Sine and Cosine Functions

Lee W Fisher, Feb 14, 2017

Define, transform and apply sinusoidal functions.

Table of Contents

  1. Definitions
    1. Chords on a Circle
    2. Sine, Cosine, Tangent 0 - 90°
    3. Sine Cosine Tangent Order to 90
    4. Sine, Cosine, Tangent 0 - 360°
    5. Sine Tangent Cosine Order to 360
    6. Drawing the sine curve: the length of a half chord
    7. Draw the Cosine Curve (0 - 360 degrees)
    8. Sine and Cosine, positive and negative
  2. Transformations
    1. Transform sine vertically, stretch and translate together
    2. Transform sine vertically, one transformation at a time
    3. Range of Sinusoidal, practice
    4. Range of a Sinusoidal Accuracy Quiz
    5. Sinusoidal Graph Match
    6. Sine Graph Match Accuracy Quiz
    7. Transform sine with period
    8. Sine and Cosine f(x)=a sin(bx)+d
    9. Transform sine horizontally, one transformation at a time
    10. Transform sine horizontally, stretch and translate together
    11. Range, Domain, Period of a Sine Function
  3. Applied Transformations
    1. London Eye
    2. ET2-05-P3a-XT2 Modelling the Motion of a Windmill Blade
    3. 4-Stroke Engine

1.

Definitions

  • 1. Chords on a Circle
  • 2. Sine, Cosine, Tangent 0 - 90°
  • 3. Sine Cosine Tangent Order to 90
  • 4. Sine, Cosine, Tangent 0 - 360°
  • 5. Sine Tangent Cosine Order to 360
  • 6. Drawing the sine curve: the length of a half chord
  • 7. Draw the Cosine Curve (0 - 360 degrees)
  • 8. Sine and Cosine, positive and negative

1.1.

Chords on a Circle

Chord Lengths
– Lee W Fisher

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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2.

Transformations

  • 1. Transform sine vertically, stretch and translate together
  • 2. Transform sine vertically, one transformation at a time
  • 3. Range of Sinusoidal, practice
  • 4. Range of a Sinusoidal Accuracy Quiz
  • 5. Sinusoidal Graph Match
  • 6. Sine Graph Match Accuracy Quiz
  • 7. Transform sine with period
  • 8. Sine and Cosine f(x)=a sin(bx)+d
  • 9. Transform sine horizontally, one transformation at a time
  • 10. Transform sine horizontally, stretch and translate together
  • 11. Range, Domain, Period of a Sine Function

2.1.

Transform sine vertically, stretch and translate together

Graph 1: Vertical Stretch, Vertical Translation
Interact with the applet to see how the values a and d transform the curve y=sin(x°).
[br][br]Notice that when d=0, the graph rises from the x axis as much as it falls. The rise and fall from the horizontal center (sometimes known as the sinusoidal axis) is known as the [url=https://www.mathsisfun.com/algebra/amplitude-period-frequency-phase-shift.html]amplitude[/url].[br][br]Notice how the y value of the local maximum and of the local minimum can be calculated by the values of a and d.
1. Order of operations
Suppose you wish to transform [math]y=\text{sin}(x°)[/math] to [math]y=3\text{sin}(x°)+5[/math]. Does it make mathematical sense to [br][br]1. stretch (multiply y values by 3) then to translate (add 5 to all new y values)?[br][br]or to [br][br]2. translate (add 5 to all y values) then to stretch (multiply new y values by 3)?[br][br](Hint - take the point [math](90,1)[/math] as an example - does the y value become [math]1\times3+5[/math] or [math]\left(1+5\right)\times3[/math]?)
Graph 2.
2(a)
Graph 2 has equation [math]y=f(x)[/math] where [math]f(x)=a\text{sin}(x°)+d[/math]. Which one of the following is a correct equation for this function?
2(b)
What is the range of the function [math]f(x)[/math], shown in graph 2?
Graph 3.
3(a)
The graph of the function [math]y=g(x)[/math] where [math]g(x)=a\text{sin}(x°)+d[/math] is shown in graph 3. What is the equation of the function [math]g(x)[/math]?
3(b)
What is the range of the function [math]g(x)[/math]?
4
The function [math]h(x)[/math] is defined [math]h(x)=7\text{sin}(x°)+10[/math]. What is the range of [math]h\left(x\right)[/math]?
– Lee W Fisher

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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2.11.

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3.

Applied Transformations

  • 1. London Eye
  • 2. ET2-05-P3a-XT2 Modelling the Motion of a Windmill Blade
  • 3. 4-Stroke Engine

3.1.

London Eye

– Lee W Fisher

3.2.

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3.3.

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Information