Inverse Trigonometric Functions — Formula Sheet
Inverse Trigonometric Functions — Formula Sheet
1. Domains and principal-value ranges
- \(\sin^{-1}x:\ [-1,1]\to\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)
- \(\cos^{-1}x:\ [-1,1]\to[0,\pi]\)
- \(\tan^{-1}x:\ \mathbb{R}\to\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)
- \(\operatorname{cosec}^{-1}x:\ \mathbb{R}\setminus(-1,1)\to\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\setminus\{0\}\)
- \(\sec^{-1}x:\ \mathbb{R}\setminus(-1,1)\to[0,\pi]\setminus\left\{\frac{\pi}{2}\right\}\)
- \(\cot^{-1}x:\ \mathbb{R}\to(0,\pi)\)
2. Complementary identities
\[\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2},\qquad \tan^{-1}x+\cot^{-1}x=\frac{\pi}{2},\qquad \sec^{-1}x+\operatorname{cosec}^{-1}x=\frac{\pi}{2}.\]
3. Reciprocal identities
\[\operatorname{cosec}^{-1}x=\sin^{-1}\frac{1}{x},\qquad \sec^{-1}x=\cos^{-1}\frac{1}{x},\qquad \cot^{-1}x=\tan^{-1}\frac{1}{x}\ \ (x\gt 0).\]
4. Negative-argument identities
\[\sin^{-1}(-x)=-\sin^{-1}x,\qquad \tan^{-1}(-x)=-\tan^{-1}x,\qquad \operatorname{cosec}^{-1}(-x)=-\operatorname{cosec}^{-1}x,\]
\[\cos^{-1}(-x)=\pi-\cos^{-1}x,\qquad \sec^{-1}(-x)=\pi-\sec^{-1}x,\qquad \cot^{-1}(-x)=\pi-\cot^{-1}x.\]
5. Cancellation laws
\[\sin(\sin^{-1}x)=x\ \ (|x|\le 1),\qquad \sin^{-1}(\sin\theta)=\theta\ \ \left(\theta\in\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]\right),\]
with the analogous statements for the other five functions, each restricted to its own principal range.
6. Conversions among inverses (for \(x\gt 0\))
\[\sin^{-1}x=\cos^{-1}\sqrt{1-x^{2}}=\tan^{-1}\frac{x}{\sqrt{1-x^{2}}}=\operatorname{cosec}^{-1}\frac{1}{x},\]
\[\cos^{-1}x=\sin^{-1}\sqrt{1-x^{2}}=\tan^{-1}\frac{\sqrt{1-x^{2}}}{x}.\]
7. Sum and difference of arctangents
\[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}\ \ (xy\lt 1),\qquad \tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy}\ \ (xy\gt -1).\]
8. Sum of arcsines and arccosines (sufficient conditions shown)
\[\sin^{-1}x+\sin^{-1}y=\sin^{-1}\!\left(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right),\quad x,y\ge 0,\ x^{2}+y^{2}\le 1,\]
\[\cos^{-1}x+\cos^{-1}y=\cos^{-1}\!\left(xy-\sqrt{1-x^{2}}\,\sqrt{1-y^{2}}\right),\quad x+y\ge 0.\]
9. Double- and triple-angle forms
\[2\tan^{-1}x=\sin^{-1}\frac{2x}{1+x^{2}}=\cos^{-1}\frac{1-x^{2}}{1+x^{2}}=\tan^{-1}\frac{2x}{1-x^{2}}\quad(\text{valid for suitable }x,\ \text{e.g. }|x|\le 1),\]
\[2\sin^{-1}x=\sin^{-1}\!\left(2x\sqrt{1-x^{2}}\right),\qquad 3\sin^{-1}x=\sin^{-1}(3x-4x^{3}),\qquad 3\cos^{-1}x=\cos^{-1}(4x^{3}-3x).\]
10. Handy boundary values
\[\sin^{-1}0=0,\ \ \sin^{-1}1=\frac{\pi}{2},\ \ \cos^{-1}1=0,\ \ \cos^{-1}0=\frac{\pi}{2},\ \ \tan^{-1}0=0,\ \ \tan^{-1}1=\frac{\pi}{4},\ \ \cot^{-1}1=\frac{\pi}{4}.\]