Inverse Trigonometric Functions — Formulas, Domain, Range

#Trigonometry

TL;DR

The inverse trigonometric functions — arcsin, arccos, arctan, arccsc, arcsec, arccot — undo the standard trig functions. Each takes a ratio and returns an angle. The trick is that sine, cosine, and tangent each map many angles to the same ratio, so their inverses only work on restricted "principal" intervals: arcsin on [−π/2,π/2], arccos on [0,π], arctan on (−π/2,π/2).

A Function That Had to Pick a Lane

Sine is periodic — it sends infinitely many angles to the same ratio. If you ask "what angle has sine 1/2?" — there's no single answer. To define an inverse function that gives back exactly one angle, mathematicians had to pick a lane: a single interval on which sine is one-to-one. That interval is the principal branch, and it's the reason every inverse trig function has a restricted range.

What Are Inverse Trigonometric Functions?

Inverse trigonometric functions are the functions that "undo" the basic trig functions. Each takes a real number (a trig ratio) as input and returns an angle as output:

arcsin⁡x=θ⟺sin⁡θ=x;(with θ in principal range)

The six functions are written either with "arc" notation (arcsinx, arccosx, etc.) or with the "−1" superscript (sin⁡−1x, cos⁡−1x, etc.). They are functions, so each input gives exactly one output — which is the whole point of restricting the range.

The Six Inverse Trigonometric Formulas

The pattern. Domain of each inverse function = range of the original function. Range of each inverse function = the principal interval where the original is one-to-one. The "−1" in sin⁡−1x is not a reciprocal — it's the inverse-function label.

Why the Range Is Restricted

Take arcsine. Sine, viewed as a function of all real numbers, is not one-to-one — infinitely many angles share the same sine value. To define an inverse, we restrict sine to the interval [−π/2,π/2], on which sine is one-to-one and continuous. The inverse of that restricted sine is what we call arcsine, with range [−π/2,π/2].

Similarly:

• Cosine is restricted to [0,π] — that's arccos's range.
• Tangent is restricted to (−π/2,π/2) — that's arctan's range.

These choices aren't arbitrary; they're the largest connected intervals containing zero on which each function is monotonic.

Three Worked Examples — Quick, Standard, Stretch

Quick

Find arcsin(1/2). We want the angle θ∈[−π/2,π/2] with sin⁡θ=1/2. From the unit circle, sin(π/6)=1/2 and π/6 is in the principal range.

arcsin(1/2)=π/6=30°

Wrong Path First — Then the Right One — Standard Example

Find arcsin(sin(2π/3)). The wrong path. A student writes: "Arcsine undoes sine, so arcsin(sin(2π/3))=2π/3." Done.

That answer is wrong, even though it looks like an "inverse undoing the original" must give back the input.

Sanity check. Arcsine returns values only in [−π/2,π/2]. The angle 2π/3 is outside that range (2π/3≈120° — past π/2=90°). So arcsin(sin(2π/3)) cannot possibly equal 2π/3.

The correct path. Compute sin(2π/3) first — using the reference angle, sin(2π/3)=sin(π−2π/3)=sin(π/3)=√3/2 (positive because 2π/3 is in quadrant II).

Then arcsin(√3/2)=π/3 — the angle in [−π/2,π/2] whose sine is √3/2.

arcsin(sin(2π/3))=π/3, not 2π/3.

Stretch

Find the exact value of sin(2arctan(3/4)).

Let θ=arctan(3/4), so tan⁡θ=3/4 and θ∈(−π/2,π/2). Imagine a right triangle with opposite side 3, adjacent side 4 — then hypotenuse is √(3²+4²)=5.

So sin⁡θ=3/5 and cos⁡θ=4/5.

Using the double-angle formula: sin(2θ)=2sin(θ)cos(θ)=2⋅(3/5)⋅(4/5)=24/25.

sin(2arctan(3/4))=24/25.

The trick — turning an inverse-trig expression into a right triangle — works for any composition like sin(arctan(x)), cos(arcsin(x)), etc. Worth memorising.

Graphs of Inverse Trigonometric Functions

Each inverse-function graph is the reflection of the corresponding restricted forward function across the line y=x.

• Arcsine. S-shaped curve from (−1,−π/2) to (1,π/2), passing through the origin.
• Arccosine. Mirror image of arcsine — from (−1,π) to (1,0), passing through (0,π/2). Strictly decreasing.
• Arctangent. Smooth S-curve passing through the origin, with horizontal asymptotes at ±π/2.

Derivatives of Inverse Trigonometric Functions

These are the most-used inverse-trig formulas in calculus:

[ \frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}} \qquad \frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}} ] [ \frac{d}{dx}\arctan x = \frac{1}{1+x^2} \qquad \frac{d}{dx}\arccot x = -\frac{1}{1+x^2} ] [ \frac{d}{dx}\arcsec x = \frac{1}{|x|\sqrt{x^2-1}} \qquad \frac{d}{dx}\arccsc x = -\frac{1}{|x|\sqrt{x^2-1}} ]

The pattern. Each "co-" inverse has the negative of its partner's derivative — because arccos x + arcsin x = π/2 for all x∈[−1,1]. Differentiate both sides and the negative sign drops out.

Identities Worth Knowing

[ , arcsin x + arccos x = \frac{\pi}{2} \quad \arctan x + \arccot x = \frac{\pi}{2} ]
[ , arcsin(-x) = -arcsin x \quad arctan(-x) = -arctan x ]
[ , arccos(-x) = \pi - arccos x \quad arctan x + arctan(1/x) = \pm\frac{\pi}{2} ]

The first row is the most useful — it lets you convert between any two complementary inverses in one step.

Why Inverse Trig Functions Show Up Everywhere

The forward trig functions answer "given an angle, what's the ratio?" The inverse functions answer the more practical question: "given a measured ratio, what angle produced it?"

• Surveying. A surveyor measures horizontal and vertical distances and needs the angle — arctan(rise/run).
• Robotics and inverse kinematics. When a robot arm needs to reach a target position, the controller solves for the joint angles by inverting the forward-kinematics equations.
• GPS and navigation. Latitude is recovered from celestial measurements using arcsin. Bearing angles use arctan.
• Computer vision. The orientation of a detected line in an image is arctan(slope).
• Physics — refraction. Snell's law gives the angle of refraction as arcsin(n₁sinθ₁/n₂).

The forward trig functions matter; the inverse ones matter more in any setting where the measurement comes first and the angle has to be recovered.

The Mathematicians Who Shaped Inverse Trigonometric Functions

Inverse trig functions developed alongside their forward counterparts, but the principal-branch convention came surprisingly late.

• Daniel Bernoulli (1700–1782, Swiss) — first used the notation A.sin A for arcsine in 1729.
• Leonhard Euler (1707–1783, Swiss) — formalised inverse trig functions in 1748.
• Augustin-Louis Cauchy (1789–1857, French) — formalised the principal-branch concept in the 1820s.

Why it matters: every time a calculator returns arcsin(0.5)=π/6 instead of, say, 5π/6 — it's executing Cauchy's principal-branch convention from 1821.

Key Takeaways

• The six inverse trigonometric functions undo the six standard trig functions, returning an angle from a ratio.
• Each has a restricted "principal" range because the original function is periodic and not one-to-one.
• The biggest exam slip is assuming arcsin(sin(x))=x for every x — it only holds when x is already in [−π/2,π/2].
• Inverse trig functions power surveying, robotics, GPS, computer vision, and refraction calculations.

Try It Yourself — Three Problems

1. Compute arccos(cos(7π/4)). (Check whether 7π/4 is in [0,π]).
2. Evaluate sin(arccos(0.6)) by drawing a right triangle.
3. Differentiate f(x)=arctan(2x) using the chain rule.