Trigonometric Functions

Evaluating Inverse Trig Functions

Evaluating arcsin, arccos, and arctan with restricted domains.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Graphing Tangent & Transformations

Making Trig Functions Invertible

Recall from earlier in this unit: a function needs to be one-to-one to have a genuine inverse. Before reading on, think about sin(x) — since it repeats forever, with countless different x-values sharing the same output, can it possibly have a real inverse at all?

Not over its full domain — but restricting sine to a single, carefully chosen interval where it's genuinely one-to-one solves the problem, the same trick used elsewhere whenever a function needs an inverse but isn't naturally one-to-one everywhere.

Definition — Inverse Trigonometric Functions

arcsin(x) (or sin⁻¹(x)) is defined using sine restricted to [−π/2, π/2]. arccos(x) uses cosine restricted to [0, π]. arctan(x) uses tangent restricted to (−π/2, π/2). Each restricted interval is chosen to be the smallest continuous range where that trig function is one-to-one while still covering every possible output value.

Worked Example — Evaluating arcsin and arccos

Evaluate arcsin(1/2) and arccos(−√2/2). For arcsin(1/2): find the angle in [−π/2, π/2] whose sine is 1/2 — that's π/6. For arccos(−√2/2): find the angle in [0, π] whose cosine is −√2/2 — that's 3π/4 (in the second quadrant, since cosine is negative there and 0 to π is entirely available to arccos, unlike arcsin's more restricted range).

Worked Example — Evaluating arctan

Evaluate arctan(−1). Find the angle in (−π/2, π/2) whose tangent is −1 — that's −π/4.

Tip

An inverse trig function's answer always falls inside its specific restricted range, even when another angle outside that range would also satisfy the original ratio — always report the one value the restriction actually allows.

Common Mistakes

Giving an answer outside the inverse function's restricted range, even if that other angle also has the correct sine, cosine, or tangent value.
Every inverse trig function has exactly one valid output range — arcsin only returns values in [−π/2, π/2], for example, even though other angles elsewhere also share the same sine value.
Using the same restricted range for all three inverse trig functions, rather than each one's own specific interval.
arcsin, arccos, and arctan each have their own distinct restricted range — [−π/2, π/2] for arcsin and arctan (open for arctan), and [0, π] for arccos — matching whichever interval makes that specific function one-to-one.

Key Takeaways

Inverse trig functions exist because sine, cosine, and tangent are first restricted to intervals where they're one-to-one.
Each inverse trig function has its own specific output range, matching the interval its original function was restricted to.
An inverse trig function's answer is always the one value inside its restricted range, even when other equivalent angles exist elsewhere.

Summary

Restricting a trig function's domain makes a genuine inverse possible, at the cost of only returning one specific angle per value. The next unit moves from evaluating trig functions to proving relationships between them — trigonometric identities.

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