How to Use This Tool
This interactive visualizer helps you understand trigonometric functions through real-time visualization. Here's how to use it:
- Select a Function: Click on sin(x), cos(x), or tan(x) buttons to switch between functions
- Adjust the Angle: Use the angle slider to move from 0° to 360° and watch the point move around the unit circle
- Modify Amplitude (A): Change how tall or short the wave appears (vertical stretch/compression)
- Modify Frequency (B): Control how many wave cycles fit in the same space (horizontal compression)
- Adjust Phase Shift (C): Slide the entire wave left or right along the x-axis
- Observe Both Views: The unit circle shows the geometric meaning, while the graph shows the function behavior
What Are Trigonometric Functions?
Trigonometric functions relate angles to ratios of sides in right triangles and points on the unit circle. They're fundamental in mathematics, physics, engineering, and many real-world applications.
The Unit Circle
The unit circle is a circle with radius 1 centered at the origin. Any point on this circle can be described by an angle θ (theta) measured from the positive x-axis.
Angle Conversions: Angles can be expressed in degrees or radians. Since one full rotation = 360° = 2π radians, we can convert between them. The tool shows angles in three formats: degrees (180°), decimal radians (3.142), and as a fraction of π (π, π/2, 2π/3, etc.) for exact mathematical representation.
The Three Primary Functions
Function Transformations
The general form of a transformed trigonometric function is:
- A (Amplitude): Vertical stretch/compression. |A| = maximum distance from center line
- B (Frequency): Horizontal compression/stretch. Period = 2π/|B|
- C (Phase Shift): Horizontal shift. Shift = -C/B units
- D (Vertical Shift): Moves the entire graph up or down
Periodic Behavior
Trigonometric functions are periodic, meaning they repeat their values at regular intervals:
- sin(x) and cos(x): Period = 2π (360°). They repeat every full rotation
- tan(x): Period = π (180°). It repeats twice as often and has vertical asymptotes
- Key property: sin(x + 2π) = sin(x), cos(x + 2π) = cos(x), tan(x + π) = tan(x)
Real-World Applications
- Physics: Wave motion, oscillations, pendulums, sound and light waves
- Engineering: Signal processing, electrical circuits (AC current), mechanical vibrations
- Navigation: GPS systems, astronomy, maritime and aviation calculations
- Computer Graphics: Rotation, animation, game development
- Music: Sound waves, harmonics, audio synthesis
Special Angles to Remember
- 0° (0 or 0π): sin = 0, cos = 1, tan = 0
- 30° (π/6): sin = 1/2, cos = √3/2, tan = 1/√3
- 45° (π/4): sin = √2/2, cos = √2/2, tan = 1
- 60° (π/3): sin = √3/2, cos = 1/2, tan = √3
- 90° (π/2): sin = 1, cos = 0, tan = undefined
- 180° (π): sin = 0, cos = -1, tan = 0
- 270° (3π/2): sin = -1, cos = 0, tan = undefined
- 360° (2π): sin = 0, cos = 1, tan = 0
