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- 1. Angular Measurements<br />
- 2. Angular Measurements<br />There are two systems of angular measurements<br />
- 3. Angular Measurements<br />There are two systems of angular measurements.<br />I. The degree system II. The radian system <br />
- 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. 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. 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. 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. 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. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />
- 10. Radian Measurements<br />The unitcircle is the circle centered at (0, 0) with radius 1.<br />r = 1<br />
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<br />
- 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<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. 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<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. 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<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. 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<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. 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<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. 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 a2 = c<br />π/4<br />π/4<br />π/4<br />(21.414)<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 = a3<br />π/6<br />π/6<br />c <br />c = 2a<br />π/3<br />π/3<br />
- 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. 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. 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. 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. 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. 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. 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. Important Trigonometric Values<br />Angles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.<br />
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Important Trigonometric Values<br />c. = -11π/6<br />
- 74. Important Trigonometric Values<br />c. = -11π/6<br />-11π/6<br />1<br />
- 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. 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. 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. 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. SOHCAHTOA<br />If the angle A is placed in the standard position, <br />hypotenuse<br />opposite<br />A<br />adjacent<br />
- 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. 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. 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. SOHCAHTOA<br />Example B: Given the rt-triangle, <br />find tan(A) and sin(B)<br />a<br />
- 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. 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. 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. 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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