Modelling with Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Modelling with trigonometric functions

What can be modelled with trigonometric functions?

• Anything that oscillates (fluctuates periodically) can be modelled using a trigonometric function

  • Normally some transformation of the sine or cosine function

• Examples include:

  • is the depth of water at a shore t hours after midnight

  • is the temperature of a city d days after the 1st January

  • is vertical height above ground of a person t minutes after entering a Ferris wheel

• Notice that the x-axis will not always contain an angle

  • In the examples above time would be on the x-axis

  • Depth of the water, temperature or vertical height would be on the y-axis

What are the parameters of trigonometric models?

• A trigonometric model could be of the form 

  • 

  • 

• The value of represents the amplitude of the function

  • The bigger the value of the bigger the range of values of the function

• The value of determines the period of the function

  • Period 

  • The smaller the value of the quicker the function repeats a cycle

• The value of represents the horizontal shift

• The value of represents the vertical shift

  • This is the principal axis

• For example, if is the vertical height (in metres) above ground of a person minutes after entering a Ferris wheel

  • The highest point above ground is 21+20 = 42 m

  • The lowest point above ground is 21-20 = 1 m

  • The speed to complete one cycle is minutes

What are possible limitations of a trigonometric model?

• The model assumes that the amplitude is the same for each cycle

  • In real-life this might not be the case

  • The function might get closer to the principal axis over time

• The model assumes that the period is the same for each cycle

  • In real-life this might not be the case

  • The time to complete a cycle might change over time