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# 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 />