DP IB Maths: AA SL

Revision Notes

3.5.3 Modelling with Trigonometric Functions

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Modelling with Trigonometric Functions

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 space left parenthesis t right parenthesis is the depth of water at a shore t hours after midnight
    • T space left parenthesis d right parenthesis spaceis the temperature of a city d days after the 1st January
    • H space left parenthesis t right parenthesis 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 
    • space f open parentheses x close parentheses equals a space sin space left parenthesis b left parenthesis x minus c right parenthesis right parenthesis plus d
    • space f open parentheses x close parentheses equals a space cos space left parenthesis b left parenthesis x minus c right parenthesis right parenthesis plus d
    • space f open parentheses x close parentheses equals a space tan space left parenthesis b left parenthesis x minus c right parenthesis right parenthesis plus d
  • 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 space a space tan space left parenthesis b left parenthesis x minus c right parenthesis right parenthesis plus d the amplitude is undefined
  • The b determines the period of the function
    • Period begin mathsize 16px style equals space fraction numerator 360 degree over denominator b end fraction equals fraction numerator 2 straight pi over denominator b end fraction end style
    • 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

ib-aa-sl-3-5-3-transformations-of-trig-graphs

Exam Tip

  • The variable in these questions is often t  for time.
  • Read the question carefully to make sure you know what you are being asked to solve.

Worked example

The water depth, D, in metres, at a port can be modelled by the function

 D open parentheses t close parentheses equals 3 space sin space open parentheses 15 degree left parenthesis t minus 2 right parenthesis close parentheses plus 12 comma blank space space space space space space space 0 space less or equal than space t space less than space 24

where t is the elapsed time, in hours, since midnight.

 

a)
Write down the depth of the water at midnight.

 aa-sl-3-5-3-modelling-with-trig-functions-we-solution-part-i-png

b)
Find the minimum water depth and the number of hours after midnight that this depth occurs.

aa-sl-3-5-3-modelling-with-trig-functions-we-solution-part-ii

 

c)
Calculate how long the water depth is at least 13.5 m each day.

aa-sl-3-5-3-modelling-with-trig-functions-we-solution-part-iii

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Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.