Properties of Inverse Trig Functions
Inverse trigonometric functions — arcsin, arccos, and arctan — are not simply the reciprocals of sine, cosine, and tangent. They are true inverse functions that 'undo' their corresponding trig functions, but only on carefully restricted domains. Because sine, cosine, and tangent are periodic and therefore not one-to-one over all real numbers, we must restrict each to a principal branch before an inverse can exist. Understanding the properties and identities that govern these inverses allows us to simplify complex expressions, evaluate compositions, and solve equations efficiently. The core properties include composition identities (e.g., sin(arcsin x) = x on the appropriate interval), co-function relationships (arcsin x + arccos x = π/2), symmetry properties (arcsin is odd, arccos satisfies arccos(−x) = π − arccos x), and reflection identities. These are not mere curiosities — they appear constantly when differentiating inverse trig functions in calculus, solving physics problems involving angles, and simplifying integrals. Mastering these identities means understanding why each one is true, not just memorising the formula. Every identity comes from the definitions of the restricted domains and from right-triangle or unit-circle reasoning. This article walks through the most important properties, shows how to derive them, and demonstrates how to apply them in calculations.
- Domain and Range Restrictions
- Composition Identities (Cancellation Laws)
- Co-function Identity: arcsin x + arccos x = π/2
- Odd/Even Symmetry
- Cross-function Identities via Right Triangles
- Addition Formulas for arctan
Domain and Range Restrictions
Before any identity can hold, we must know the principal branches. arcsin: domain [−1, 1], range [−π/2, π/2]. arccos: domain [−1, 1], range [0, π]. arctan: domain (−∞, ∞), range (−π/2, π/2). These restrictions ensure each inverse is a genuine function.
Composition Identities (Cancellation Laws)
There are two directions of composition. Forward: sin(arcsin x) = x for x ∈ [−1, 1]; cos(arccos x) = x for x ∈ [−1, 1]; tan(arctan x) = x for all real x. These hold for every input in the domain because applying the function after its inverse always returns the original value. Reverse: arcsin(sin θ) = θ only if θ ∈ [−π/2, π/2]; arccos(cos θ) = θ only if θ ∈ [0, π]; arctan(tan θ) = θ only if θ ∈ (−π/2, π/2). Outside these ranges, the reverse composition adjusts the angle back into the principal branch.
Co-function Identity: arcsin x + arccos x = π/2
Let α = arcsin x, so sin α = x and α ∈ [−π/2, π/2]. Then cos(π/2 − α) = sin α = x, so arccos x = π/2 − α = π/2 − arcsin x. This gives the identity arcsin x + arccos x = π/2 for all x ∈ [−1, 1]. Geometrically, in a right triangle the two acute angles are complementary, so the angle whose sine is x and the angle whose cosine is x must add to π/2.
Odd/Even Symmetry
arcsin is an odd function: arcsin(−x) = −arcsin x for x ∈ [−1, 1]. Proof: sin(−arcsin x) = −sin(arcsin x) = −x, and −arcsin x lies in [−π/2, π/2], confirming it equals arcsin(−x). arctan is also odd: arctan(−x) = −arctan x. arccos is neither odd nor even but satisfies arccos(−x) = π − arccos x, a reflection identity derivable by noting cos(π − θ) = −cos θ.
Cross-function Identities via Right Triangles
Given arcsin x = θ, we can read off a right triangle with opposite side x and hypotenuse 1, so adjacent side = √(1 − x²). This immediately gives: cos(arcsin x) = √(1 − x²); tan(arcsin x) = x/√(1 − x²). Similarly, given arctan x = φ, the right triangle has opposite x and adjacent 1, so hypotenuse = √(1 + x²). This gives: sin(arctan x) = x/√(1 + x²); cos(arctan x) = 1/√(1 + x²). These identities are used constantly in integration (trig substitution) and calculus.
Addition Formulas for arctan
arctan x + arctan y = arctan((x + y)/(1 − xy)) + C, where C = 0 if xy < 1, C = π if xy > 1 and x > 0, and C = −π if xy > 1 and x < 0. A famous special case: arctan(1/2) + arctan(1/3) = arctan(1) = π/4, since (1/2 + 1/3)/(1 − 1/6) = (5/6)/(5/6) = 1. This identity was used historically to compute digits of π.
The Key Equation
- \arcsin x — arcsine of x (radians): The angle in [−π/2, π/2] whose sine equals x
- \arccos x — arccosine of x (radians): The angle in [0, π] whose cosine equals x
- \pi/2 — ninety degrees in radians (radians): A right angle; the two complementary angles in a right triangle always sum to this value
Worked Example
Problem. Simplify the expression sin(arctan(3/4)) without a calculator, and verify that arcsin(−1/2) = −arcsin(1/2).
Answer. sin(arctan(3/4)) = 3/5; arcsin(−1/2) = −π/6 = −arcsin(1/2)
Origins · 1 minLeonhard Euler and how this was discovered
Euler was the first mathematician to systematically treat inverse trigonometric functions as functions in their own right, introducing the notation and studying their series expansions. His 1748 work Introductio in analysin infinitorum established much of modern notation for trigonometric functions. He also discovered the arctan addition formula and used arctan identities to compute approximations of π with extraordinary efficiency. Euler lost vision in his right eye in 1738 and became almost completely blind by 1766, yet he continued to produce mathematics at the same prolific rate — reportedly dictating entire papers to a scribe from memory.
Related Concepts
- Definitions of Inverse Trigonometric Functions — The identities in this article follow directly from the restricted-domain definitions of arcsin, arccos, and arctan.
- Right Triangle Trigonometry — The right-triangle method is the primary tool for deriving cross-function identities such as sin(arctan x).
- The Unit Circle — Unit-circle reasoning underpins the symmetry properties arcsin(−x) = −arcsin x and arccos(−x) = π − arccos x.
- Derivatives of Inverse Trig Functions — The cross-function identities here (e.g., cos(arcsin x) = √(1−x²)) appear directly when differentiating arcsin and arccos using the chain rule.