t3 analytic trigonometry and trig formulas

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t3 analytic trigonometry and trig formulas

  1. 1. Analytic Trigonometry<br />
  2. 2. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />
  3. 3. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />
  4. 4. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />
  5. 5. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />
  6. 6. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />Let (x, y) be a point on the unit circle,then<br />x=cos() and y=sin()<br />as shown. <br />
  7. 7. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />Let (x, y) be a point on the unit circle,then<br />x=cos() and y=sin()<br />as shown. <br />(cos(),sin())<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />
  8. 8. Analytic Trigonometry<br />We define the following reciprocals functions that are used often in science and engineering.<br />csc() = 1/sin(), read as "cosecant of ".<br />sec() = 1/cos(), read as "secant of ".<br />cot() = 1/tan(), read as "cotangent of ".<br />Let (x, y) be a point on the unit circle,then<br />x=cos() and y=sin()<br />as shown. <br />By Pythagorean Theorem,<br />y2 + x2 = 1, so we have:<br />sin2() + cos2()=1<br />or s2 + c2 = 1<br />or all angle .<br />(cos(),sin())<br />y=sin()<br />(1,0)<br /><br />x=cos()<br />
  9. 9. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . <br />
  10. 10. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. <br />
  11. 11. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. <br />The Trig-Hexagram<br />
  12. 12. Analytic Trigonometry<br />The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities. <br />The Trig-Hexagram<br />the CO-side<br />the regular-side<br />
  13. 13. Analytic Trigonometry<br />The Division Relations:<br />
  14. 14. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />
  15. 15. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />II<br />III<br />I<br />
  16. 16. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />III<br />I<br />II<br />
  17. 17. Analytic Trigonometry<br />The Division Relations:<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br />I<br />II<br />III<br />
  18. 18. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br /> The Reciprocal Relations<br />
  19. 19. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br /> The Reciprocal Relations<br />Start from any function, going across diagonally, we always have I = II / III.<br />
  20. 20. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br />III<br />II<br />I<br /> The Reciprocal Relations<br />Start from any function, going across diagonally, we always have I = II / III.<br />Example B: sec(A) = 1/cos(A), <br />
  21. 21. Analytic Trigonometry<br />The Division Relations<br />Start from any function,<br />going around the outside,<br />we always have<br />I = II / III or I * III = II<br />Example A:<br />tan(A) = sin(A)/cos(A),<br />sec(A) = csc(A)/cot(A),<br />sin(A)cot(A) = cos(A)<br />I<br />II<br />III<br /> The Reciprocal Relations<br />Start from any function, going across diagonally, we always have I = II / III.<br />Example B: sec(A) = 1/cos(A), cot(A) = 1/tan(A) <br />
  22. 22. Analytic Trigonometry<br /> Square-Sum Relations<br />
  23. 23. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />
  24. 24. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />
  25. 25. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />
  26. 26. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A) <br />
  27. 27. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A) <br />The identities from this hexagram are called the<br />Fundamental Identities. <br />
  28. 28. Analytic Trigonometry<br /> Square-Sum Relations<br />For each of the three inverted <br />triangles, the sum of the squares of the top two is the square of the bottom one.<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A) <br />The identities from this hexagram are called the<br />Fundamental Identities. <br />Weassume these identities from here on and list the most important ones below.<br />
  29. 29. Fundamental Identities<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />
  30. 30. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />
  31. 31. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />
  32. 32. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.<br />
  33. 33. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />we have the following co-relations.<br />
  34. 34. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />we have the following co-relations.<br />sin(A) = cos(90 – A)<br />sin(A) = cos(B)<br />
  35. 35. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />w have the following co-relations.<br />sin(A) = cos(90 – A)<br />tan(A) = cot(90 – A)<br />sin(A) = cos(B)<br />tan(A) = cot(B)<br />
  36. 36. Fundamental Identities<br />Reciprocal Relations<br />sec(A)=1/C <br />csc(A)=1/S<br />cot(A)=1/T<br />Division Relations<br />tan(A)=S/C<br />cot(A)=C/S<br />Square-Sum Relations<br />sin2(A) + cos2(A)=1<br />tan2(A) + 1 = sec2(A)<br />1 + cot2(A) = csc2(A)<br />B<br />A<br />A and B are complementary<br />Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, <br />w have the following co-relations.<br />sin(A) = cos(90 – A)<br />tan(A) = cot(90 – A)<br />sec(A) = csc(90 – A)<br />sin(A) = cos(B)<br />tan(A) = cot(B)<br />sec(A) = csc(B)<br />
  37. 37. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />
  38. 38. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).<br />
  39. 39. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).<br />Therefore cosine is an even function and sine is an odd function. <br />
  40. 40. Fundamental Identities<br />The Negative Angle Relations<br />cos(-A) = cos(A),<br />sin(-A) = - sin(A)<br />A<br />A<br />-A<br />-A<br />1<br />1<br />Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).<br />Therefore cosine is an even function and sine is an odd function. <br />Since tangent and cotangent are quotients of sine <br />and cosine, they are also odd functions.<br />
  41. 41. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />
  42. 42. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> Frank Ma<br />2006<br />
  43. 43. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> Frank Ma<br />2006<br />
  44. 44. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> Frank Ma<br />2006<br />
  45. 45. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> Frank Ma<br />2006<br />
  46. 46. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1<br /> Frank Ma<br />2006<br />
  47. 47. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br /> Frank Ma<br />2006<br />
  48. 48. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br />Example D: (1 – sin(A))(1 + sin(A)) <br /> Frank Ma<br />2006<br />
  49. 49. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br />Example D: (1 – sin(A))(1 + sin(A)) <br /> = 1 – sin2(A) <br /> Frank Ma<br />2006<br />
  50. 50. Notes on Square-Sum Identities<br />Terms in the square-sum identities may be rearranged and give other versions of the identities.<br />Example C: sin2(A) – 1 = – cos2(A)<br /> sec2(A) – tan2(A) = 1<br /> cot2(A) – csc2(A) = -1 <br />Difference of squares may be factored since<br /> x2 – y2 = (x – y)(x + y)<br />Example D: (1 – sin(A))(1 + sin(A)) <br /> = 1 – sin2(A) = cos2(A)<br /> Frank Ma<br />2006<br />
  51. 51. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />
  52. 52. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) <br />
  53. 53. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) ;difference of squares <br />= sec2(x) – 1 <br />
  54. 54. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) ;difference of squares <br />= sec2(x) – 1 ;square-relation<br />= tan2(x) <br />
  55. 55. Algebraic Identities<br />Example E: Simplify (sec(x) – 1)(sec(x) + 1). <br />Express the answer in sine and cosine. <br />(sec(x) – 1)(sec(x) + 1) ;difference of squares <br />= sec2(x) – 1 ;square-relation<br />= tan2(x) ; division relation <br />= <br />sin2(x)<br />cos2(x)<br />

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