Inverse Trigonometric Functions

Inverse trigonometric functions undo the trigonometric functions: given a value, they return the angle that produced it. Because sine, cosine, and the others are not one-to-one on the whole real line, each inverse function uses a carefully chosen principal value interval.

The chapter has four threads. The first is why a principal value exists at all and which interval each of the six functions adopts. The second walks through arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant, giving the domain, range, monotonicity, and graph of each. The third is evaluation: how to read off special-angle values such as $\arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3}$ by locating the angle inside its principal interval. The fourth is the history of the notation, which grew out of the need to solve trigonometric equations and to integrate trigonometric expressions—mathematicians first wrote “A. sin” and “arc. sin”, with “arc” standing for arc length, because the value of an inverse sine can be read as the length of an arc on the unit circle. A competing form, $\sin^{-1} x$, was introduced by John Herschel in the early nineteenth century but is easily confused with $\frac{1}{\sin x}$, so the “arcsin” style became the international standard.

If you want to see the complete pattern first, read the arcsine function; the other five follow exactly the same structure.