Modelling with Trigonometric Functions (DP IB Maths: AA HL)

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 seconds after entering a Ferris wheel
• Notice that the x-axis will not always contain an angle
  • In the examples above time or number of days 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 a represents the amplitude of the function
  • The bigger the value of a the bigger the range of values of the function
  • For the function  the amplitude is undefined
• The b determines the period of the function
  • Period 
  • The bigger the value of b the quicker the function repeats a cycle
• The c represents the horizontal shift
• The d represents the vertical shift
  • This is the principal axis

What are possible limitations of a trigonometric model?

• The amplitude is the same for each cycle
  • In real-life this might not be the case
  • The function might get closer to the value of d over time
• 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