Principal inverse sine and inverse cosine

Sine is differentiable, hence continuous, and strictly increasing on $[-\pi/2,\pi/2]$; its endpoint values are $-1$ and $1$ (The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$, Signs, monotonicity intervals, and ranges of sine and cosine, Quarter-turn values and shifts by pi/2 and pi, Parity and the Pythagorean identity for sine and cosine). The intermediate value theorem therefore makes its restricted image exactly $[-1,1]$ (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$). Likewise, cosine is continuous and strictly decreasing on $[0,\pi]$, with endpoint values $1$ and $-1$, so its restricted image is $[-1,1]$ (Signs, monotonicity intervals, and ranges of sine and cosine, The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$, Quarter-turn values and shifts by pi/2 and pi, Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$). Their principal inverses are denoted

$\arcsin:[-1,1]\to[-\pi/2,\pi/2],\qquad\arccos:[-1,1]\to[0,\pi],$

and are characterised by

$\sin(\arcsin y)=y,\qquad\cos(\arccos y)=y\qquad(-1\le y\le1).$

The chosen target intervals are part of the notation: without them, inverse sine and inverse cosine would be multivalued relations rather than functions. Their continuity follows from Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g : f[I] \to I$ is continuous and strictly monotone in the same sense as $f$.