11X1 T08 05 t results (2010)

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  1. 1. The t results
  2. 2. The t results Let t tan 2
  3. 3. The t results Let t tan2 2 tan tan 2 2 1 tan 2 2t tan 1 t 2
  4. 4. The t results Let t tan2 2t 2 tan tan 2 21 t 2 1 tan 2 2t tan 1 t 2
  5. 5. The t results Let t tan2 1 t2 2t 2 tan tan 2 21 t 2 1 tan 2h 2t 1 t2 2 2 2 2t tan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 1 2 2
  6. 6. The t results Let t tan2 1 t2 2t 2 tan tan 2 21 t 2 1 tan 2h 2t 1 t2 2 2 2 2t tan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 1 2 If t tan ; 222ttan 1 t 2
  7. 7. The t results Let t tan2 1 t2 2t 2 tan tan 2 21 t 2 1 tan 2 h 2t 1 t 2 222 2t tan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 1 2 If t tan ; 222t 2ttan sin 1 t 2 1 t 2
  8. 8. The t results Let t tan21 t2 2t 2 tan tan 2 2 1 t 2 1 tan 2h 2t 1 t 2 2 22 2t tan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 12 If t tan ;22 2t 2t 1 t2tan sin cos 1 t 2 1 t 21 t2
  9. 9. The t resultsLet t tan21 t2 2t 2 tan tan 2 2 1 t 2 1 tan 2h 2t 1 t 2 2 22 2t tan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 12 If t tan ;22 2t 2t 1 t2tan sin cos 1 t 2 1 t 21 t2 Note: 2tIf t tan ; tan 2 1 t2
  10. 10. The t resultsLet t tan21 t2 2t 2 tan tan 2 2 1 t 2 1 tan 2h 2t 1 t 2 2 22 2t tan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 12 If t tan ;22 2t 2t 1 t2tan sin cos 1 t 2 1 t 21 t2 Note: 2t 2tIf t tan ; tan 2 If t tan 2 ; tan 4 1 t21 t2
  11. 11. 1 cos x x e.g. i Show that t , where t tansin x 2
  12. 12. 1 cos xx e.g. i Show that t , where t tansin x2 1 t2 1 cos x 1 1 t 2x x is double , 2 sin x 2t 1 t2 so t results can be used for sin x,cos x
  13. 13. 1 cos x x e.g. i Show that t , where t tansin x 2 1 t2 1x 1 cos x 1 t 2x is double , 2 sin x 2t 1 t2 so t results can be used for sin x,cos x 1 t 2 1 t 2 2t
  14. 14. 1 cos x x e.g. i Show that t , where t tansin x 2 1 t2 1x 1 cos x 1 t 2x is double , 2 sin x 2t 1 t2 so t results can be used for sin x,cos x 1 t 2 1 t 2 2t2t 22t t
  15. 15. 2 tan 75 ii Use t tan to simplify 21 tan 2 75
  16. 16. 2 tan 75 ii Use t tan to simplify 21 tan 2 75 Let t tan 75 ;
  17. 17. 2 tan 75 ii Use t tan to simplify 21 tan 2 75 Let t tan 75 ; i.e. 75 2 150
  18. 18. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; i.e. 75 2 150
  19. 19. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin i.e. 75 2 150
  20. 20. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin sin150i.e. 75 1 2 1502
  21. 21. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin sin150 i.e. 7512 150 2 1 sin cos iii Prove tan 1 sin cos 2
  22. 22. 2 tan 752 tan 75 2t ii Use t tan to simplify 21 tan 2 751 tan 2 75 1 t 2 Let t tan 75 ; sin sin150i.e. 751 2 1502 1 sin cos iii Prove tan 1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2Let t tan ; 2 1 sin cos 2t 1 t 2 11 t 2 1 t2
  23. 23. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin sin150i.e. 75 1 2 150 2 1 sin cos iii Prove tan 1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2Let t tan ; 2 1 sin cos 2t 1 t 2 1 1 t 1 t22 1 t 2 2t 1 t 2 1 t 2 2t 1 t 2
  24. 24. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin sin150i.e. 75 1 2 150 2 1 sin cos iii Prove tan 1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2Let t tan ; 2 1 sin cos 2t 1 t 2 1 1 t 1 t22 1 t 2 2t 1 t 2 1 t 2 2t 1 t 2 2t 2 2t 2 2t
  25. 25. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin sin150i.e. 75 1 2 150 2 1 sin cos iii Prove tan 1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2Let t tan ; 2 1 sin cos 2t 1 t 2 1 1 t 1 t22 1 t 2 2t 1 t 2 1 t 2 2t 1 t 2 2t 2 2t 2 2t 2t t 1 2 1 t
  26. 26. 2 tan 75 2 tan 75 2t ii Use t tan to simplify 21 tan 2 75 1 tan 2 75 1 t 2 Let t tan 75 ; sin sin150i.e. 75 1 2 150 2 1 sin cos iii Prove tan 1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2Let t tan ; 2 1 sin cos 2t 1 t 2 1 1 t 1 t22 1 t 2 2t 1 t 2 1 t 2 2t 1 t 2 2t 2 2t 2 2t 2t t 1 t tan 2 1 t 2
  27. 27. (iv) By making the substitution t tan or otherwise, 2 show that cosec cot cot2005 Extension 1 HSC Q4b)2
  28. 28. (iv) By making the substitution t tan or otherwise, 2 show that cosec cot cot2005 Extension 1 HSC Q4b)21 t2 1 t2 cosec cot 2t2t
  29. 29. (iv) By making the substitution t tan or otherwise, 2 show that cosec cot cot2005 Extension 1 HSC Q4b)21 t2 1 t2 cosec cot 2t 2t 22t1t
  30. 30. (iv) By making the substitution t tan or otherwise, 2 show that cosec cot cot2005 Extension 1 HSC Q4b)21 t2 1 t2 cosec cot 2t 2t 22t1t cot 2
  31. 31. (iv) By making the substitution t tan or otherwise, 2 show that cosec cot cot2005 Extension 1 HSC Q4b)21 t2 1 t2 cosec cot 2t 2t 22t1t cot 2 Exercise 2B; 1def, 3ace, 4bcf, 5aceg, 6, 8bd, 10a