Inverse Trigonometric Functions

Why We Need Inverse Trigonometric Functions

The standard trigonometric functions sin, cos and tan are periodic — they repeat their values infinitely often. This means they are not one-to-one, and a straightforward inverse function does not exist over their full domain. To define arcsin, arccos and arctan as proper functions, we restrict the domains of sin, cos and tan to intervals on which they are one-to-one (monotone), and then take the inverse on those restricted functions.

Defining arcsin, arccos, arctan

The function arcsin(x) is the inverse of sin restricted to [−π/2, π/2]. Its domain is [−1, 1] (the range of sin) and its range is [−π/2, π/2]. It is an increasing function — larger x gives larger arcsin(x). Similarly, arccos(x) inverts cos restricted to [0, π], giving a decreasing function with domain [−1, 1] and range [0, π]. The function arctan(x) inverts tan restricted to (−π/2, π/2), has domain ℝ and range (−π/2, π/2), with horizontal asymptotes approaching ±π/2.

Exact Values

Since arcsin is the inverse of sin restricted to [−π/2, π/2], to find arcsin(k) you ask: “which angle in [−π/2, π/2] has sine equal to k?” Knowing the standard triangle exact values (30–60–90 and 45–45–90) allows you to evaluate all common cases without a calculator. A common error is giving arccos(−1/2) = −π/3 — this is wrong because the range of arccos is [0, π], not [−π/2, π/2]. The correct answer is arccos(−1/2) = 2π/3 (in Q2).

Composition Identities

Because arcsin is the inverse of (restricted) sin, we have sin(arcsin x) = x for all x ∈ [−1, 1]. This is straightforward. The reverse composition arcsin(sin x) = x is only guaranteed when x ∈ [−π/2, π/2] — the restricted domain. Outside this interval, arcsin(sin x) still returns a value, but it “folds” x back into the range [−π/2, π/2]. For example, arcsin(sin(5π/6)) = arcsin(1/2) = π/6 (not 5π/6). Understanding this folding behaviour is essential when solving equations.

Solving Equations Using Inverse Functions

Inverse trig functions allow us to solve equations of the form sin(f(x)) = k. Apply arcsin to both sides to get f(x) = arcsin(k) (the principal value), then use periodicity to find additional solutions. For example, to solve arcsin(2x − 1) = π/6: take sin of both sides to get 2x − 1 = sin(π/6) = 1/2, so x = 3/4. The key is that arcsin, arccos and arctan return a single value (the principal value), and additional solutions are found via the periodicity of the original trig function.

Graphical Interpretation

The graph of y = arcsin(x) is the reflection of the restricted sine curve y = sin(x) (for x ∈ [−π/2, π/2]) in the line y = x. It passes through (−1, −π/2), (0, 0) and (1, π/2), is concave down for x > 0 and concave up for x < 0. Similarly, y = arctan(x) passes through the origin, is always increasing and concave, and approaches π/2 as x → +∞ and −π/2 as x → −∞. These asymptotes are horizontal, making arctan fundamentally different from a polynomial or exponential function.