8. analytic trigonometry and trig formulas-x
- 2. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). (1,0) The unit circle x2 + y2 = 1 (x, y)
- 3. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). Here is the picture of the geometric measurements, related to the unit circle, of the trig–functions. cos() sin() (1,0) The unit circle x2 + y2 = 1 (x, y)
- 4. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). Here is the picture of the geometric measurements, related to the unit circle, of the trig–functions. cos() sin() (1,0) tan() The unit circle x2 + y2 = 1 (x, y)
- 5. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). Here is the picture of the geometric measurements, related to the unit circle, of the trig–functions. cos() sin() (1,0) tan() cot() The unit circle x2 + y2 = 1 (x, y)
- 6. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). Here is the picture of the geometric measurements, related to the unit circle, of the trig–functions. cos() sin() (1,0) tan() cot() The following reciprocal functions are useful in science and engineering: csc() = 1/sin() = 1/y, the “cosecant of ". sec() = 1/cos() = 1/x, the "secant of " (x, y) The unit circle x2 + y2 = 1
- 7. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). Here is the picture of the geometric measurements, related to the unit circle, of the trig–functions. cos() sin() (1,0) tan() cot() The following reciprocal functions are useful in science and engineering: csc() = 1/sin() = 1/y, the “cosecant of ". sec() = 1/cos() = 1/x, the "secant of " Since y2 + x2 = 1, we’ve sin2() + cos2() = 1 or s()2 + c()2 = 1 or simply s2 + c2 = 1 all angles . This is the most important identity in trigonometry.. (x, y) The unit circle x2 + y2 = 1
- 8. Analytic Trigonometry Let (x, y) be a point on the unit circle, then x = cos() and y = sin(). Here is the picture of the geometric measurements, related to the unit circle, of the trig–functions. cos() sin() (1,0) tan() cot() The following reciprocal functions are useful in science and engineering: csc() = 1/sin() = 1/y, the “cosecant of ". sec() = 1/cos() = 1/x, the "secant of " Since y2 + x2 = 1, we’ve sin2() + cos2() = 1 or s()2 + c()2 = 1 or simply s2 + c2 = 1 all angle . This is the most important identity in trigonometry. (x, y) The unit circle x2 + y2 = 1
- 9. Analytic Trigonometry If we divide the identity s2 + c2 = 1 by s2, we obtain 1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
- 10. Analytic Trigonometry If we divide the identity s2 + c2 = 1 by s2, we obtain 1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2(). If we divide the identity s2 + c2 = 1 by c2, we obtain (s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
- 11. Analytic Trigonometry If we divide the identity s2 + c2 = 1 by s2, we obtain 1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2(). If we divide the identity s2 + c2 = 1 by c2, we obtain (s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2(). The Trig–HexagramThe hexagram shown here is a useful visual tool for recalling various relations.
- 12. Analytic Trigonometry If we divide the identity s2 + c2 = 1 by s2, we obtain 1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2(). If we divide the identity s2 + c2 = 1 by c2, we obtain (s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2(). The Trig–HexagramThe hexagram shown here is a useful visual tool for recalling various relations. The right side of the hexagram is the “co”–side listing all the co–functions. The “co”–side
- 13. Analytic Trigonometry If we divide the identity s2 + c2 = 1 by s2, we obtain 1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2(). If we divide the identity s2 + c2 = 1 by c2, we obtain (s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2(). The Trig–HexagramThe hexagram shown here is a useful visual tool for recalling various relations. The right side of the hexagram is the “co”–side listing all the co–functions. The following three groups of formulas extracted from this hexagram are referred to as the fundamental trig–identities. The “co”–side
- 14. Analytic Trigonometry The Division Relations
- 15. Analytic Trigonometry The Division Relations Starting from any function I, going around the perimeter to functions II and III, then I = II / III or I * III = II
- 16. Analytic Trigonometry The Division Relations Starting from any function I, going around the perimeter to functions II and III, then I = II / III or I * III = II Example A: tan(A) = sec(A) = sin(A)cot(A) =
- 17. I II III Analytic Trigonometry The Division Relations Starting from any function I, going around the perimeter to functions II and III, then I = II / III or I * III = II Example A: tan(A) = sin(A)/cos(A), sec(A) = sin(A)cot(A) = ÷
- 18. I II III Analytic Trigonometry The Division Relations Starting from any function I, going around the perimeter to functions II and III, then 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) = ÷
- 19. I II III Analytic Trigonometry The Division Relations Starting from any function I, going around the perimeter to functions II and III, then 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)
- 20. Analytic Trigonometry The Reciprocal Relations The Division Relations Starting from any function I, going around the perimeter to functions II and III, then 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)
- 21. Analytic Trigonometry The Reciprocal Relations Start from any function, going across diagonally, we always have I = 1 / III. The Division Relations Starting from any function I, going around the perimeter to functions II and III, then 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)
- 22. I III Analytic Trigonometry The Reciprocal Relations Start from any function, going across diagonally, we always have I = 1 / III. Example B: sec(A) = 1/cos(A), ÷II The Division Relations Starting from any function I, going around the perimeter to functions II and III, then 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)
- 23. IIII Analytic Trigonometry The Reciprocal Relations Start from any function, going across diagonally, we always have I = 1 / III. Example B. sec(A) = 1/cos(A), cot(A) = 1/tan(A) The Division Relations Starting from any function I, going around the perimeter to functions II and III, then 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) ÷ II
- 25. 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. Analytic Trigonometry
- 26. 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 Analytic Trigonometry
- 27. 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) Analytic Trigonometry
- 28. 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) Analytic Trigonometry
- 29. 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) Analytic Trigonometry Identities from this hexagram in the above manner are called the fundamental Identities. We assume these identities from here on and list the most important ones below.
- 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 C = cos(A) S = sin(A) T = tan(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) C = cos(A) S = sin(A) T = tan(A)
- 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) Two angles are complementary if their sum is 90o. A B A and B are complementary C = cos(A) S = sin(A) T = tan(A)
- 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) Two angles are complementary if their sum is 90o. Let A and B = 90 – A be complementary angles, we have the following co–relations. A B A and B are complementary C = cos(A) S = sin(A) T = tan(A)
- 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) Two angles are complementary if their sum is 90o. Let A and B = 90 – A be complementary angles, we have the following co–relations. S(A) = C(B) = C(90 – A) T(A) = cot(B) = cot(90 – A) sec(A) = csc(B) = csc(90 – A) A B A and B are complementary C = cos(A) S = sin(A) T = tan(A)
- 36. cos(–A) = cos(A), sin(–A) = – sin(A) The Negative Angle Relations Fundamental Identities 1 A –A 1 A –A sin(A) sin(–A)
- 37. cos(–A) = cos(A), sin(–A) = – sin(A) The Negative Angle Relations Fundamental Identities Recall that f(x) is even if and only if f(–x) = f(x), and that f(x) is odd if and only if f(–x) = –f(x). 1 A –A 1 A –A sin(A) sin(–A)
- 38. cos(–A) = cos(A), sin(–A) = – sin(A) The Negative Angle Relations Fundamental Identities 1 A –A 1 A –A Therefore cosine is an even function and sine is an odd function. Recall that f(x) is even if and only if f(–x) = f(x), and that f(x) is odd if and only if f(–x) = –f(x). sin(A) sin(–A)
- 39. cos(–A) = cos(A), sin(–A) = – sin(A) The Negative Angle Relations Fundamental Identities 1 A –A 1 A –A Therefore cosine is an even function and sine is an odd function. Since tangent and cotangent are quotients of sine and cosine, so they are also odd. Recall that f(x) is even if and only if f(–x) = f(x), and that f(x) is odd if and only if f(–x) = –f(x). sin(A) sin(–A)
- 40. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas.
- 41. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas. Example C. a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
- 42. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas. Example C. a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A)) b. sec2(A) – tan2(A) = 1 = (sec(A) – tan(A))(sec(A) + tan(A))
- 43. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas. Example C. a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A)) b. sec2(A) – tan2(A) = 1 = (sec(A) – tan(A))(sec(A) + tan(A)) c. cot2(A) – csc2(A) = –1 = (cot(A) – csc(A))(cot(A) + csc(A))
- 44. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas. Example C. a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A)) b. sec2(A) – tan2(A) = 1 = (sec(A) – tan(A))(sec(A) + tan(A)) c. cot2(A) – csc2(A) = –1 = (cot(A) – csc(A))(cot(A) + csc(A)) Example D. Simplify (sec(x) – 1)(sec(x) + 1) And express the answer in sine and cosine.
- 45. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas. Example C. a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A)) b. sec2(A) – tan2(A) = 1 = (sec(A) – tan(A))(sec(A) + tan(A)) c. cot2(A) – csc2(A) = –1 = (cot(A) – csc(A))(cot(A) + csc(A)) Example D. Simplify (sec(x) – 1)(sec(x) + 1) And express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) = sec2(x) – 1
- 46. Notes on Square–Sum Identities The square–sum identities may be rearranged into the difference-of-squares identities which have factored versions of the formulas. Example C. a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A)) b. sec2(A) – tan2(A) = 1 = (sec(A) – tan(A))(sec(A) + tan(A)) c. cot2(A) – csc2(A) = –1 = (cot(A) – csc(A))(cot(A) + csc(A)) Example D. Simplify (sec(x) – 1)(sec(x) + 1) And express the answer in sine and cosine. (sec(x) – 1)(sec(x) + 1) = sec2(x) – 1 = T2(x) = S2(x) C2(x)
- 47. Notes on Square–Sum Identities Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π.
- 48. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π.
- 49. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π.
- 50. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Hence 2S = 1, or S(x) = ½ Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π.
- 51. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6, Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π.
- 52. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6, or that S = 1, so x = π/2. Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π.
- 53. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6, or that S = 1, so x = π/2. Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π. b. Find all solutions for x.
- 54. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6, or that S = 1, so x = π/2. Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π. b. Find all solutions for x. By adding 2nπ where n is an integer, to the solutions from a. we obtain all the solutions for x.
- 55. Notes on Square–Sum Identities Writing S(x) as S and replacing C2 as 1 – S2, we’ve S2 – (1 – S2) = S or 2S2 – S – 1 = 0 so (2S + 1)(S – 1) = 0 Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6, or that S = 1, so x = π/2. Example E. Solve the equation S2(x) – C2(x) = S(x). a. Find all solutions x between 0 and 2π. b. Find all solutions for x. By adding 2nπ where n is an integer, to the solutions from a. we obtain all the solutions for x. Here is a list: {π/6 or π/2 or 5π/6 + 2nπ, where n is an integer}.