三角函数
理解三角函数及其性质
Interactive Trigonometric Functions Visualization
Welcome to the Trigonometric Functions Explorer
This interactive visualization helps you understand the graphs and properties of trigonometric functions. Explore how each function behaves, their periods, domains, ranges, and key characteristics.
What you can explore:
- Sine and Cosine - Continuous, periodic functions with range [-1, 1]
- Tangent - Periodic with vertical asymptotes, range (-∞, ∞)
- Cosecant - Reciprocal of sine, has vertical asymptotes
- Secant - Reciprocal of cosine, has vertical asymptotes
- Cotangent - Reciprocal of tangent, periodic with vertical asymptotes
How to Use This Visualization
Interactive Features:
- • Select Function - Choose from 6 trigonometric functions
- • Adjust Periods - Control how many periods to display
- • Toggle Visualizations - Show/hide asymptotes and key points
- • Explore Properties - See domain, range, and period information
What You'll See:
- • Function Curve (colored) - The graph of the trigonometric function
- • Vertical Asymptotes (red, dashed) - Where function is undefined
- • Key Points (colored dots) - Important points on the curve
- • Grid Lines - Help visualize the function's behavior
- • Properties Panel - Domain, range, period, and characteristics
Sine function - periodic with period 2π
More periods show more of the function's behavior
Function Properties
Period
2π
Function repeats every 2π radians
Domain
(-∞, ∞)
All valid input values
Range
[-1, 1]
All possible output values
Amplitude
1
Maximum distance from center
Function Characteristics
Function: f(x) = sin(x)
Period: 2π radians
Continuous: Yes
Bounded: Yes (between -1 and 1)
Key Concepts
Period: The smallest positive value p such that f(x + p) = f(x) for all x in the domain.
Asymptotes: Vertical lines where the function approaches infinity. Occur where the function is undefined.
Reciprocal Functions: csc, sec, and cot are reciprocals of sin, cos, and tan respectively, creating different graph shapes.
Trigonometric Functions Overview
Basic Functions:
- sin(x) - period: 2π
- cos(x) - period: 2π
- tan(x) - period: π
Reciprocal Functions:
- csc(x) = 1/sin(x)
- sec(x) = 1/cos(x)
- cot(x) = 1/tan(x)
Key Properties:
- sin and cos: continuous, bounded
- tan and cot: period π, vertical asymptotes
- csc and sec: period 2π, vertical asymptotes
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