X2 t01 10 complex & trig (2013)

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X2 t01 10 complex & trig (2013)

  1. 1. Complex Numbers & Trig Identities (i) Express cos 2 and sin 2 in terms of cos  and sin 
  2. 2. Complex Numbers & Trig Identities (i) Express cos 2 and sin 2 in terms of cos  and sin  cos 2  i sin 2  cos  i sin   2
  3. 3. Complex Numbers & Trig Identities (i) Express cos 2 and sin 2 in terms of cos  and sin  cos 2  i sin 2  cos  i sin    cos 2   2i sin  cos  sin 2  2
  4. 4. Complex Numbers & Trig Identities (i) Express cos 2 and sin 2 in terms of cos  and sin  cos 2  i sin 2  cos  i sin    cos 2   2i sin  cos  sin 2  By equating real and imaginary parts 2
  5. 5. Complex Numbers & Trig Identities (i) Express cos 2 and sin 2 in terms of cos  and sin  cos 2  i sin 2  cos  i sin    cos 2   2i sin  cos  sin 2  By equating real and imaginary parts 2 cos 2  cos 2   sin 2  sin 2  2 sin  cos
  6. 6. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan 
  7. 7. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin   3
  8. 8. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  3
  9. 9. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin   3
  10. 10. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts 3 sin 3  3sin  cos 2   sin 3 
  11. 11. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts 3 sin 3  3sin  cos 2   sin 3   3sin  1  sin 2    sin 3   3sin   3sin 3   sin 3   3sin   4sin 3 
  12. 12. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts 3 sin 3  3sin  cos 2   sin 3   3sin  1  sin 2    sin 3   3sin   3sin 3   sin 3   3sin   4sin 3  cos3  cos3   3sin 2  cos 
  13. 13. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts 3 sin 3  3sin  cos 2   sin 3   3sin  1  sin 2    sin 3   3sin   3sin 3   sin 3   3sin   4sin 3  cos3  cos3   3sin 2  cos   cos3   3 1  cos 2   cos   cos3   3cos   3cos3   4cos3   3cos 
  14. 14. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts sin 3 tan 3  sin 3  3sin  cos 2   sin 3  cos3  3sin  1  sin 2    sin 3  3sin  cos 2   sin 3   3 3 cos3   3sin 2  cos   3sin   3sin   sin  3  3sin   4sin 3  cos3  cos3   3sin 2  cos   cos3   3 1  cos 2   cos   cos3   3cos   3cos3   4cos3   3cos 
  15. 15. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts sin 3 tan 3  sin 3  3sin  cos 2   sin 3  cos3  3sin  1  sin 2    sin 3  3sin  cos 2   sin 3   3 3 cos3   3sin 2  cos   3sin   3sin   sin  3 3sin  cos 2   sin 3   3sin   4sin  cos3   3 2 cos3   3sin 2  cos  cos3  cos   3sin  cos  cos3  3 2  cos   3 1  cos   cos  3sin  sin 3    cos3   3cos   3cos3  cos  cos3   3 cos3  3sin 2   4cos   3cos   3 cos  cos 2  3
  16. 16. (ii) Express sin3 ,cos3 and tan 3 in terms of sin ,cos  and tan  cos3  i sin 3   cos   i sin    cos3   3i sin  cos 2   3sin 2  cos   i sin 3  By equating real and imaginary parts sin 3 tan 3  sin 3  3sin  cos 2   sin 3  cos3  3sin  1  sin 2    sin 3  3sin  cos 2   sin 3   3 3 cos3   3sin 2  cos   3sin   3sin   sin  3 3sin  cos 2   sin 3   3sin   4sin  cos3   3 2 cos3   3sin 2  cos  cos3  cos   3sin  cos  cos3  3 2  cos   3 1  cos   cos  3sin  sin 3   3  cos3   3cos   3cos3  cos  cos3   3tan   tan   3 cos3  3sin 2  1  3tan 2   4cos   3cos   3 cos  cos 2  3
  17. 17. (iii) Show that x  cos  9 is a solution to 8 x 3  6 x  1  0
  18. 18. (iii) Show that x  cos 8 x3  6 x  1  0 2(4 x 3  3 x)  1  0  9 is a solution to 8 x 3  6 x  1  0
  19. 19. (iii) Show that x  cos 8 x3  6 x  1  0 2(4 x 3  3 x)  1  0 let x  cos   9 is a solution to 8 x 3  6 x  1  0
  20. 20. (iii) Show that x  cos  9 is a solution to 8 x 3  6 x  1  0 8 x3  6 x  1  0 2(4 x 3  3 x)  1  0 let x  cos  2(4cos3   3cos  )  1  0 2cos3  1  0 1 cos3  2
  21. 21. (iii) Show that x  cos  9 is a solution to 8 x 3  6 x  1  0 8 x3  6 x  1  0 2(4 x 3  3 x)  1  0 let x  cos  2(4cos3   3cos  )  1  0 2cos3  1  0 1 cos3  2  5 7 3  , , 3 3 3  5 7  , , 9 9 9
  22. 22. (iii) Show that x  cos  9 is a solution to 8 x 3  6 x  1  0 8 x3  6 x  1  0 2(4 x 3  3 x)  1  0 let x  cos  2(4cos3   3cos  )  1  0 2cos3  1  0 1 cos3  2  5 7 3  , , 3 3 3  5 7  , , 9 9 9  5 7 x  cos ,cos ,sin 9 9 9
  23. 23. (iii) Show that x  cos  9 is a solution to 8 x 3  6 x  1  0 8 x3  6 x  1  0 2(4 x 3  3 x)  1  0 let x  cos  2(4cos3   3cos  )  1  0 2cos3  1  0 1 cos3  2  5 7 3  , , 3 3 3  5 7  , , 9 9 9  5 7 x  cos ,cos ,sin 9 9 9  x  cos  9 is a solution
  24. 24. (iii) Evaluate cos  9 cos 2 5 cos 9 9
  25. 25. (iii) Evaluate cos   9 cos 2 5 cos 9 9 5 7 x  cos ,cos ,sin are the roots of 8x 3  6 x  1  0 9 9 9
  26. 26. (iii) Evaluate cos  9 cos 2 5 cos 9 9  5 7 x  cos ,cos ,sin are the roots of 8x 3  6 x  1  0 9 9 9 1   8  5 7 1 cos cos cos  9 9 9 8 product of roots
  27. 27. (iii) Evaluate cos  9 cos 2 5 cos 9 9  5 7 x  cos ,cos ,sin are the roots of 8x 3  6 x  1  0 9 9 9 1   8  5 7 1 cos cos cos  9 9 9 8 but cos 7 2   cos 9 9 product of roots
  28. 28. (iii) Evaluate cos  9 cos 2 5 cos 9 9  5 7 x  cos ,cos ,sin are the roots of 8x 3  6 x  1  0 9 9 9 1   8  5 7 1 cos cos cos  9 9 9 8 but cos  7 2   cos 9 9 2 5 1 cos  cos cos  9 9 9 8 product of roots
  29. 29.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n
  30. 30.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z z n  cos n  i sin n n
  31. 31.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn
  32. 32.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn  cos n  i sin n cos is even function  cos x   cos x    sin is odd function  sin  x    sin x 
  33. 33.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn  cos n  i sin n cos is even function  cos x   cos x    sin is odd function  sin  x    sin x  1 z  n  2 cos n z n zn  1  2i sin n n z
  34. 34.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn  cos n  i sin n cos is even function  cos x   cos x    sin is odd function  sin  x    sin x  1 z  n  2 cos n z n v Express cos3  in terms of cos n zn  1  2i sin n n z
  35. 35.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn  cos n  i sin n cos is even function  cos x   cos x    sin is odd function  sin  x    sin x  1 z  n  2 cos n z n v Express cos3  in terms of cos n 3 3 z  1 2 cos     z  3 1 3 3 8 cos   z  3 z   3 z z zn  1  2i sin n n z
  36. 36.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn  cos n  i sin n cos is even function  cos x   cos x    sin is odd function  sin  x    sin x  1 z  n  2 cos n z n v Express cos3  in terms of cos n 3 3 z  1 2 cos     z  3 1 3 3 8 cos   z  3 z   3 z z 1 1   z 3  3   3 z       z   z  zn  1  2i sin n n z
  37. 37.  iv  1 1 n If z  cos   i sin  , find z  n and z  n z z n z n  cos n  i sin n 1  z n  cos n   i sin  n  zn  cos n  i sin n cos is even function  cos x   cos x    sin is odd function  sin  x    sin x  1 z  n  2 cos n z n v zn  1  2i sin n n z Express cos3  in terms of cos n 3 3 z  1 2 cos     z  3 1 3 3 8 cos   z  3 z   3 z z 1 1   z 3  3   3 z       z   z  8cos3   2cos3  6cos  1 3  cos   cos 3  cos 4 4 3
  38. 38. Cambridge: Exercise 7B; 2 to 5, 7 to 12

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