Inverse Trigonometric Functions

Graphs of Inverse Trigonometric Functions

Understand

The graph of \(y=f^{-1}(x)\) is obtained by reflecting the graph of \(y=f(x)\) (on its principal branch) in the line \(y=x\): the point \((a,b)\) becomes \((b,a)\). So domain and range swap places.

\(y=\sin^{-1}x\): increasing, from \((-1,-\frac\pi2)\) to \((1,\frac\pi2)\).

\(y=\cos^{-1}x\): decreasing, from \((-1,\pi)\) to \((1,0)\).

\(y=\tan^{-1}x\): increasing, with horizontal asymptotes \(y=\pm\frac\pi2\).

Reading the graphs

\(\sin^{-1}\) and \(\tan^{-1}\) are increasing and odd (symmetric about the origin).
\(\cos^{-1}\) and \(\cot^{-1}\) are decreasing; \(\cos^{-1}(0)=\cot^{-1}(0)=\frac\pi2\).
\(\tan^{-1}x\to\pm\frac\pi2\) and \(\cot^{-1}x\to0,\pi\) as \(x\to\pm\infty\) – horizontal asymptotes.
\(\sec^{-1}\) and \(\csc^{-1}\) have a gap for \(-1<x<1\) and skip the values \(\frac\pi2\) and \(0\) respectively.

Key Concepts

Inverse graph = reflection in y = x.
Endpoints of sin⁻¹: (−1, −π/2), (1, π/2); of cos⁻¹: (−1, π), (1, 0).
tan⁻¹ has asymptotes y = ±π/2.

Key Points

Monotonicity

\(\sin^{-1},\tan^{-1}\) increase; \(\cos^{-1},\cot^{-1}\) decrease. Extreme values of expressions follow from the endpoints of the ranges.

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Topic Summary

Know the shape, endpoints, monotonicity and asymptotes of each inverse function; use them to read off ranges and extreme values.

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