Inverse Trigonometric Functions

Understand inverse trigonometric operations. See how restricting domains makes trigonometric functions one-to-one, mapping their coordinate reflections symmetrically across the diagonal line $y = x$.

Interactive Sandbox

Reflection Symmetry Grapher

Select a function. Use the slider to vary input $x$. Notice how trig point $P(a, b)$ (cyan) mirrors onto inverse trig point $P'(b, a)$ (rose) perpendicularly across the diagonal line $y = x$ (purple).

Reflection: y = sin(x) vs y = arcsin(x)

Active Function Mapping

θ = arcsin(0.50) ⇔ sin(θ) = 0.50
Input Value (x) 0.50
The purple dotted segment connects coordinates (a, b) and (b, a), highlighting reflection symmetry.
Algebraic Solver

Domain & Range constraints

Trace the mathematical properties and principal branch values for the selected input.

Principal Branch Evaluation


Principal Branch Constraints


Domain (Restricted Input)

[-1.00, 1.00]

Range (Principal Output)

[-π/2, π/2] ([-1.57, 1.57])

Evaluation

Conceptual Quiz

Test your understanding of domain restrictions, principal outputs, and coordinate reflections.

Question 1 of 5 Score: 0/0

What is the principal range of y = arcsin(x)?

Correct Answer!

Explanation text...

Inverse Trig Summary


  • sin⁻¹(x) range: [-π/2, π/2]
  • cos⁻¹(x) range: [0, π]
  • tan⁻¹(x) range: (-π/2, π/2)
  • Reflections: Mirrored across y = x line