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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

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445.102 Mathematics 2

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  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 ........

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