1. 445.102 Mathematics 2 Module 4 Cyclic Functions Lecture 2 Reciprocal Relationships

2. 445.102 Lecture 4/2  Administration  Last Lecture  Looking Again at the Unit Circle  Some Other Functions  Equations with Many Solutions  Summary

3. Administration  Chinese Tutorials  Text Handouts Modules 0, 1, 2—> p52 Module 3—> pp87 - 109 Module 4—> pp77 - 88  This Week’s Tutorial Assignment 4 & Working Together

4. 445.102 Lecture 4/2  Administration  Last Lecture  Looking Again at the Unit Circle  Some Other Functions  Equations with Many Solutions  Summary

5. Radians A mathematical measure of angle is defined using the radius of a circle. 1 radian

6. sin(ø) ø 1

7. Post-Lecture Exercise 1 45° = π / 4 radians 60° = π / 3 radians 80° = 4π / 9 radians 2 full turns = 4π radians 270° = 3π / 2 radians 2π radians = 180°3 radians = 171.9° 6π radians = 3 turns 3f(x) = sin x is an ODD function. 4f(2.5) = 0.598f( π / 4 ) = 0.707 f(20) = 0.913f(–4) = 0.757 f –1 (0.5) = 0.524 f –1 (0.3) = 0.305 f –1 (–0.6) = –0.644 5The domain of f(x) = sin x is the Real Numbers 6The domain of the inverse function is –1 ≤ x ≤ 1

8. Lecture 4/1 – Summary  There are many functions where the variable can be regarded as an ANGLE.  One way of measuring an angle is that derived from the radius of the circle. This is called RADIAN measure.  From the UNIT CIRCLE, we can see that the SINE of an angle is the height of a triangle drawn inside the circle. Sine(ø) then becomes a function depending on the size of the angle ø.

9. The Sine Function (Many Rotations)

10. Preliminary Exercise

11. 445.102 Lecture 4/2  Administration  Last Lecture  Looking Again at the Unit Circle  Some Other Functions  Equations with Many Solutions  Summary

12. C(ø) ø 1

13. cos(ø) ø 1

14. tan(ø) ø 1

15. Constructions on the Unit Circle ø cos(ø) 1 sin(ø) tan(ø)

16. The Cosine Function (Many Rotations)

17. The Tangent Function (Many Rotations)

18. 445.102 Lecture 4/2  Administration  Last Lecture  Looking Again at the Unit Circle  Some Other Functions  Equations with Many Solutions  Summary

19. The Secant Function

20. sec ø / 1 = sec ø = 1 / cos ø cos(ø) 1 sec ø 1

21. Inverse Functions The sine function maps an angle to a number. e.g. sin π / 4 =0.707 The inverse sine function maps a number to an angle.e.g. sin -1 0.707 = π / 4 Note the difference between: The inverse sine: sin -1 0.707 = π / 4 The reciprocal of sine: (sin π / 4 ) -1 = 1 / (sin π / 4 ) = 1 / 0.707 = 1.414

22. Inverse Functions Here is a quick exercise.......... (remember to give your answers in radians): 1.What angle has a sine of 0.25 ? 2.What angle has a tangent of 3.5 ? 3.What angle has a cosine of –0.4 ? 4.What is sec π / 2 ? 5.What is cot 5π / 3 ? 6.What is arctan 10 ?

23. 445.102 Lecture 4/2  Administration  Last Lecture  Looking Again at the Unit Circle  Some Other Functions  Equations with Many Solutions  Summary

24. An Equation 2cos ø – 0.6 = 0 2cos ø = 0.6 cos ø = 0.3

25. An Example.... 4 sin ø + 3 = 1 4sin ø = –2 sin ø = –0.5 ø = sin -1 (–0.5) = –0.524 –0.524, π+0.524, 2π–0.524, 3π+0.524,.... nπ+0.524 (n = 1,3,5,7,....) nπ–0.524 (n = 0,2,4,6,....) nπ+0.524 (n =...-5,-3,-1,1,3,5,7,....) nπ–0.524 (n =...-6,-4,-2,0,2,4,6,....)

26. An Example.... 4 sin ø + 3 = 1 4sin ø = –2 sin ø = –0.5 ø = sin -1 (–0.5) = –0.524 –0.524, π+0.524, 2π–0.524, 3π+0.524,.... nπ+0.524 (n = 1,3,5,7,....) nπ–0.524 (n = 0,2,4,6,....) nπ+0.524 (n =...-5,-3,-1,1,3,5,7,....) nπ–0.524 (n =...-6,-4,-2,0,2,4,6,....)

27. A Special Triangle 1 unit

28. A Special Triangle 1 1

29. 1 1 √2 π/4π/4

30. A Special Triangle 1 1 √2 π/4π/4 sin π / 4 = 1 / √2 cos π / 4 = 1 / √2 tan π / 4 = 1 / 1 = 1

31. Another Special Triangle 2 units

32. Another Special Triangle 2 √3 1

33. Another Special Triangle 2 π/3π/3 π/6π/6 √3 1

34. Another Special Triangle 2 π/3π/3 π/6π/6 √3 1 sin π / 6 = 1 / 2 cos π / 6 = √3 / 2 tan π / 6 = 1 / √3 sin π / 3 = √3 / 2 cos π / 3 = 1 / 2 tan π / 3 = √3 / 1 =√3

35. 445.102 Lecture 4/2  Administration  Last Lecture  Looking Again at the Unit Circle  Some Other Functions  Equations with Many Solutions  Summary

36. Lecture 4/2 – Summary  Sine, cosine and tangent can be seen as lengths on the Unit Circle that depend on the angle under consideration.  So sine, cosine and tangent are functions where the angle is the variable.  For each of these there is a reciprocal function.  The graphs of these functions can be used to “see” the solutions of trigonometric equations

37. 445.102 Lecture 4/2  Before the next lecture........ Go over Lecture 4/2 in your notes Do the Post-Lecture exercise p84 Do the Preliminary Exercise p85  See you tomorrow........