/3 =3/3 <br />SOHCAHTOA<br />Given arighttriangle and one of the small angles, say A, the adjacent and the opposite of the angle A are as shown:<br />
  79. 79. SOHCAHTOA<br />If the angle A is placed in the standard position, <br />hypotenuse<br />opposite<br />A<br />adjacent<br />
  80. 80. SOHCAHTOA<br />If the angle A is placed in the standard position, then the trig-values of A are:<br />opposite <br />O <br />Sin(A) = <br />= <br />hypotenuse <br />H <br />hypotenuse<br />opposite<br />adjacent <br />A <br />A<br />Cos(A) = <br />= <br />adjacent<br />hypotenuse <br />H <br />opposite <br />O <br />= <br />Tan(A) = <br />adjacent <br />A <br />
  81. 81. SOHCAHTOA<br />If the angle A is placed in the standard position, then the trig-values of A are:<br />opposite <br />O <br />Sin(A) = <br />= <br />hypotenuse <br />H <br />hypotenuse<br />opposite<br />adjacent <br />A <br />A<br />Cos(A) = <br />= <br />adjacent<br />hypotenuse <br />H <br />opposite <br />O <br />= <br />Tan(A) = <br />adjacent <br />A <br />One checks easily that these trig-values are the same as the ones defined via the unit circle. <br />
  82. 82. SOHCAHTOA<br />If the angle A is placed in the standard position, then the trig-values of A are:<br />opposite <br />O <br />Sin(A) = <br />= <br />hypotenuse <br />H <br />hypotenuse<br />opposite<br />adjacent <br />A <br />A<br />Cos(A) = <br />= <br />adjacent<br />hypotenuse <br />H <br />opposite <br />O <br />= <br />Tan(A) = <br />adjacent <br />A <br />One checks easily that these trig-values are the same as the ones defined via the unit circle. <br />Hence we use "SOHCAHTOA" to remember the definition of trig-functions for positive angles that are smaller than 90o. <br />
  83. 83. SOHCAHTOA<br />Example B: Given the rt-triangle, <br />find tan(A) and sin(B)<br />a<br />
  84. 84. SOHCAHTOA<br />Example B: Given the rt-triangle, <br />find tan(A) and sin(B)<br />a<br />To find a first, <br />
  85. 85. SOHCAHTOA<br />Example B: Given the rt-triangle, <br />find tan(A) and sin(B)<br />a<br />To find a first, we've<br />a2 + 82 = 112<br />
  86. 86. SOHCAHTOA<br />Example B: Given the rt-triangle, <br />find tan(A) and sin(B)<br />a<br />To find a first, we've<br />a2 + 82 = 112<br />a2 = 121 – 64 = 57<br />a = 57<br />
  87. 87. SOHCAHTOA<br />Example B: Given the rt-triangle, <br />find tan(A) and sin(B)<br />a<br />To find a first, we've<br />a2 + 82 = 112<br />a2 = 121 – 64 = 57<br />a = 57<br />Opp <br />57 <br />Hence tan(A) =<br />= <br />Adj <br />8 <br />8 <br />Opp <br />= <br />sin(B) =<br />11 <br />Hyp <br />

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