Principal Values: Defining Inverse Trig & Restricted Domains

arccos(x) returns angles from 0 to π (endpoints included).

Among the listed, it’s the inverse trig function with outputs spanning 0 to π.

tan(0) = 0, so arctan(0) = 0.

Only numbers between −1 and 1 can be cosines of real angles,

so arccos(x) accepts inputs in [−1, 1].

arcsin outputs principal angles from −π/2 to π/2, inclusive.

As x → ±∞, arctan(x) → ±π/2.

These horizontal lines bound the outputs but are never reached.

For any x ∈ [−1, 1], sin(arcsin(x)) = x.

The reverse, arcsin(sin(x)) = x, only holds when x ∈ [−π/2, π/2].

cos(x) is strictly decreasing on [0, π].

Restricting the domain to this interval allows a unique inverse, arccos(x).

sin(x) is not one-to-one over ℝ because it repeats every 2π.

Restricting to [−π/2, π/2] makes sin(x) strictly increasing and one-to-one, so it has an inverse, arcsin(x).

The standard name is arcsin(x) (also written sin⁻¹(x)).

It returns the angle θ ∈ [−π/2, π/2] with sin(θ) = x.

arcsin(x) requires x ∈ [−1, 1], so arcsin(2) is undefined.

arctan(2), arccot(2), and arctan(−2) are defined.

arctan outputs angles strictly between −π/2 and π/2.

For any valid sine/cosine value x, the complementary angles satisfy

arcsin(x) + arccos(x) = π/2.

On that restricted domain, sin(x) is one-to-one, and its inverse is arcsin(x).

arccos(x) returns angles from 0 to π, inclusive.

With the proper domain restriction, sin(x) inverts to arcsin(x).

The other pairings mismatch functions.

On (−π/2, π/2), tan(x) is one-to-one and its inverse is arctan(x).

sin(−π/2) = −1, and −π/2 is in the arcsin range.

Thus arcsin(−1) = −π/2.

arctan(x) approaches π/2 but never reaches it.

π/2 is the maximum limiting value (horizontal asymptote).

cos(π/3) = 1/2, and π/3 ∈ [0, π], the arccos range.

Inverse functions must be one-to-one.

So we restrict each trig function’s domain to an interval where it’s injective.