AP Precalculus: Sine & Cosine Functions
Master graphing, transformations, and writing equations for sinusoidal functions
📚 Understanding Sinusoidal Functions
Sine and cosine functions are periodic functions that model wave-like behavior. They're used to describe oscillations, sound waves, tides, and countless other natural phenomena. This guide covers the general form, all transformation parameters, and how to graph and write equations.
1 General Form of Sine & Cosine Functions
Both sine and cosine functions follow the same general transformed form, with parameters that control the shape and position of the graph.
- Starts at midline
- Goes up first (if \(a > 0\))
- Key point at \((0, d)\)
- Starts at maximum (if \(a > 0\))
- Goes down first
- Key point at \((0, d + a)\)
You may also see: \(y = a\sin(bx + c) + d\). In this case, phase shift = \(-\frac{c}{b}\). The form \(y = a\sin(b(x-c)) + d\) makes the phase shift \(c\) visible directly.
2 Understanding the Parameters
Each parameter in \(y = a\sin(b(x - c)) + d\) controls a specific transformation of the graph.
Min: \(d - |a|\)
If \(a < 0\), the graph is reflected over the midline. Sine starts going down instead of up; cosine starts at minimum instead of maximum.
3 Graphing Transformations
To graph a transformed sine or cosine function, apply transformations in order: horizontal stretch/compress → horizontal shift → vertical stretch/compress → vertical shift.
Shift Direction Guide
Key Points for One Cycle
| Function | Start | 1/4 Period | 1/2 Period | 3/4 Period | Full Period |
|---|---|---|---|---|---|
| \(y = \sin x\) | 0 (midline) | 1 (max) | 0 (midline) | -1 (min) | 0 (midline) |
| \(y = \cos x\) | 1 (max) | 0 (midline) | -1 (min) | 0 (midline) | 1 (max) |
Draw the midline at \(y = d\) first. Mark max at \(d + |a|\) and min at \(d - |a|\). Then plot 5 key points across one period, shifted by \(c\).
4 Writing Equations from Graphs
Given a graph of a sinusoidal function, extract the parameters to write the equation.
Step-by-Step Process
- Find max and min values from the graph
- Calculate amplitude: \(|a| = \frac{\text{max} - \text{min}}{2}\)
- Calculate midline (vertical shift): \(d = \frac{\text{max} + \text{min}}{2}\)
- Find the period: Distance from peak to peak (or trough to trough)
- Calculate \(b\): \(|b| = \frac{2\pi}{\text{period}}\)
- Find phase shift \(c\): How far is the starting point from the origin?
- Choose sine or cosine: based on where the graph starts
Use Sine When...
Graph starts at the midline and goes up (or down if negative)
Use Cosine When...
Graph starts at a maximum (or minimum if negative)
Given: Max = 5, Min = 1, Period = \(\pi\), starts at max when \(x = \frac{\pi}{3}\)
Amplitude: \(|a| = \frac{5-1}{2} = 2\)
Midline: \(d = \frac{5+1}{2} = 3\)
Find \(b\): Period = \(\pi\), so \(|b| = \frac{2\pi}{\pi} = 2\)
Phase shift: Starts at max when \(x = \frac{\pi}{3}\), so \(c = \frac{\pi}{3}\)
Equation: \(y = 2\cos\left(2\left(x - \frac{\pi}{3}\right)\right) + 3\)
5 Sine & Cosine Relationship
Sine and cosine are the same wave, just shifted horizontally by \(\frac{\pi}{2}\) (90°).
Converting Between Forms
Sine → Cosine
\(\sin(x) = \cos\left(x - \frac{\pi}{2}\right)\)
Shift cosine right by \(\frac{\pi}{2}\)
Cosine → Sine
\(\cos(x) = \sin\left(x + \frac{\pi}{2}\right)\)
Shift sine left by \(\frac{\pi}{2}\)
Any sinusoidal function can be written as either a sine or cosine function — just adjust the phase shift! Use whichever form is more convenient.
6 Domain, Range & Periodic Properties
Sine and cosine functions are defined for all real numbers and repeat infinitely.
7 Real-World Applications
Sinusoidal functions model many periodic phenomena in nature and science.
- Sound waves: Frequency = pitch, amplitude = volume
- Tides: Daily water level changes modeled with period ≈ 12 hours
- Temperature: Daily and seasonal temperature cycles
- Ferris wheels: Height over time as wheel rotates
- Alternating current (AC): Voltage varies sinusoidally
Problem: A Ferris wheel has radius 25 ft, center 30 ft high, rotates once every 40 seconds. Model height \(h(t)\).
Amplitude: 25 (radius)
Midline: 30 (center height)
Period: 40 seconds, so \(b = \frac{2\pi}{40} = \frac{\pi}{20}\)
Starting at bottom (min): Use negative cosine
Equation: \(h(t) = -25\cos\left(\frac{\pi}{20}t\right) + 30\)
📋 Quick Reference
Amplitude
\(|a|\) or \(\frac{\text{max}-\text{min}}{2}\)
Period
\(\frac{2\pi}{|b|}\)
Midline
\(d\) or \(\frac{\text{max}+\text{min}}{2}\)
Max Value
\(d + |a|\)
Min Value
\(d - |a|\)
Sine ↔ Cosine
Shift by \(\frac{\pi}{2}\)