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Understanding Inverse Circular Functions: Key Concepts and Evaluation Techniques

This chapter delves into the fundamental aspects of inverse circular functions, focusing on sine and cosine. It explains the significance of principal values, how to evaluate inverse trigonometric expressions, and the conditions for 1-1 functions in these contexts. Additionally, it covers inverse trigonometric identities and provides a series of examples and solutions to illustrate these concepts. Ideal for learners seeking a thorough understanding of inverse circular functions and their applications in various mathematical scenarios.

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Understanding Inverse Circular Functions: Key Concepts and Evaluation Techniques

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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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