Graphing Trigonometric Functions
Trigonometric functions are fundamental mathematical tools that model periodic phenomena—events that repeat at regular intervals. Graphing these functions allows us to visualize oscillatory behavior found throughout nature, engineering, and physics.
The Basic Trigonometric Functions
The three primary trigonometric functions we will focus on are:
- Sine function: \( y = \sin(x) \)
- Cosine function: \( y = \cos(x) \)
- Tangent function: \( y = \tan(x) \)
In their standard forms, these functions have the following characteristics:
| Function | Domain | Range | Period | Key Features |
|---|---|---|---|---|
| \( y = \sin(x) \) | All real numbers | \( [-1,\, 1] \) | \( 2\pi \) | Passes through the origin; odd function |
| \( y = \cos(x) \) | All real numbers | \( [-1,\, 1] \) | \( 2\pi \) | \( y \)-intercept at \( (0,\,1) \); even function |
| \( y = \tan(x) \) | All real numbers except \( x = \tfrac{\pi}{2} + n\pi \) | All real numbers | \( \pi \) | Vertical asymptotes at \( x = \tfrac{\pi}{2} + n\pi \) |
Interactive: Fundamental Trigonometric Functions Explorer
How to use: Select a function and adjust the sliders to modify parameters A, B, C, and D to see how they transform the graph. Toggle visual aids to understand amplitude, period, phase shift, and vertical shift.
Interactive visualization — available in the full lesson
Instructions:
- Click SIN, COS, or TAN to switch functions
- Drag sliders to adjust: Amplitude A, Period parameter B, Phase shift C, Vertical shift D
- Toggle visual aids on/off by clicking the checkboxes
- For tangent, asymptotes appear at x = (π/2 + nπ - C)/B
- Use navigation controls to zoom and pan