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

  • $D\left(t\right)$ is the depth of water at a shore t hours after midnight

  • $T\left(d\right)$is the temperature of a city d days after the 1st January

  • $H\left(t\right)$ 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 

  • $f\left(x\right)=a\sin \left(b\right(x-c\left)\right)+d$

  • $f\left(x\right)=a\cos \left(b\right(x-c\left)\right)+d$

• The value of $a$ represents the amplitude of the function

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

• The value of $b$ determines the period of the function

  • Period $=\frac{360°}{b}=\frac{2\mathrm{\pi }}{b}$

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

• The value of $c$ represents the horizontal shift

• The value of $d$ represents the vertical shift

  • This is the principal axis

• For example, if $H\left(t\right)=-20\cos \left(2t\right)+21$ is the vertical height (in metres) above ground of a person $t$ 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 $\frac{2\mathrm{\pi }}{2}=3.141...$ 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