The document defines several trigonometric reciprocal functions including cosecant, secant, and cotangent. It then presents trigonometric identities relating these functions through division, reciprocal, and square-sum relations. These fundamental identities are considered important relationships that hold for all angles and are used throughout analytic trigonometry.

1. Analytic Trigonometry

2. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering.

3. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering. csc() = 1/sin(), read as "cosecant of ".

4. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering. csc() = 1/sin(), read as "cosecant of ". sec() = 1/cos(), read as "secant of ".

5. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering. csc() = 1/sin(), read as "cosecant of ". sec() = 1/cos(), read as "secant of ". cot() = 1/tan(), read as "cotangent of ".

6. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering. csc() = 1/sin(), read as "cosecant of ". sec() = 1/cos(), read as "secant of ". cot() = 1/tan(), read as "cotangent of ". Let (x, y) be a point on the unit circle,then x=cos() and y=sin() as shown.

7. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering. csc() = 1/sin(), read as "cosecant of ". sec() = 1/cos(), read as "secant of ". cot() = 1/tan(), read as "cotangent of ". Let (x, y) be a point on the unit circle,then x=cos() and y=sin() as shown. (cos(),sin()) y=sin() (1,0)  x=cos()

8. Analytic Trigonometry We define the following reciprocals functions that are used often in science and engineering. csc() = 1/sin(), read as "cosecant of ". sec() = 1/cos(), read as "secant of ". cot() = 1/tan(), read as "cotangent of ". Let (x, y) be a point on the unit circle,then x=cos() and y=sin() as shown. By Pythagorean Theorem, y2 + x2 = 1, so we have: sin2() + cos2()=1 or s2 + c2 = 1 or all angle . (cos(),sin()) y=sin() (1,0)  x=cos()

9. Analytic Trigonometry The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles .

10. Analytic Trigonometry 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.

11. Analytic Trigonometry 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. The Trig-Hexagram

12. Analytic Trigonometry 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. The Trig-Hexagram the CO-side the regular-side

13. Analytic Trigonometry The Division Relations:

14. Analytic Trigonometry The Division Relations: Start from any function, going around the outside, we always have I = II / III or I * III = II

15. Analytic Trigonometry The Division Relations: Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), II III I

16. Analytic Trigonometry The Division Relations: Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = csc(A)/cot(A), III I II

17. Analytic Trigonometry The Division Relations: Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = csc(A)/cot(A), sin(A)cot(A) = cos(A) I II III

18. Analytic Trigonometry The Division Relations Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = csc(A)/cot(A), sin(A)cot(A) = cos(A) The Reciprocal Relations

19. Analytic Trigonometry The Division Relations Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = csc(A)/cot(A), sin(A)cot(A) = cos(A) The Reciprocal Relations Start from any function, going across diagonally, we always have I = II / III.

20. Analytic Trigonometry The Division Relations Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = csc(A)/cot(A), sin(A)cot(A) = cos(A) III II I The Reciprocal Relations Start from any function, going across diagonally, we always have I = II / III. Example B: sec(A) = 1/cos(A),

21. Analytic Trigonometry The Division Relations Start from any function, going around the outside, we always have I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = csc(A)/cot(A), sin(A)cot(A) = cos(A) I II III The Reciprocal Relations Start from any function, going across diagonally, we always have I = II / III. Example B: sec(A) = 1/cos(A), cot(A) = 1/tan(A)

22. Analytic Trigonometry Square-Sum Relations

23. Analytic Trigonometry Square-Sum Relations For each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one.

24. Analytic Trigonometry Square-Sum Relations For each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one. sin2(A) + cos2(A)=1

25. Analytic Trigonometry Square-Sum Relations For each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one. sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A)

26. Analytic Trigonometry Square-Sum Relations For each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one. sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A)

27. Analytic Trigonometry Square-Sum Relations For each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one. sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) The identities from this hexagram are called the Fundamental Identities.

28. Analytic Trigonometry Square-Sum Relations For each of the three inverted triangles, the sum of the squares of the top two is the square of the bottom one. sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) The identities from this hexagram are called the Fundamental Identities. Weassume these identities from here on and list the most important ones below.

29. Fundamental Identities Division Relations tan(A)=S/C cot(A)=C/S

30. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S

31. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S Square-Sum Relations sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A)

32. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S Square-Sum Relations sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) B A A and B are complementary Two angles are complementary if their sum is 90o.

33. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S Square-Sum Relations sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) B A A and B are complementary Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, we have the following co-relations.

34. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S Square-Sum Relations sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) B A A and B are complementary Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, we have the following co-relations. sin(A) = cos(90 – A) sin(A) = cos(B)

35. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S Square-Sum Relations sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) B A A and B are complementary Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, w have the following co-relations. sin(A) = cos(90 – A) tan(A) = cot(90 – A) sin(A) = cos(B) tan(A) = cot(B)

36. Fundamental Identities Reciprocal Relations sec(A)=1/C csc(A)=1/S cot(A)=1/T Division Relations tan(A)=S/C cot(A)=C/S Square-Sum Relations sin2(A) + cos2(A)=1 tan2(A) + 1 = sec2(A) 1 + cot2(A) = csc2(A) B A A and B are complementary Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles, w have the following co-relations. sin(A) = cos(90 – A) tan(A) = cot(90 – A) sec(A) = csc(90 – A) sin(A) = cos(B) tan(A) = cot(B) sec(A) = csc(B)

37. Fundamental Identities The Negative Angle Relations cos(-A) = cos(A), sin(-A) = - sin(A) A A -A -A 1 1

38. Fundamental Identities The Negative Angle Relations cos(-A) = cos(A), sin(-A) = - sin(A) A A -A -A 1 1 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).

39. Fundamental Identities The Negative Angle Relations cos(-A) = cos(A), sin(-A) = - sin(A) A A -A -A 1 1 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). Therefore cosine is an even function and sine is an odd function.

40. Fundamental Identities The Negative Angle Relations cos(-A) = cos(A), sin(-A) = - sin(A) A A -A -A 1 1 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). Therefore cosine is an even function and sine is an odd function. Since tangent and cotangent are quotients of sine and cosine, they are also odd functions.

41. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities.

42. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A)  Frank Ma 2006

43. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A)  Frank Ma 2006

44. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1  Frank Ma 2006

45. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1  Frank Ma 2006

46. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1  Frank Ma 2006

47. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y)  Frank Ma 2006

48. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y) Example D: (1 – sin(A))(1 + sin(A))  Frank Ma 2006

49. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y) Example D: (1 – sin(A))(1 + sin(A)) = 1 – sin2(A)  Frank Ma 2006

50. Notes on Square-Sum Identities Terms in the square-sum identities may be rearranged and give other versions of the identities. Example C: sin2(A) – 1 = – cos2(A) sec2(A) – tan2(A) = 1 cot2(A) – csc2(A) = -1 Difference of squares may be factored since x2 – y2 = (x – y)(x + y) Example D: (1 – sin(A))(1 + sin(A)) = 1 – sin2(A) = cos2(A)  Frank Ma 2006

51. Algebraic Identities Example E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine.

52. Algebraic Identities Example E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1)

53. Algebraic Identities Example E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1

54. Algebraic Identities Example E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1 ;square-relation = tan2(x)

55. Algebraic Identities Example E: Simplify (sec(x) – 1)(sec(x) + 1). Express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) ;difference of squares = sec2(x) – 1 ;square-relation = tan2(x) ; division relation = sin2(x) cos2(x)