1 / 37

445.102 Mathematics 2

445.102 Mathematics 2. Module 4 Cyclic Functions Lecture 2 Reciprocal Relationships. 445.102 Lecture 4/2. Administration Last Lecture Looking Again at the Unit Circle Some Other Functions Equations with Many Solutions Summary. Administration. Chinese Tutorials

vsteve
Download Presentation

445.102 Mathematics 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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. RadiansA mathematical measure of angle is defined using the radius of a circle. 1 radian

  6. sin(ø) 1 sin(ø) ø

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

  13. cos(ø) 1 ø cos(ø)

  14. tan(ø) tan(ø) ø 1

  15. Constructions on the Unit Circle tan(ø) 1 sin(ø) ø cos(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 sec ø 1 cos(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-10.707 = π/4 • Note the difference between: The inverse sine: sin-10.707 = 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 = 02cos ø = 0.6cos ø = 0.3

  25. An Example .... 4sin ø + 3 = 14sin ø = –2sin 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 .... 4sin ø + 3 = 14sin ø = –2sin 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 1 unit

  28. A Special Triangle 1 1

  29. A Special Triangle √2 1 π/4 1

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

  31. Another Special Triangle 2 units 2 units

  32. Another Special Triangle 2 √3 1

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

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

  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 ........

More Related