Master inverse trig functions easily

You need the angle that produces a given sine, cosine, or tangent. But trig functions repeat their values every full cycle, so a single input matches infinitely many angles. Inverse trig functions solve that by using a restricted domain: each inverse returns exactly one principal value per input. The three fundamental inverse trig functions are arcsin (or sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹). Their domains, ranges, graphs, exact value tables, and common traps like confusing sin⁻¹(x) with 1/sin(x) are covered here.

The function sin(x) repeats every 2π radians. It fails the horizontal line test: a horizontal line at y = 0.5 hits the sine curve at x = π/6, 5π/6, and infinitely many other points. Because it is not one-to-one, sin(x) has no inverse function unless you restrict its domain. By convention, the restriction is [-π/2, π/2] (or [-90°, 90°]). On that interval the sine curve is strictly increasing and passes the horizontal line test. The inverse of sin(x) on that restricted domain is arcsin(x). The same logic applies to cos(x) and tan(x), each with its own restricted domain.

OpenStax Precalculus 2e (2021, Rice University) covers this material in sections 3.7 (Inverse Functions) and 5.7 (Inverse Trigonometric Functions). The one-to-one function horizontal line test is the gatekeeper: if the function fails it, an inverse does not exist without a domain restriction.

The arcsine function, also written sin⁻¹(x), answers the question: what angle in [-π/2, π/2] has a sine equal to x?

Domain

The domain of arcsin is [-1, 1]. Inputs outside this interval produce no real output because the sine of a real angle never exceeds 1 in magnitude.

Range

The range of arcsin is [-π/2, π/2] (or [-90°, 90°]). This is the principal value range. An angle of, say, 2π/3 has sine √3/2, but arcsin(√3/2) returns π/3, not 2π/3, because 2π/3 lies outside the range.

Graph

The graph of y = arcsin(x) is the reflection of the restricted sine curve across the line y = x. It passes through (-1, -π/2), (0, 0), and (1, π/2). The curve is increasing and symmetric about the origin: arcsin(-x) = -arcsin(x). The graph never extends beyond x = ±1 or y = ±π/2.

The arccosine function, cos⁻¹(x), returns the angle in [0, π] whose cosine is x. Its domain is also [-1, 1], but its range is [0, π] (0° to 180°).

Domain and Range

Domain: [-1, 1]. Range: [0, π]. This is the principal value range for arccos. For example, arccos(0) = π/2, and arccos(-1) = π. Unlike arcsin, arccos is decreasing on its domain: as x increases, the output angle decreases.

Graph Notes

The graph of arccos is not symmetric; arccos(-x) = π - arccos(x). It passes through (-1, π), (0, π/2), and (1, 0). The output never goes below 0 or above π.

The arctangent function, tan⁻¹(x), returns the angle in (-π/2, π/2) whose tangent is x. Unlike arcsin and arccos, arctan accepts all real numbers as input.

Domain and Range

Domain: all real numbers (−∞, ∞). Range: (-π/2, π/2) (open interval, excluding the endpoints). The principal value for arctan is always in Quadrants I or IV. For example, arctan(1) = π/4, not 5π/4.

Graph and Asymptotes

The graph of y = arctan(x) has horizontal asymptotes at y = -π/2 and y = π/2. As x → ∞, arctan(x) approaches π/2 from below; as x → -∞, it approaches -π/2 from above. The function is increasing and odd: arctan(-x) = -arctan(x).

Function	Domain	Range (Principal Value)	Common Notation
arcsin (sin⁻¹)	[-1, 1]	[-π/2, π/2]	arcsin x, sin⁻¹ x
arccos (cos⁻¹)	[-1, 1]	[0, π]	arccos x, cos⁻¹ x
arctan (tan⁻¹)	All real numbers	(-π/2, π/2)	arctan x, tan⁻¹ x

The table below lists the exact outputs for common inputs. Memorise these for tests and for evaluating compositions.

x	arcsin(x)	arccos(x)	arctan(x)
0	0	π/2	0
½	π/6	π/3	arctan(½) not standard
√2/2	π/4	π/4	π/4 (x=1)
√3/2	π/3	π/6	π/3 (x=√3)
1	π/2	0	π/4
-½	-π/6	2π/3	-π/6
-1	-π/2	π	-π/4

For x values not in this table, a calculator or the arcsin, arccos, arctan keys on a scientific calculator will give a decimal approximation.

The most frequent error with inverse trig functions is confusing the notation sin⁻¹(x) with the reciprocal 1/sin(x). The superscript -1 in sin⁻¹(x) means the inverse function, not the multiplicative inverse. The reciprocal of sine is cosecant: csc(x) = 1/sin(x). For example, sin⁻¹(0.5) = π/6, while 1/sin(π/6) = 2. These are completely different numbers. The same applies to cos⁻¹(x) and tan⁻¹(x): they are never equal to 1/cos(x) or 1/tan(x). If you see sin⁻¹(x) in a problem, you are finding an angle; if you see csc(x) or 1/sin(x), you are taking a reciprocal.

For any x in the domain of the inner inverse function, the composition sin(arcsin x) equals x, provided the result stays inside the domain of the outer function. Specifically:

• sin(arcsin x) = x for all x in [-1, 1].
• cos(arccos x) = x for all x in [-1, 1].
• tan(arctan x) = x for all real x.

The reverse composition, arcsin(sin θ), is trickier. It equals θ only if θ lies within the principal value range of arcsin, [-π/2, π/2]. For example, arcsin(sin(5π/6)) = arcsin(½) = π/6, not 5π/6, because 5π/6 is outside the range. The same holds for arccos(cos θ) and arctan(tan θ): the output is the principal value, which may differ from the original angle. Always check whether the angle is within the principal value range before simplifying.