6. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) +
7. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4.
8. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. 3π 8π 11π = + 12 12 12
9. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. 3π 8π π 2π 11π = + = + ; 12 12 12 4 3
10. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. π π π 3π 8π π 2π 11π = = – + = + ; 12 12 12 12 4 3 4 3
11. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. π π π 3π 8π π 2π 11π = = – + = + ; 12 12 12 12 4 3 4 3 Example A: Find cos(11π/12) without a calculator.
12. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. π π π 3π 8π π 2π 11π = = – + = + ; 12 12 12 12 4 3 4 3 Example A: Find cos(11π/12) without a calculator. 11π π 2π cos( ) = cos( ) + 12 4 3
13. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. π π π 3π 8π π 2π 11π = = – + = + ; 12 12 12 12 4 3 4 3 Example A: Find cos(11π/12) without a calculator. 11π π 2π π 2π π 2π cos( ) = cos( ) c( ) s( ) = c( ) + s( ) – 12 4 3 4 3 4 3 Cosine-Sum Formulas
14. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. π π π 3π 8π π 2π 11π = = – + = + ; 12 12 12 12 4 3 4 3 Example A: Find cos(11π/12) without a calculator. 11π π 2π π 2π π 2π cos( ) = cos( ) c( ) s( ) = c( ) + s( ) – 12 4 3 4 3 4 3 2 2 3 (-1) = – Cosine-Sum Formulas 2 2 2 2
15. Cosine-Sum-Difference Formulas cos(A + B) = cos(A)cos(B) – sin(A)sin(B) cos(A – B) = cos(A)cos(B) + sin(A)sin(B) – Short version: C(A±B) = C(A)C(B) S(A)S(B) + All fractions with denominator 12 may be written as sum or difference of fractions with denominators 3, 6 and 4. π π π 3π 8π π 2π 11π = = – + = + ; 12 12 12 12 4 3 4 3 Example A: Find cos(11π/12) without a calculator. 11π π 2π π 2π π 2π cos( ) = cos( ) c( ) s( ) = c( ) + s( ) – 12 4 3 4 3 4 3 2 2 3 -2 – 6 (-1) -0.966 = – = Cosine-Sum Formulas 2 2 2 2 4
18. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
19. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get:
20. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B)
21. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B) Short version: S(A±B) = S(A)C(B) ± C(A)S(B)
22. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B) Short version: S(A±B) = S(A)C(B) ± C(A)S(B) Example B: Find sin(– π/12) without a calculator.
23. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B) Short version: S(A±B) = S(A)C(B) ± C(A)S(B) Example B: Find sin(– π/12) without a calculator. –π π π sin( ) = sin( ) – 12 4 3
24. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B) Short version: S(A±B) = S(A)C(B) ± C(A)S(B) Example B: Find sin(– π/12) without a calculator. –π π π π π π π sin( ) = sin( ) c( ) s( ) = s( ) c( ) – – 12 4 3 4 3 4 3 Sum Formulas
25. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B) Short version: S(A±B) = S(A)C(B) ± C(A)S(B) Example B: Find sin(– π/12) without a calculator. –π π π π π π π sin( ) = sin( ) c( ) s( ) = s( ) c( ) – – 12 4 3 4 3 4 3 2 2 3 1 = – = Sum Formulas 2 2 2 2
26. Sine-Sum-Difference Formulas From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Write sin(A – B) = sin(A + (-B)), expand we get: sin(A – B) = sin(A)cos(B) – cos(A)sin(B) Short version: S(A±B) = S(A)C(B) ± C(A)S(B) Example B: Find sin(– π/12) without a calculator. –π π π π π π π sin( ) = sin( ) c( ) s( ) = s( ) c( ) – – 12 4 3 4 3 4 3 2 2 3 2 – 6 1 -0.259 = – = Sum Formulas 2 2 2 2 4
27. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here,
28. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A)
29. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)
30. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A)
31. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A)
32. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A)
33. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A) cos2(A) –(1 – cos2(A))
34. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A) cos2(A) –(1 – cos2(A)) = 2cos2(A) – 1
35. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A) cos2(A) –(1 – cos2(A)) = 2cos2(A) – 1 sin(2A) = sin(A + A)
36. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A) cos2(A) –(1 – cos2(A)) = 2cos2(A) – 1 sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A)
37. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A) cos2(A) –(1 – cos2(A)) = 2cos2(A) – 1 sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A) sin(2A) = 2sin(A)cos(A)
38. Double Angle Formulas From the sum-of-angle formulas, we obtain the double-angle formulas by setting A = B shown here, cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A) cos(2A) = cos2(A) – sin2(A) (1 – sin2(A)) – sin2(A) = 1 – 2sin2(A) cos2(A) –(1 – cos2(A)) = 2cos2(A) – 1 sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A) sin(2A) = 2sin(A)cos(A) Cosine Double Angle Formulas: Sine Double Angle Formulas: cos(2A) = cos2(A) – sin2(A) = 1 – 2sin2(A) = 2cos2(A) – 1 sin(2A) = 2sin(A)cos(A)
39. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A).
40. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1,
41. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1
42. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A)
43. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A)
44. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A)
45. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A) Since A is in 2nd quad.=> cos(A) = - 5/7 Frank Ma 2006
46. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A) Since A is in 2nd quad.=> cos(A) = - 5/7 Frank Ma 2006 y A
47. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A) Since A is in 2nd quad.=> cos(A) = - 5/7 Frank Ma 2006 y2 + (-5)2 = (7)2 y A
48. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A) Since A is in 2nd quad.=> cos(A) = - 5/7 Frank Ma 2006 y2 + (-5)2 = (7)2 y2 = 2 y A
49. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A) Since A is in 2nd quad.=> cos(A) = - 5/7 Frank Ma 2006 y2 + (-5)2 = (7)2 y2 = 2 y A y = ±2 y = 2
50. Double Angle Formulas Example C: Given angle A in the 2nd quad. and cos(2A)= 3/7, find tan(A). Use the formula cos(2A) = 2cos2(A) – 1, we get 3/7 = 2cos2(A) – 1 10/7 = 2cos2(A) 5/7 = cos2(A) ±5/7 = cos(A) Since A is in 2nd quad.=> cos(A) = - 5/7 Frank Ma 2006 y2 + (-5)2 = (7)2 y2 = 2 y A y = ±2 y = 2 2 Therefore tan(A) = – -0.632 5
53. Half-angle Formulas From cos(2A) = 2cos2(A) – 1, we get 1+cos(2A) cos2(A) = 2 In the square root form, we get 1+cos(2A) ± cos(A) = 2
54. Half-angle Formulas From cos(2A) = 2cos2(A) – 1, we get 1+cos(2A) cos2(A) = 2 In the square root form, we get 1+cos(2A) ± cos(A) = 2 if we replace A by B/2 so that 2A = B,
55. Half-angle Formulas From cos(2A) = 2cos2(A) – 1, we get 1+cos(2A) cos2(A) = 2 In the square root form, we get 1+cos(2A) ± cos(A) = 2 if we replace A by B/2 so that 2A = B, we get the half-angle formula of cosine: B 1+cos(B) ± cos( ) = 2 2
56. Half-angle Formulas From cos(2A) = 2cos2(A) – 1, we get 1+cos(2A) cos2(A) = 2 In the square root form, we get 1+cos(2A) ± cos(A) = 2 if we replace A by B/2 so that 2A = B, we get the half-angle formula of cosine: B 1+cos(B) ± cos( ) = 2 2 Similarly, we get the half-angle formula of sine: B 1 – cos(B) ± sin( ) = 2 2
57. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2.
58. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).
59. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2). -7 -3 A
60. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2). -7 -3 A 58
61. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2). Since –π < A < –π /2, so –π/2 < A/2 < –π/4, -7 -3 A 58
62. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2). Since –π < A < –π /2, so –π/2 < A/2 < –π/4, we have that A/2 is in the 4th quadrant. -7 -3 A 58
63. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2). Since –π < A < –π /2, so –π/2 < A/2 < –π/4, we have that A/2 is in the 4th quadrant. -7 -3 1 + cos(A) A A cos( ) = Hence, 58 2 2
64. Half-angle Formulas 1 – cos(B) B B ± ± 1+cos(B) cos( ) = sin( ) = and 2 2 2 2 The ± are to be determined by the position of the angle B/2. Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2). Since –π < A < –π /2, so –π/2 < A/2 < –π/4, we have that A/2 is in the 4th quadrant. -7 -3 1 + cos(A) A A cos( ) = Hence, 58 2 2 1 – 7 /58 0.201 = 2
65. Sum of Angles Formulas ± Double Angle Formulas Half Angle Formulas sin(2A) = 2sin(A)cos(A) 1+cos(B) B ± cos( ) = 2 2 cos(2A) = cos2(A) – sin2(A) = 2cos2(A) – 1 = 1 – 2sin2(A) Frank Ma 2006 1 – cos(B) B ± sin( ) = 2 2
