t4 sum and double half-angle formulas

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t4 sum and double half-angle formulas

  1. 1. Sum and Difference Formulas Double-Angle, and the Half-Angle Formulas<br />
  2. 2. Difference-Sum of Angles Formulas<br />–<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />S(A±B) = S(A)C(B) ± S(B)C(A)<br />Double-Angle Formulas<br />Half-Angle Formulas<br />S(2A) = 2S(A)C(A)<br /><br />1 + C(B)<br />B<br />±<br />C( ) =<br />2<br />2<br />C(2A) = C2(A) – S2(A)<br /> = 2C2(A) – 1<br /> = 1 – 2S2(A) <br /> Frank Ma<br />2006<br /><br />1 – C(B)<br />B<br />±<br />S( ) =<br />2<br />2<br />
  3. 3. Difference-Sum of Angles Formulas<br />–<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />S(A±B) = S(A)C(B) ± S(B)C(A)<br />Double-Angle Formulas<br />Half-Angle Formulas<br />S(2A) = 2S(A)C(A)<br /><br />1 + C(B)<br />B<br />±<br />C( ) =<br />2<br />2<br />C(2A) = C2(A) – S2(A)<br /> = 2C2(A) – 1<br /> = 1 – 2S2(A) <br /> Frank Ma<br />2006<br /><br />1 – C(B)<br />B<br />±<br />S( ) =<br />2<br />2<br />The cosine-difference formula is the basis for all the other formulas listed above.<br />
  4. 4. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />
  5. 5. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />
  6. 6. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />
  7. 7. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />
  8. 8. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />3π<br />8π<br />11π<br />=<br />+<br />12<br />12<br />12<br />
  9. 9. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />4<br />3<br />
  10. 10. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />π<br />π<br />π<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />=<br />– <br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />12<br />4<br />3<br />4<br />3<br />
  11. 11. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />π<br />π<br />π<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />=<br />– <br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />12<br />4<br />3<br />4<br />3<br />Example A: Find cos(11π/12) without a calculator.<br />
  12. 12. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />π<br />π<br />π<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />=<br />– <br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />12<br />4<br />3<br />4<br />3<br />Example A: Find cos(11π/12) without a calculator.<br />11π<br />π<br />2π<br />cos( ) <br />=<br />cos( ) <br />+<br />12<br />4<br />3<br />
  13. 13. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />π<br />π<br />π<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />=<br />– <br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />12<br />4<br />3<br />4<br />3<br />Example A: Find cos(11π/12) without a calculator.<br />11π<br />π<br />2π<br />π<br />2π<br />π<br />2π<br />cos( ) <br />=<br />cos( ) <br />c( ) <br />s( ) <br />=<br />c( ) <br />+<br />s( ) <br />–<br />12<br />4<br />3<br />4<br />3<br />4<br />3<br />Cosine-Sum Formulas<br />
  14. 14. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />π<br />π<br />π<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />=<br />– <br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />12<br />4<br />3<br />4<br />3<br />Example A: Find cos(11π/12) without a calculator.<br />11π<br />π<br />2π<br />π<br />2π<br />π<br />2π<br />cos( ) <br />=<br />cos( ) <br />c( ) <br />s( ) <br />=<br />c( ) <br />+<br />s( ) <br />–<br />12<br />4<br />3<br />4<br />3<br />4<br />3<br />2<br />2<br />3<br />(-1)<br />=<br />– <br />Cosine-Sum Formulas<br />2<br />2<br />2<br />2<br />
  15. 15. Cosine-Sum-Difference Formulas<br />cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<br />cos(A – B) = cos(A)cos(B) + sin(A)sin(B)<br />–<br />Short version:<br />C(A±B) = C(A)C(B) S(A)S(B)<br />+<br />All fractions with denominator 12 may be written as sum or difference of fractions with denominators <br />3, 6 and 4.<br />π<br />π<br />π<br />3π<br />8π<br />π<br />2π<br />11π<br />=<br />=<br />– <br />+<br />=<br />+<br />;<br />12<br />12<br />12<br />12<br />4<br />3<br />4<br />3<br />Example A: Find cos(11π/12) without a calculator.<br />11π<br />π<br />2π<br />π<br />2π<br />π<br />2π<br />cos( ) <br />=<br />cos( ) <br />c( ) <br />s( ) <br />=<br />c( ) <br />+<br />s( ) <br />–<br />12<br />4<br />3<br />4<br />3<br />4<br />3<br />2<br />2<br />3<br />-2 – 6<br />(-1)<br /> -0.966<br />=<br />– <br />=<br />Cosine-Sum Formulas<br />2<br />2<br />2<br />2<br />4<br />
  16. 16. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B))<br />
  17. 17. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, <br />
  18. 18. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />
  19. 19. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />
  20. 20. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />
  21. 21. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />Short version:<br />S(A±B) = S(A)C(B) ± C(A)S(B)<br />
  22. 22. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />Short version:<br />S(A±B) = S(A)C(B) ± C(A)S(B)<br />Example B: Find sin(– π/12) without a calculator.<br />
  23. 23. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />Short version:<br />S(A±B) = S(A)C(B) ± C(A)S(B)<br />Example B: Find sin(– π/12) without a calculator.<br />–π<br />π<br />π<br />sin( ) <br />=<br />sin( ) <br />–<br />12<br />4<br />3<br />
  24. 24. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />Short version:<br />S(A±B) = S(A)C(B) ± C(A)S(B)<br />Example B: Find sin(– π/12) without a calculator.<br />–π<br />π<br />π<br />π<br />π<br />π<br />π<br />sin( ) <br />=<br />sin( ) <br />c( ) <br />s( ) <br />=<br />s( ) <br />c( ) <br />–<br />–<br />12<br />4<br />3<br />4<br />3<br />4<br />3<br />Sum Formulas<br />
  25. 25. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />Short version:<br />S(A±B) = S(A)C(B) ± C(A)S(B)<br />Example B: Find sin(– π/12) without a calculator.<br />–π<br />π<br />π<br />π<br />π<br />π<br />π<br />sin( ) <br />=<br />sin( ) <br />c( ) <br />s( ) <br />=<br />s( ) <br />c( ) <br />–<br />–<br />12<br />4<br />3<br />4<br />3<br />4<br />3<br />2<br />2<br />3<br />1<br />=<br />– <br />=<br />Sum Formulas<br />2<br />2<br />2<br />2<br />
  26. 26. Sine-Sum-Difference Formulas<br />From the co-relation sin(A + B) = cos(π/2 – (A+B)) = cos((π/2 – A) – B) and exapnd, we get: <br />sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<br />Write sin(A – B) = sin(A + (-B)), expand we get: <br />sin(A – B) = sin(A)cos(B) – cos(A)sin(B)<br />Short version:<br />S(A±B) = S(A)C(B) ± C(A)S(B)<br />Example B: Find sin(– π/12) without a calculator.<br />–π<br />π<br />π<br />π<br />π<br />π<br />π<br />sin( ) <br />=<br />sin( ) <br />c( ) <br />s( ) <br />=<br />s( ) <br />c( ) <br />–<br />–<br />12<br />4<br />3<br />4<br />3<br />4<br />3<br />2<br />2<br />3<br />2 – 6<br />1<br /> -0.259<br />=<br />– <br />=<br />Sum Formulas<br />2<br />2<br />2<br />2<br />4<br />
  27. 27. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />
  28. 28. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) <br />
  29. 29. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />
  30. 30. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />
  31. 31. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />
  32. 32. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />
  33. 33. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />cos2(A) –(1 – cos2(A))<br />
  34. 34. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />cos2(A) –(1 – cos2(A)) <br />= 2cos2(A) – 1<br />
  35. 35. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />cos2(A) –(1 – cos2(A)) <br />= 2cos2(A) – 1<br />sin(2A) = sin(A + A) <br />
  36. 36. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />cos2(A) –(1 – cos2(A)) <br />= 2cos2(A) – 1<br />sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A)<br />
  37. 37. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />cos2(A) –(1 – cos2(A)) <br />= 2cos2(A) – 1<br />sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A)<br />sin(2A) = 2sin(A)cos(A) <br />
  38. 38. Double Angle Formulas<br />From the sum-of-angle formulas, we obtain the <br />double-angle formulas by setting A = B shown here,<br />cos(2A) = cos(A + A) = cos(A)cos(A) – sin(A)sin(A)<br />cos(2A) = cos2(A) – sin2(A) <br />(1 – sin2(A)) – sin2(A)<br />= 1 – 2sin2(A)<br />cos2(A) –(1 – cos2(A)) <br />= 2cos2(A) – 1<br />sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A)<br />sin(2A) = 2sin(A)cos(A) <br />Cosine Double Angle Formulas:<br />Sine Double Angle Formulas:<br />cos(2A) = cos2(A) – sin2(A) <br /> = 1 – 2sin2(A) <br /> = 2cos2(A) – 1 <br />sin(2A) = 2sin(A)cos(A) <br />
  39. 39. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />
  40. 40. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, <br />
  41. 41. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1<br />
  42. 42. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br />
  43. 43. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />
  44. 44. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />
  45. 45. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />Since A is in 2nd quad.=> cos(A) = - 5/7 <br /> Frank Ma<br />2006<br />
  46. 46. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />Since A is in 2nd quad.=> cos(A) = - 5/7 <br /> Frank Ma<br />2006<br />y<br />A<br />
  47. 47. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />Since A is in 2nd quad.=> cos(A) = - 5/7 <br /> Frank Ma<br />2006<br />y2 + (-5)2 = (7)2<br />y<br />A<br />
  48. 48. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />Since A is in 2nd quad.=> cos(A) = - 5/7 <br /> Frank Ma<br />2006<br />y2 + (-5)2 = (7)2<br />y2 = 2<br />y<br />A<br />
  49. 49. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />Since A is in 2nd quad.=> cos(A) = - 5/7 <br /> Frank Ma<br />2006<br />y2 + (-5)2 = (7)2<br />y2 = 2<br />y<br />A<br />y = ±2 <br />y = 2 <br />
  50. 50. Double Angle Formulas<br />Example C: Given angle A in the 2nd quad. and <br />cos(2A)= 3/7, find tan(A). <br />Use the formula cos(2A) = 2cos2(A) – 1, we get<br /> 3/7 = 2cos2(A) – 1 <br /> 10/7 = 2cos2(A)<br /> 5/7 = cos2(A)<br />±5/7 = cos(A)<br />Since A is in 2nd quad.=> cos(A) = - 5/7 <br /> Frank Ma<br />2006<br />y2 + (-5)2 = (7)2<br />y2 = 2<br />y<br />A<br />y = ±2 <br />y = 2 <br />2<br /><br />Therefore tan(A) = <br />–<br /> -0.632<br />5<br />
  51. 51. Half-angle Formulas<br />From cos(2A) = 2cos2(A) – 1, we get <br />
  52. 52. Half-angle Formulas<br />From cos(2A) = 2cos2(A) – 1, we get <br />1+cos(2A)<br />cos2(A) =<br />2<br />
  53. 53. Half-angle Formulas<br />From cos(2A) = 2cos2(A) – 1, we get <br />1+cos(2A)<br />cos2(A) =<br />2<br />In the square root form, we get<br /><br />1+cos(2A)<br />±<br />cos(A) =<br />2<br />
  54. 54. Half-angle Formulas<br />From cos(2A) = 2cos2(A) – 1, we get <br />1+cos(2A)<br />cos2(A) =<br />2<br />In the square root form, we get<br /><br />1+cos(2A)<br />±<br />cos(A) =<br />2<br />if we replace A by B/2 so that 2A = B, <br />
  55. 55. Half-angle Formulas<br />From cos(2A) = 2cos2(A) – 1, we get <br />1+cos(2A)<br />cos2(A) =<br />2<br />In the square root form, we get<br /><br />1+cos(2A)<br />±<br />cos(A) =<br />2<br />if we replace A by B/2 so that 2A = B, we get the <br />half-angle formula of cosine:<br />B<br /><br />1+cos(B)<br />±<br />cos( ) =<br />2<br />2<br />
  56. 56. Half-angle Formulas<br />From cos(2A) = 2cos2(A) – 1, we get <br />1+cos(2A)<br />cos2(A) =<br />2<br />In the square root form, we get<br /><br />1+cos(2A)<br />±<br />cos(A) =<br />2<br />if we replace A by B/2 so that 2A = B, we get the <br />half-angle formula of cosine:<br />B<br /><br />1+cos(B)<br />±<br />cos( ) =<br />2<br />2<br />Similarly, we get the half-angle formula of sine: <br />B<br /><br />1 – cos(B)<br />±<br />sin( ) =<br />2<br />2<br />
  57. 57. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />
  58. 58. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />
  59. 59. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />-7<br />-3<br />A<br />
  60. 60. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />-7<br />-3<br />A<br />58<br />
  61. 61. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />Since –π < A < –π /2, so <br />–π/2 < A/2 < –π/4, <br />-7<br />-3<br />A<br />58<br />
  62. 62. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />Since –π < A < –π /2, so <br />–π/2 < A/2 < –π/4, we have that<br />A/2 is in the 4th quadrant. <br />-7<br />-3<br />A<br />58<br />
  63. 63. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />Since –π < A < –π /2, so <br />–π/2 < A/2 < –π/4, we have that<br />A/2 is in the 4th quadrant. <br />-7<br />-3<br />1 + cos(A)<br /><br />A<br />A<br />cos( ) =<br />Hence,<br />58<br />2<br />2<br />
  64. 64. Half-angle Formulas<br /><br /><br />1 – cos(B)<br />B<br />B<br />±<br />±<br />1+cos(B)<br />cos( ) =<br />sin( ) =<br />and<br />2<br />2<br />2<br />2<br />The ± are to be determined by the position of the <br />angle B/2. <br />Example D: Given A where –π < A < –π /2, and tan(A) = 3/7, draw A and find cos(A/2).<br />Since –π < A < –π /2, so <br />–π/2 < A/2 < –π/4, we have that<br />A/2 is in the 4th quadrant. <br />-7<br />-3<br />1 + cos(A)<br /><br />A<br />A<br />cos( ) =<br />Hence,<br />58<br />2<br />2<br /><br />1 – 7 /58<br /> 0.201<br />=<br />2<br />
  65. 65. Sum of Angles Formulas<br />±<br />Double Angle Formulas<br />Half Angle Formulas<br />sin(2A) = 2sin(A)cos(A)<br /><br />1+cos(B)<br />B<br />±<br />cos( ) =<br />2<br />2<br />cos(2A) = cos2(A) – sin2(A)<br /> = 2cos2(A) – 1<br /> = 1 – 2sin2(A) <br /> Frank Ma<br />2006<br /><br />1 – cos(B)<br />B<br />±<br />sin( ) =<br />2<br />2<br />

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