Inverse Trigonometric Functions¶
Recall (doc/postdoc). Not a first course.
Job: principal branches of multi-valued inverses of periodic maps; \(\arcsin:[-1,1]\to[-\pi/2,\pi/2]\), etc.
Core¶
| Function | Domain | Principal range |
|---|---|---|
| \(\arcsin x\) | \([-1,1]\) | \([-\pi/2,\pi/2]\) |
| \(\arccos x\) | \([-1,1]\) | \([0,\pi]\) |
| \(\arctan x\) | \(\mathbb{R}\) | \((-\pi/2,\pi/2)\) |
Relations: \(\arcsin x+\arccos x=\pi/2\); \(\tan(\arctan x)=x\); \(\sin(\arcsin x)=x\) on domains.
Derivatives:
\[
(\arcsin x)'=\frac{1}{\sqrt{1-x^2}},\quad
(\arctan x)'=\frac{1}{1+x^2}
\]
Multi-valued: full inverse relations are \(\arcsin x+2\pi k\) or \(\pi-\arcsin x+2\pi k\) for \(\sin\).
Used later for¶
- Integrals: \(\int dx/\sqrt{1-x^2}\), \(\int dx/(1+x^2)\)
- Complex analysis: branches of \(\operatorname{Log}\), \(\operatorname{Arcsin}\)
- Geometry: polar angle \(\operatorname{atan2}\)
- Differential geometry: angle functions, charts
Check¶
Is \(\arcsin(\sin\theta)=\theta\) for all real \(\theta\)?
No — only for \(\theta\in[-\pi/2,\pi/2]\). Outside, projects to principal value (e.g. \(\arcsin(\sin\frac{2\pi}{3})=\frac{\pi}{3}\)).