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Chapter 35

Chapter 35. Inverse Circular Functions. Prepared by : Tan Chor How (B.Sc). Some fundamental concepts. Let. y = sinx. then we have. or. i.e. is the inverse function of. Iff y is the 1-1 function!. doesn’t mean. Also doesn’t mean. 1. 0. -1. In this region,

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Chapter 35

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  1. Chapter 35 Inverse Circular Functions Prepared by : Tan Chor How (B.Sc) By Chtan FYHS-Kulai

  2. Some fundamental concepts By Chtan FYHS-Kulai

  3. Let y = sinx then we have or i.e. By Chtan FYHS-Kulai

  4. is the inverse function of Iff y is the 1-1 function! By Chtan FYHS-Kulai

  5. doesn’t mean Also doesn’t mean By Chtan FYHS-Kulai

  6. 1 0 -1 In this region, , y is 1-1 function. By Chtan FYHS-Kulai

  7. Now, if you flip the previous graph, Principal values -1 The principal values of y is defined as that value lying between . 0 1 By Chtan FYHS-Kulai

  8. Similarly, check the cosine graph 1 0 -1 In this region, , y is 1-1 function. By Chtan FYHS-Kulai

  9. Now, if you flip the previous graph, Principal values -1 The principal values of y is defined as that value lying between 0 and ∏ . 0 1 By Chtan FYHS-Kulai

  10. 0 By Chtan FYHS-Kulai

  11. Graph of Principal values The principal values of y is defined as that value lying between . 0 By Chtan FYHS-Kulai

  12. Some books write as . Domain of y is Range of y is By Chtan FYHS-Kulai

  13. By Chtan FYHS-Kulai

  14. In general, we have By Chtan FYHS-Kulai

  15. We also have : By Chtan FYHS-Kulai

  16. e.g. 1 Evaluate . Soln : By Chtan FYHS-Kulai

  17. e.g. 2 Evaluate . Soln : By Chtan FYHS-Kulai

  18. e.g. 3 Evaluate . Soln : By Chtan FYHS-Kulai

  19. e.g. 4 Evaluate . Soln : Let By Chtan FYHS-Kulai

  20. Now, let see Same as . Domain of y is Range of y is By Chtan FYHS-Kulai

  21. In general, we have By Chtan FYHS-Kulai

  22. We also have : By Chtan FYHS-Kulai

  23. e.g. 5 Evaluate . Soln : Between gives . By Chtan FYHS-Kulai

  24. Now, let see Same as . Domain of y is Range of y is By Chtan FYHS-Kulai

  25. Now, let see Same as . Domain of y is Range of y is By Chtan FYHS-Kulai

  26. By Chtan FYHS-Kulai

  27. e.g. 6 Evaluate . Soln : By Chtan FYHS-Kulai

  28. e.g. 7 Evaluate . Soln : Let 5 4 3 By Chtan FYHS-Kulai

  29. e.g. 8 Find the value of the following Expression : By Chtan FYHS-Kulai

  30. Soln : Let and 2 5 3 1 4 By Chtan FYHS-Kulai

  31. By Chtan FYHS-Kulai

  32. e.g. 9 Find the value of the following Expression : By Chtan FYHS-Kulai

  33. Soln : Let By Chtan FYHS-Kulai

  34. There are 2 possible answers. [because a and b are both positive values, a+b must be positive value.] By Chtan FYHS-Kulai

  35. Inverse trigonometric identities By Chtan FYHS-Kulai

  36. Identity (1) By Chtan FYHS-Kulai

  37. Identity (2) By Chtan FYHS-Kulai

  38. Let prove the identity #1 To prove : Same as to prove : A By Chtan FYHS-Kulai

  39. Check slide #14 LHS of A : RHS of A : We have, and B [ x(-1) ] By Chtan FYHS-Kulai

  40. C B and C state that both and are . By Chtan FYHS-Kulai

  41. i.e. By Chtan FYHS-Kulai

  42. Let prove the identity #2 To prove : Same as to prove : A By Chtan FYHS-Kulai

  43. But and [ x(-1) ] By Chtan FYHS-Kulai

  44. Both and By Chtan FYHS-Kulai

  45. e.g. 10 Prove that By Chtan FYHS-Kulai

  46. Soln : Let then By Chtan FYHS-Kulai

  47. i.e. By Chtan FYHS-Kulai

  48. e.g. 11 Prove that Soln : Let LHS: By Chtan FYHS-Kulai

  49. RHS: 2 B 3 By Chtan FYHS-Kulai

  50. Inverse trigonometric equations By Chtan FYHS-Kulai

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