Sine, cosine and tangent all fail to be one-to-one over their full domains, so each is first restricted to an interval on which it is a bijection, and the inverse is defined only on that restricted piece.
Inverse sine. Restrict sinx to [−2π,2π] (the interval on which it is one-to-one AND onto [−1,1]). Then sin−1:[−1,1]→[−2π,2π] is defined by sin−1x=y⟺siny=x and y∈[−2π,2π]. Its graph is the mirror image of the restricted sine curve in the line y=x: an increasing S-shaped curve through the origin, from (−1,−2π) to (1,2π). Since sine is odd on its restricted domain, sin−1x is also odd: sin−1(−x)=−sin−1x.
Inverse cosine. Restrict cosx to [0,π] (one-to-one, onto [−1,1]). Then cos−1:[−1,1]→[0,π] is defined by cos−1x=y⟺cosy=x and y∈[0,π]. Its graph falls from (−1,π) through (0,2π) to (1,0). Cosine on [0,π] is neither even nor odd (evenness would need a domain symmetric about 0 where it stays one-to-one, which is impossible for a periodic function away from the origin), so cos−1x is neither even nor odd either; a useful substitute identity is cos−1(−x)=π−cos−1x.
Inverse tangent. Restrict tanx to (−2π,2π) (one-to-one, onto all of R since tangent has vertical asymptotes exactly at the endpoints). Then tan−1:R→(−2π,2π) is defined by tan−1x=y⟺tany=x and y∈(−2π,2π). Its graph is a flattening S-curve through the origin that hugs the two horizontal asymptotes y=±2π without ever touching them, since tan−1x is defined for every real x (unlike sin−1,cos−1, whose domain stops at ±1). Tangent is odd on its restricted domain, so tan−1x is odd.
The interval each function is restricted to is called its principal domain; the corresponding restricted output interval is its range (and the value of the inverse function AT a point is called that point's principal value — developed fully in §4.9). None of these three functions is periodic once inverted, and their monotonicity carries over: sin−1x and tan−1x are strictly increasing, while cos−1x is strictly decreasing.
| Domain | Range (principal value branch) |
|---|
| sin−1x | [−1,1] | [−2π,2π] |
| cos−1x | [−1,1] | [0,π] |
| tan−1x | R | (−2π,2π) |
Working with a domain question. Because sin−1 and cos−1 only accept inputs in [−1,1], a composite expression like sin−1(g(x)) has domain exactly {x:−1≤g(x)≤1}, solved as a compound inequality. tan−1(g(x)), by contrast, places no restriction on g(x) itself (any real value is accepted) — the only restriction comes from whatever makes g(x) itself well defined (e.g. a square root needing a non-negative radicand).