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Inverse Trigonometric Functions - Theory & Concepts - Trigonometry Inverse Trig Graphs

Deep dive into arcsin, arccos, arctan, domain restrictions, principal values, and composition.

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Inverse Trig Graphs & Restrictions

Observe the reflection symmetry over the line y=xy = x and the crucial domain restriction required to define the inverse.

Value (xx)0.50
-101

Domain Restrictions

Base Function:f(x)=sin⁡(x)f(x) = \sin(x)
Restricted Domain of Base:[−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}](Highlights in thick blue on the graph)
Inverse Function:f−1(x)=arcsin⁡(x)f^{-1}(x) = \arcsin(x)
INVERSE DOMAIN:
[−1,1][-1, 1]
INVERSE RANGE:
[−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}]

Reflected Points (y=xy = x)

Inverse Point:(0.50, 0.52)
Base Point:(0.52, 0.50)
By reflecting any point (x,y)(x, y) on the inverse graph across the line y=xy = x, we get (y,x)(y, x)which lies exactly on the restricted base function's graph.
Overlay plotting of trigonometric function, inverse, and line of symmetry y = xHorizontal AxisVertical Axis-3-3-2-2-1-1112233Line y = xFull Base Trig WaveRestricted Domain Wave HighlightInverse Trig GraphReflection ConnectorInverse point (x, y)Base function point (y, x)
Base: sin(x)Inverse: arcsinSymmetry: y = x

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