The trigonometric functions are periodic, and thus get all their values infinitely many times. Therefore their inverse functions, the cyclometric functions, are multivalued, but the values within suitable chosen intervals are unique; they form single-valued functions, called the branches of the multivalued functions.

The principal branches of the most used cyclometric functions are as follows:

• $\arcsin{x}$ is the angle $y$ satisfying $\sin y = x$ and $-\frac{\pi}{2} < y \leqq \frac{\pi}{2}$ (defined for $-1 \leqq x \leqq 1$ )
• $\arccos{x}$ is the angle $y$ satisfying $\cos y = x$ and $0 \leqq y < \pi$ (defined for $-1 \leqq x \leqq 1$ )
• $\arctan{x}$ is the angle $y$ satisfying $\tan y = x$ and $-\frac{\pi}{2} < y < \frac{\pi}{2}$ (defined in the whole $\mathbb{R}$ )
• $\arccot\,{x}$ is the angle $y$ satisfying $\cot y = x$ and $0 < y < \pi$ (defined in the whole $\mathbb{R}$ )

Those functions are denoted also $\sin^{-1}x$ , $\cos^{-1}x$ , $\tan^{-1}x$ and $\cot^{-1}x$ . We here use these notations temporarily for giving the corresponding multivalued functions ($n = 0,\, \pm1,\, \pm2,\, ...$ ): $$\sin^{-1}x = n\pi+(-1)^n\arcsin{x}$$ $$\cos^{-1}x = 2n\pi\pm\arccos{x}$$ $$\tan^{-1}x = n\pi+\arctan{x}$$ $$\cot^{-1}x = n\pi+\arccot\,{x}$$

Some formulae $$\arcsin{x}+\arccos{x} = \frac{\pi}{2}$$ $$\arctan{x}+\arccot\,{x} = \frac{\pi}{2}$$ $$\arcsin{x} = \int_0^x\frac{dt}{\sqrt{1-t^2}}\,dt$$ $$\arctan{x} = \int_0^x\frac{dt}{1+t^2}\,dt$$ $$\arcsin{x} = x+\frac{1}{2}\!\cdot\!\frac{x^3}{3}+ \frac{1\!\cdot\!3}{2\!\cdot\!4}\!\cdot\!\frac{x^5}{5}+ \frac{1\!\cdot\!3\!\cdot\!5}{2\!\cdot\!4\!\cdot\! 6}\!\cdot\!\frac{x^7}{7}+\ldots\quad(|x|\leqq 1)$$ $$\arctan{x} = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+-\ldots \quad(|x| \leqq 1)$$ $$\frac{d}{dx}\arccos{x} = -\frac{1}{\sqrt{1-x^2}}\quad(|x| < 1)$$ $$\frac{d}{dx}\arccot\,{x} = -\frac{1}{1+x^2}\quad(\forall x\in \mathbb{R})$$

The classic abbreviations of the cyclometric functions are usually explained as follows. The values of the trigonometric functions are got from the unit circle by means of its arc (in Latin arcus) with starting point (1,0). The arc represents the angle (which may be thought as a central angle of the circle), and its end point $(\xi,\,\eta)$ is achieved when the starting point has circulated along the circumference anticlockwise for positive angle (and clockwise for negative angle). Then the cosine of the arc (i.e. angle) is the abscissa $\xi$ of the end point, the sine of the arc is the ordinate $\eta$ of it. Correspondingly, one can get the tangent and cotangent of the arc by using certain points on the tangent lines $x = 1$ and $y = 1$ of the unit circle.

The word cosine is in Latin cosinus, its genitive form is cosini. So e.g. ``$\arccos$ '' of a given real number $x$ means the `arc of the cosine value $x$ ', in Latin arcus cosini x.