7.6 Modeling with Trigonometric Functions

Sinusoidal Functions

Sinusoidal functions describe any quantity that follows a smooth, wave-like pattern over time or distance. They're the go-to tool for modeling anything that repeats: tides, temperatures, sound waves, hours of daylight. The core idea is that once you know a few parameters (amplitude, period, and shifts), you can write an equation that captures the entire repeating pattern.

more resources to help you study

practice questions

Amplitude and Period of Sinusoids

Amplitude is the vertical distance from the midline to the maximum (or minimum) of the function. It tells you how "tall" the wave is. You find it by taking the absolute value of the coefficient in front of the trig function.

• For $y = 3\sin(x)$, the amplitude is $3$, meaning the wave reaches 3 units above and 3 units below the midline.

Period is the horizontal length of one complete cycle. It tells you how "wide" each repetition is. You calculate it with:

$T = \frac{2\pi}{|B|}$

where $B$ is the coefficient of $x$ inside the trig function.

• For $y = \sin(2x)$, $B = 2$, so the period is $\frac{2\pi}{2} = \pi$. The wave completes a full cycle every $\pi$ units instead of the usual $2\pi$.

Graphing and Equations for Sinusoids

The general form of a sinusoidal function is:

$y = A\sin(B(x - C)) + D \quad \text{or} \quad y = A\cos(B(x - C)) + D$

• $A$ = amplitude (vertical stretch)
• $B$ = determines the period via $T = \frac{2\pi}{|B|}$
• $C$ = phase shift (horizontal shift; positive $C$ shifts right)
• $D$ = vertical shift, which sets the midline at $y = D$

Graphing from an equation:

1. Find $A$, $B$, $C$, and $D$ directly from the equation.
2. Calculate the period: $T = \frac{2\pi}{|B|}$.
3. Mark the midline at $y = D$.
4. Starting at $x = C$ (the phase shift), plot one full cycle using the amplitude above and below the midline.
5. Extend the pattern in both directions as needed.

Writing an equation from a graph or data:

1. Find the midline: $D = \frac{\text{max} + \text{min}}{2}$.

2. Find the amplitude: $A = \frac{\text{max} - \text{min}}{2}$.

3. Find the period $T$ by measuring the distance for one full cycle, then solve $B = \frac{2\pi}{T}$.

4. Determine the phase shift $C$ by identifying where a standard cycle begins (a maximum for cosine, a midline crossing going up for sine).

5. Plug everything into the general form.

Modeling with Trigonometric Functions

Periodic phenomena are events that repeat at regular intervals. Trig functions let you capture these patterns with a single equation.

Common examples:

• Tides rising and falling roughly every 12.4 hours
• Average monthly temperature cycling over 12 months
• Hours of daylight varying over a 365-day year

Steps to build a model:

1. Identify the midline ($D$): Average the maximum and minimum data values.
2. Find the amplitude ($A$): Half the difference between the max and min.
3. Determine the period and calculate $B$: Use the natural cycle length of the phenomenon (e.g., 12 months for temperature). Then $B = \frac{2\pi}{T}$.
4. Choose sine or cosine: If your data starts at a maximum or minimum, cosine is often simpler. If it starts at the midline, sine works well.
5. Find the phase shift ($C$): Adjust so the equation lines up with where the cycle actually starts in your data.
6. Check your equation against a few known data points and adjust if needed.

Harmonic Motion

Harmonic Motion in Trigonometry

Harmonic motion describes an object oscillating back and forth around a resting point. Think of a pendulum swinging, a weight bouncing on a spring, or a guitar string vibrating. The displacement follows a sinusoidal pattern over time.

The simple harmonic motion equation is:

$x(t) = A\cos(\omega t + \phi) \quad \text{or} \quad x(t) = A\sin(\omega t + \phi)$

• $A$ = amplitude (maximum displacement from equilibrium)
• $\omega$ = angular frequency in radians per unit time, related to the period by $\omega = \frac{2\pi}{T}$
• $\phi$ = phase shift (adjusts the starting position at $t = 0$)
• $t$ = time

Frequency ($f$) is the number of complete cycles per unit time: $f = \frac{1}{T} = \frac{\omega}{2\pi}$.

Velocity and acceleration are found by differentiating the position function. For a cosine position function $x(t) = A\cos(\omega t + \phi)$:

• Velocity: $v(t) = -A\omega\sin(\omega t + \phi)$
• Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi)$

Notice that the acceleration is always proportional to the displacement but in the opposite direction. That's the defining feature of simple harmonic motion: the restoring force always pulls the object back toward equilibrium.

Oscillation and Equilibrium

The equilibrium position is the center point of the motion, where the object would sit at rest if undisturbed. In the equation, this corresponds to $x = 0$ (or whatever the vertical shift $D$ is, if one is included).

Oscillation is the repetitive back-and-forth movement around that equilibrium. A few things to keep in mind:

• At maximum displacement ($x = \pm A$), velocity is zero and the object momentarily stops before reversing direction.
• At equilibrium ($x = 0$), velocity is at its maximum and the object moves fastest.
• The total distance from one extreme to the other is $2A$, but the amplitude itself is just $A$.

These relationships between position, velocity, and acceleration at different points in the cycle are a common source of exam questions, so make sure you can connect the graph of each quantity to the physical motion.