Identità Trigonometriche Inverse
Comprendere e applicare identità trigonometriche inverse
Interactive Inverse Trigonometric Identities Visualization
Welcome to the Inverse Trigonometric Identities Explorer
This interactive visualization helps you understand inverse trigonometric identities and their domain restrictions. Explore how inverse functions interact with their original functions and discover complementary relationships.
What you can explore:
- Composition Identities - arcsin(sin(x)), sin(arcsin(x)), etc.
- Complementary Identities - arcsin(x) + arccos(x) = π/2
- Sum Formulas - arctan(x) + arctan(y) identities
- Domain Restrictions - Why identities only hold for certain values
How to Use This Visualization
Interactive Features:
- • Select Identity - Choose from 9 different identities
- • Adjust x Value - Change input to see when identity holds
- • Adjust y Value - For sum formulas, change second input
- • Toggle Graph - Show/hide the graphical representation
What You'll See:
- • Left Side (Green) - The left-hand side of the identity
- • Right Side (Blue, Dashed) - The right-hand side
- • Equality Check - Whether LHS = RHS at current x
- • Domain Restrictions - When the identity holds
arcsin undoes sin, but only for restricted domain
⚠ Only equals x when x ∈ [-π/2, π/2]
Identity Evaluation
Identity:
arcsin(sin(x)) = x (for x ∈ [-π/2, π/2])
Left Side (LHS)
0.5000
≈ 28.65°
Right Side (RHS)
0.5000
≈ 28.65°
✓ Identity Holds
Key Concepts
Composition: f(f⁻¹(x)) = x always holds, but f⁻¹(f(x)) = x only when x is in the restricted domain.
Domain Restrictions: Inverse functions are defined on restricted domains to ensure they are one-to-one.
Complementary: arcsin(x) + arccos(x) = π/2 because sin and cos are complementary in a right triangle.
Common Identities
• sin(arcsin(x)) = x, x ∈ [-1, 1]
• arcsin(sin(x)) = x, x ∈ [-π/2, π/2]
• arccos(cos(x)) = x, x ∈ [0, π]
• arctan(tan(x)) = x, x ∈ (-π/2, π/2)
• arcsin(x) + arccos(x) = π/2
• arctan(x) + arctan(1/x) = π/2, x > 0
Understanding Domain Restrictions
Why arcsin(sin(x)) ≠ x for all x?
The sine function is not one-to-one (multiple x values give the same sin(x)). arcsin is defined to return values in [-π/2, π/2]. So arcsin(sin(π)) = arcsin(0) = 0, not π.
Why sin(arcsin(x)) = x always?
arcsin(x) always returns a value in [-π/2, π/2], and sin is one-to-one on this interval, so sin(arcsin(x)) = x for all x in the domain of arcsin (which is [-1, 1]).
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