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Analytic Trigonometry

Analytic Trigonometry . Barnett Ziegler Bylean. Graphs of trig functions. Chapter 3. Basic graphs. Ch 1 - section 1. Why study graphs?. Assignment. Be able to sketch the 6 basic trig functions WITHOUT referencing notes or using a graphing calculator.

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Analytic Trigonometry

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  1. Analytic Trigonometry Barnett Ziegler Bylean

  2. Graphs of trig functions Chapter 3

  3. Basic graphs Ch 1 - section 1

  4. Why study graphs?

  5. Assignment • Be able to sketch the 6 basic trig functions WITHOUT referencing notes or using a graphing calculator. • Be able to answer questions concerning: domain/range x-int/y=int increasing/decreasing symmetry asmptote without notes or calculator.

  6. Hints for hand graphs • X-axis - count by π/2 with domain [-2π, 3π] • Y-axis – count by 1’s with a range of [-5,5]

  7. Defining trig functions in terms of (x,y) Input x (cos(ө),sin(ө) ) ө Output cos()=, sin(x)=, tan(x) ,sec(x), csc(x), cot(x)

  8. y=sin(x) Input x • Domain/range • X-intercept • Y-intercept • Other points • Periodic/period • Increase • Decrease • Symmetry (odd) (cos(ө),sin(ө) ) ө Output cos()=,sin(x)= , tan(x) ,sec(x), csc(x), cot(x) Output y = sin(x) – using π/2 for the x-scale

  9. y = cos(x) Input x (cos(ө),sin(ө) ) Output cos()= ө Domain/range X-intercept Y-intercept Other points Periodic/period Increase Decrease Symmetry (odd)

  10. y = tan(x) and y = cot(x) Input x y = tan(x) y = cot(x) restricted/asymptotes: Range ? y-intercept x-intercept (cos(ө),sin(ө) ) ө

  11. y = sec(x) and y = csc(x) Input x sec(x) csc(x) restricted/asymptotes? range? (cos(ө),sin(ө) ) ө

  12. Transformations of sin and cos Chapter 3 – section 2

  13. Review transformations • Given f(x) • What do you know about the following • f(x-3) f(x + 5) • f(3x) f(x/7) • f(x) + 6 f(x) – 4 • 3f(x) f(x)/3

  14. Trigonometric Transformations - dilations • Y = Acos(Bx) y = Asin(Bx) • Multiplication causes a scale change in the graph • The graph appears to stretch or compress

  15. Vertical dilation : y = Af(x) • If the multiplication is external (A) it multiplies the y-co-ordinate (stretches vertically) – the x intercepts are stable (y=0), the y intercept is not stable for cosine • The height of a wave graph is referred to as the amplitude (direct correlation to physics wave theory) - It is how much impact the x has on the y value - louder sound, harder heartbeat etc. • Amplitude is measured from axis to max. and from axis to min.

  16. Examples of some graphs y=3(sin(x) y=sin(x) y=-2sin(x)

  17. y = 3cos(x) 1 Scale π/2

  18. Horizontal Dilations • If the multiplication is inside the function it compresses horizontally against the y-axis – the x-intercepts are compressed – the y- intercept is stable – this affects the period of the function • Period – the length of the domain interval that covers a full rotation – The period for sine and cosine is 2π – multiplying the x – coordinates speeds up the rotation thereby compressing the period - • New period is 2π/multiplier • Frequency – the reciprocal of the period-

  19. Examples of some graphs y=cos(2x) y= cos(x/2) y=cos(x)

  20. Sketch a graph (without a calculator) • y = 3cos(2x) • y = - sin(πx)

  21. Transformations - Vertical shifts • Adding “outside” the function shifts the graph up or down – think of it like moving the x-axis • f(x) = sin(x) + 2 g(x) = cos(x) - 4

  22. Pertinent information affected by shift • the amplitude and period are not affected by a vertical shift • The x and y intercepts are affected by shift – • The maximum and minimum values are affected by vertical shift

  23. Finding max/min values • Max/min value for both sin(x) and cos(x) are 1 and -1 respectively • Amplitude changes these by multiplying • Shift change changes them by adding • Ex: k(x)= 4cos(3x -5) – 2 • the max value is now 4(1)- 2= 2 • the min value is now 4(-1) – 2 =-6

  24. Example • Graph k(x) = 4 + 2cos(𝛑x)

  25. Summary

  26. Writing equations • Identify amplitude • Identify period • Identify axis shift

  27. Horizontal shifts Chapter 3 – section 3

  28. Simple Harmonics • f(x) = Asin(Bx + C) or g(x) = Acos(Bx + C) are referred to as Simple Harmonics. • These include horizontal shifts referred to as phase shifts • The shift is -C units horizontally followed by a compression of 1/B - thus the phase shift is -C/B units • The amplitude and period are not affected by the phase shift

  29. Horizontal shift • f(x) = cos(x + ) • g(x) = cos(2x – )

  30. find amplitude, max, min, period and phase shift • f(x) = 3cos(2x – π/3) • y = 2 – 4sin(πx + π/5)

  31. Tangent/cotangent/secant/cosecant revisited Chapter 3 – section 6

  32. Basic graphs • asymptotes • Period • Increasing/decreasing • tan(x) cot(x) • sec(x) csc(x)

  33. k + A tan(Bx+C) or k + A cot(Bx+C) • No max or min - effect of A is minimal • Period is π/B instead of 2π/B • Phase shift is still -C/B and affects the x intercepts and asymptotes • k moves the x and y intercepts

  34. Examples • y = 3 + 2tan(3x) • y = cot()

  35. k+ Asec(Bx+ C) or k + Acsc(Bx + C) • local maxima and minima affected by k and A • Directly based on sin and cos so Period is 2π/B • Shift is still -C/B

  36. Examples • y = 3 + 2sec(3πx) • y = 1 – csc (2x + π/3)

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