Modelling with Trigonometric Functions

• Boundedness of Sine and Cosine: The fundamental functions $\sin(\theta)$ and $\cos(\theta)$ are strictly bounded between $-1$ and $1$. This property is the basis for finding the maximum and minimum values of any trigonometric model.

• Linear Transformations: The parameters in a model like $y = a + b\cos(qx + r)$ act as transformations. The value $a$ shifts the graph vertically, $b$ scales it vertically (amplitude), and $q$ scales it horizontally (affecting the period).

• Continuity: Trigonometric models assume the transition between states is smooth and continuous. This makes them ideal for physical motions where objects do not teleport but move through every intermediate point.

Finding Extrema

• Maximum Value: To find the maximum of $y = a + b\sin(f(x))$, set the sine term to $1$ if $b > 0$, or $-1$ if $b < 0$. The resulting maximum is $a + |b|$.
• Minimum Value: Similarly, set the sine or cosine term to the value that results in the lowest overall output. The resulting minimum is $a - |b|$.

Calculating the Period

• Formula: The period $T$ is the interval required for one full cycle. It is calculated as $T = \frac{360}{q}$ for degrees or $T = \frac{2\pi}{q}$ for radians, where $q$ is the coefficient of the independent variable.

Solving for Time or Position

• Inverse Functions: To find when a specific value occurs (e.g., when a tide reaches 5 meters), substitute the value for $y$ and solve for $x$ using inverse trigonometric functions. Always check the required domain for multiple solutions within one cycle.

• Degrees vs. Radians: Models involving time and physical rates almost exclusively use radians. Ensure the calculator is in the correct mode before performing any inverse operations or evaluations.

• Identify the Variable: In most modelling questions, the independent variable is $t$ (time). Always check if $t$ represents seconds, minutes, or hours to ensure your period calculation matches the context.

• Check the Midline First: When given a graph or a set of data, the midline is the average of the maximum and minimum values: $a = \frac{max + min}{2}$. This is often the easiest parameter to find first.

• Verify the Sign of 'b': If a model starts at a minimum value at $t=0$ but uses a cosine function, the coefficient $b$ must be negative. A common mistake is assuming $b$ is always positive.

• Sanity Check: If you are calculating the height of a Ferris wheel and get a negative value, re-check your midline and amplitude. Physical heights relative to the ground cannot be negative.

• Incorrect Period Calculation: Students often confuse the period $T$ with the coefficient $q$. Remember that $q$ is the 'speed' of the cycle, while $T$ is the 'length' of the cycle; they are inversely proportional.

• Ignoring the Phase Shift: The term $r$ in $\sin(qx + r)$ shifts the graph horizontally. Forgetting this shift will lead to incorrect time predictions even if the amplitude and period are correct.

• Calculator Mode Errors: This is the most frequent source of lost marks. If the formula contains $\pi$ or refers to circular motion in a standard scientific context, use radian mode.