11 x1 t08 05 t results (2012)

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<ol><li> 1. The t results </li><li> 2. The t resultsLet t tan2 </li><li> 3. The t resultsLet t tan 22 tantan 21 tan 222ttan 1 t 2 </li><li> 4. The t resultsLet t tan 22t2 tantan 2 1 t 21 tan 222ttan 1 t 2 </li><li> 5. The t resultsLet t tan 2 1 t2 2t2 tantan 2 1 t 21 tan 22h 2t 1 t 2 22 22ttan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 12 2 </li><li> 6. The t resultsLet t tan 2 1 t2 2t2 tantan 2 1 t 21 tan 22h 2t 1 t 2 22 22ttan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 12If t tan ;22 2t tan 1 t 2 </li><li> 7. The t resultsLet t tan 2 1 t2 2t2 tantan 2 1 t 21 tan 22 h 2t 1 t 2 2 222ttan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 12If t tan ;22 2t 2t tan sin 1 t 2 1 t 2 </li><li> 8. The t resultsLet t tan 21 t22t2 tantan 2 1 t 21 tan 22h 2t 1 t 2 2222ttan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 1 2If t tan ; 222t 2t 1 t2 tan sin cos 1 t 2 1 t 21 t2 </li><li> 9. The t results Let t tan 21 t22t2 tantan 2 1 t 21 tan 22h 2t 1 t 2 2222ttan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 1 2If t tan ; 222t 2t 1 t2 tan sin cos 1 t 2 1 t 21 t2Note:2t If t tan ; tan 2 1 t2 </li><li> 10. The t results Let t tan 21 t22t2 tantan 2 1 t 21 tan 22h 2t 1 t 2 2222ttan 1 t 2 4t 2 1 2t 2 t 4 t 4 2t 2 1 t 1 2If t tan ; 222t 2t 1 t2 tan sin cos 1 t 2 1 t 21 t2Note:2t 2t If t tan ; tan 2 If t tan 2 ; tan 4 1 t21 t2 </li><li> 11. 1 cos x xe.g. i Show that t , where t tan sin x 2 </li><li> 12. 1 cos xxe.g. i Show that t , where t tan sin x21 t21 cos x 1 1 t 2xx is double , 2 sin x 2t 1 t2 so t results can be used for sin x,cos x </li><li> 13. 1 cos x xe.g. i Show that t , where t tan sin x 21 t21x 1 cos x 1 t 2 x is double , 2 sin x 2t1 t2 so t results can be used for sin x,cos x 1 t 2 1 t 2 2t </li><li> 14. 1 cos x xe.g. i Show that t , where t tan sin x 21 t21x 1 cos x 1 t 2 x is double , 2 sin x 2t1 t2 so t results can be used for sin x,cos x 1 t 2 1 t 2 2t 2t 2 2tt </li><li> 15. 2 tan 75 ii Use t tan to simplify21 tan 2 75 </li><li> 16. 2 tan 75 ii Use t tan to simplify21 tan 2 75Let t tan 75 ; </li><li> 17. 2 tan 75 ii Use t tan to simplify21 tan 2 75Let t tan 75 ; i.e. 752 150 </li><li> 18. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; i.e. 752 150 </li><li> 19. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin i.e. 752 150 </li><li> 20. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin sin150 i.e. 75 12 1502 </li><li> 21. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin sin150i.e. 751 2 150 21 sin cos iii Prove tan1 sin cos 2 </li><li> 22. 2 tan 752 tan 75 2t ii Use t tan to simplify 21 tan 2 751 tan 2 75 1 t 2Let t tan 75 ; sin sin150 i.e. 7512 15021 sin cos iii Prove tan1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2 Let t tan ; 2 1 sin cos 2t 1 t 21 1 t 21 t2 </li><li> 23. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin sin150 i.e. 75 12 150 21 sin cos iii Prove tan1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2 Let t tan ; 2 1 sin cos 2t 1 t 21 1 t 1 t2 21 t 2 2t 1 t 2 1 t 2 2t 1 t 2 </li><li> 24. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin sin150 i.e. 75 12 150 21 sin cos iii Prove tan1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2 Let t tan ; 2 1 sin cos 2t 1 t 21 1 t 1 t2 21 t 2 2t 1 t 2 1 t 2 2t 1 t 22t 2 2t 2 2t </li><li> 25. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin sin150 i.e. 75 12 150 21 sin cos iii Prove tan1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2 Let t tan ; 2 1 sin cos 2t 1 t 21 1 t 1 t2 21 t 2 2t 1 t 2 1 t 2 2t 1 t 22t 2 2t 2 2t2t t 12 1 t </li><li> 26. 2 tan 75 2 tan 75 2t ii Use t tan to simplify21 tan 2 75 1 tan 2 75 1 t 2Let t tan 75 ; sin sin150 i.e. 75 12 150 21 sin cos iii Prove tan1 sin cos 22t 1 t 2 1 sin cos 1 1 t 2 1 t 2 Let t tan ; 2 1 sin cos 2t 1 t 21 1 t 1 t2 21 t 2 2t 1 t 2 1 t 2 2t 1 t 22t 2 2t 2 2t2t t 1 t tan2 1 t 2 </li><li> 27. (iv) By making the substitution t tan or otherwise,2 show that cosec cot cot2005 Extension 1 HSC Q4b) 2 </li><li> 28. (iv) By making the substitution t tan or otherwise,2 show that cosec cot cot2005 Extension 1 HSC Q4b) 2 1 t2 1 t2cosec cot 2t2t </li><li> 29. (iv) By making the substitution t tan or otherwise,2 show that cosec cot cot2005 Extension 1 HSC Q4b) 2 1 t2 1 t2cosec cot 2t 2t2 2t 1 t </li><li> 30. (iv) By making the substitution t tan or otherwise,2 show that cosec cot cot2005 Extension 1 HSC Q4b) 2 1 t2 1 t2cosec cot 2t 2t2 2t 1 t cot2 </li><li> 31. (iv) By making the substitution t tan or otherwise,2 show that cosec cot cot2005 Extension 1 HSC Q4b) 2 1 t2 1 t2cosec cot 2t 2t2 2t 1 t cot2 Exercise 2B; 1def, 3ace, 4bcf, 5aceg, 6, 8bd, 10a</li></ol>