Modelling with Trigonometric Functions

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Anything that oscillates (fluctuates periodically) can be modelled using a trigonometric function
- Normally some transformation of the sine or cosine function
Examples include:
- $D (t)$ is the depth of water at a shore t hours after midnight
- $T (d)$ is the temperature of a city d days after the 1st January
- $H (t)$ 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

A trigonometric model could be of the form
- $f (x) = a \sin (b (x - c)) + d$
- $f (x) = a \cos (b (x - c)) + d$
- $f (x) = a \tan (b (x - c)) + d$
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 $a \tan (b (x - c)) + d$ the amplitude is undefined
The b determines the period of the function
- Period $= \frac{360 °}{b} = \frac{2 π}{b}$
- 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

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

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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.

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

$D (t) = 3 \sin (15 ° (t - 2)) + 12, 0 \leq t < 24$

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

Write down the depth of the water at midnight.

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

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

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