Funzioni Trigonometriche Inverse
Lavorare con funzioni trigonometriche inverse e le loro applicazioni
Interactive Inverse Trigonometric Functions Visualization
Welcome to the Inverse Trigonometric Functions Explorer
This interactive visualization helps you understand inverse trigonometric functions (arcsin, arccos, arctan, etc.) and their properties. Inverse functions are created by restricting the domain of the original trigonometric functions to make them one-to-one.
What you can explore:
- arcsin(x) - Inverse of sine, domain [-1, 1], range [-π/2, π/2]
- arccos(x) - Inverse of cosine, domain [-1, 1], range [0, π]
- arctan(x) - Inverse of tangent, domain (-∞, ∞), range (-π/2, π/2)
- arccsc(x) - Inverse of cosecant, domain |x| ≥ 1
- arcsec(x) - Inverse of secant, domain |x| ≥ 1
- arccot(x) - Inverse of cotangent, domain (-∞, ∞), range (0, π)
How to Use This Visualization
Interactive Features:
- • Select Function Type - Choose from 6 inverse trigonometric functions
- • Adjust x Value - Change the input value to see the output
- • Toggle Visualizations - Show/hide original function and reflection line
- • Explore Properties - See domain, range, and key characteristics
What You'll See:
- • Inverse Function Curve (colored) - The graph of the inverse function
- • Original Function (purple, dashed) - Reflected version of original
- • Reflection Line (gray, dashed) - y = x line showing reflection
- • Point at x (colored) - Shows input and output values
- • Properties Panel - Domain, range, and function characteristics
Inverse of sine - restricted domain [-1, 1]
Adjust to see different input/output pairs
Function Evaluation
Input (x)
0.5000
Output f(x)
0.5236
≈ 30.00°
Function Properties
Domain
[-1, 1]
Valid input values
Range
[-π/2, π/2]
Possible output values
Original Function
sin(x)
Inverse relationship
Key Properties
Function: f(x) = arcsin(x)
Domain: [-1, 1]
Range: [-π/2, π/2]
One-to-One: Yes (restricted domain)
Key Concepts
Inverse Function: If f(x) = y, then f⁻¹(y) = x. The graph of f⁻¹ is the reflection of f across y = x.
Restricted Domain: Trigonometric functions are not one-to-one, so we restrict their domains to create inverses.
Range Restriction: The range of the inverse is the restricted domain of the original function.
Inverse Trigonometric Functions
Basic Inverses:
- arcsin(x): domain [-1, 1], range [-π/2, π/2]
- arccos(x): domain [-1, 1], range [0, π]
- arctan(x): domain (-∞, ∞), range (-π/2, π/2)
Reciprocal Inverses:
- arccsc(x): domain |x| ≥ 1, range [-π/2, 0) ∪ (0, π/2]
- arcsec(x): domain |x| ≥ 1, range [0, π/2) ∪ (π/2, π]
- arccot(x): domain (-∞, ∞), range (0, π)
Lezioni
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