Inverse trigonometric functions

Inverse trigonometric functions are the inverse relations of the fundamental trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—defined by restricting the domains of these periodic functions to principal intervals that make them bijective, thereby ensuring a unique output angle for each input value within the specified domain.^[1]^[2] These functions, commonly denoted as arcsin(x), arccos(x), arctan(x), and their counterparts for the reciprocal trigonometric functions, allow the determination of angles θ such that sin(θ) = x, cos(θ) = x, or tan(θ) = x, respectively, and are essential in contexts like solving right triangles, integration in calculus, and signal processing.^[3]^[4] The notation for inverse trigonometric functions was standardized in modern form by John Herschel in 1813, using superscripts like sin⁻¹(x) to indicate the inverse (not exponentiation), though the "arc" prefix—such as arcsin(x)—is preferred in international standards like ISO 80000-2 for clarity.^[4] For the principal branches, the domains and ranges are precisely defined to yield values in radians: arcsin(x) has domain [-1, 1] and range [-π/2, π/2]; arccos(x) has domain [-1, 1] and range [0, π]; arctan(x) has domain (-∞, ∞) and range (-π/2, π/2); while arccsc(x), arcsec(x), and arccot(x) follow analogous restrictions, such as arccsc(x) with domain (-∞, -1] ∪ [1, ∞) and range [-π/2, 0) ∪ (0, π/2].^[1]^[2]^[4] Key properties include identities like sin(arcsin(x)) = x for x in [-1, 1], and relationships between functions, such as arccos(x) = π/2 - arcsin(x), which stem from the complementary nature of sine and cosine.^[3]^[4] In calculus, their derivatives are well-defined, for instance, the derivative of arcsin(x) is 1/√(1 - x²), enabling antiderivatives of expressions like 1/√(a² - x²) and applications in physics for modeling oscillatory motion.^[4] These functions extend to complex numbers but are primarily used in real analysis for their monotonicity and continuity within principal ranges.^[1]

Notation and Definitions

Standard Notation

Inverse trigonometric functions are the inverse relations of the six basic trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—serving primarily to recover angles from given ratios of sides in right triangles or from trigonometric values in broader contexts. The most common notations employ either an "arc" prefix or a superscript exponent: \arcsin x or \sin^{-1} x for the inverse sine, \arccos x or \cos^{-1} x for the inverse cosine, \arctan x or \tan^{-1} x for the inverse tangent, \arccot x or \cot^{-1} x for the inverse cotangent, \arcsec x or \sec^{-1} x for the inverse secant, and \arccsc x or \csc^{-1} x for the inverse cosecant.^[5] The arc- prefix notation originated in the 18th century with contributions from mathematicians like Euler, who used "A sin" in 1737, but the superscript notation was introduced by John Herschel in 1813; the former gained broader adoption in European texts by the early 20th century to mitigate confusion between inverse functions and trigonometric reciprocals (such as \csc x = 1/\sin x).^[6]^[6] In older mathematical literature and certain specialized fields like computer programming, abbreviated forms such as \asin x, \acos x, and \atan x are used for the principal inverse functions.^[7]

Principal Branches

Trigonometric functions like sine, cosine, and tangent are periodic with period $2\pi and possess symmetries, such as \sin(\theta) = \sin(\pi - \theta), rendering them non-injective over the real numbers and thus multi-valued when inverted.^[8] To define single-valued inverse functions, principal branches are established by restricting the output to specific intervals where the original functions are bijective on their domains.^[8] These principal ranges are selected to ensure the inverse functions are continuous, to maintain symmetries (e.g., odd functions for arcsin and arctan), and to correspond to angles in the first and fourth quadrants for alignment with right-triangle conventions where angles are non-negative and acute or obtuse as needed.^[9] The standard principal branches, as defined in authoritative mathematical references, are as follows:

Function	Principal Range
\arcsin x	[- \pi/2, \pi/2]
\arccos x	[0, \pi]
\arctan x	(-\pi/2, \pi/2)
\arccot x	(0, \pi)
\arcsec x	[0, \pi] \setminus \{\pi/2\}
\arccsc x	[- \pi/2, 0) \cup (0, \pi/2]

^[9] For the less frequently used functions \arccot x, \arcsec x, and \arccsc x, conventions vary across textbooks and computational systems; for instance, some sources adopt (-\pi/2, \pi/2] for \arccot x to align with \arctan(1/x), or [0, \pi/2) \cup (\pi, 3\pi/2) for \arcsec x to simplify differentiation, though the ranges listed above represent the most common choices for continuity on the real line.^[10]^[11]

Fundamental Properties

Domains and Ranges

The domains of inverse trigonometric functions are determined by the ranges of their corresponding trigonometric functions, ensuring that the inverses are well-defined and single-valued over the real numbers. Specifically, the arcsine function, \arcsin(x), and arccosine function, \arccos(x), are defined for inputs x in the closed interval [-1, 1], as these are the possible output values of the sine and cosine functions over their full real domains.^[12] In contrast, the arctangent function, \arctan(x), and arccotangent function, \arccot(x), accept all real numbers as inputs (x \in \mathbb{R}), reflecting the unbounded range of the tangent and cotangent functions within their principal periods. The arcsecant function, \arcsec(x), and arccosecant function, \arccsc(x), are defined for |x| \geq 1, corresponding to the ranges of secant and cosecant outside the open interval (-1, 1).^[13] These domain restrictions arise from the periodic and non-injective nature of the trigonometric functions, which would otherwise yield multi-valued inverses without careful limitation to ensure real, unique outputs.^[12] The ranges of these inverse functions, known as principal branches, are selected intervals where the original trigonometric functions are bijective, guaranteeing monotonicity and continuity within the defined domains. For instance, \arcsin(x) outputs values in [- \pi/2, \pi/2], increasing monotonically from -\pi/2 at x = -1 to \pi/2 at x = 1. Similarly, \arccos(x) ranges over [0, \pi], decreasing from \pi to $0 as x goes from -1 to $1. The \arctan(x) function produces outputs in (-\pi/2, \pi/2), approaching the asymptotes \pm \pi/2 as x tends to \pm \infty, while maintaining strict increase across all real inputs. The \arccot(x) range is (0, \pi), decreasing from \pi to $0 over \mathbb{R}. For \arcsec(x), the principal range is [0, \pi/2) \cup (\pi/2, \pi], and for \arccsc(x), it is [- \pi/2, 0) \cup (0, \pi/2], both exhibiting discontinuities at points where the original functions are undefined, but monotonic in their respective subintervals.^[13]^[12] Graphically, these domains and ranges illustrate the inverses as reflections of the restricted trigonometric graphs over the line y = x, emphasizing how input limitations prevent outputs outside the real numbers and enforce one-to-one mappings. The principal branches ensure the functions are strictly monotonic—either increasing or decreasing—within their domains, with horizontal asymptotes for \arctan(x) and \arccot(x) highlighting their behavior at infinity, unlike the bounded ranges of the other inverses.^[12] The following table summarizes the domains and principal ranges for the six inverse trigonometric functions, underscoring that \arctan(x) and \arccot(x) are uniquely defined over all reals, in contrast to the bounded or exterior domains of the others:

Function	Domain	Principal Range
\arcsin(x)	[-1, 1]	[- \pi/2, \pi/2]
\arccos(x)	[-1, 1]	[0, \pi]
\arctan(x)	(-\infty, \infty)	(-\pi/2, \pi/2)
\arccot(x)	(-\infty, \infty)	(0, \pi)
\arcsec(x)	(-\infty, -1] \cup [1, \infty)	[0, \pi/2) \cup (\pi/2, \pi]
\arccsc(x)	(-\infty, -1] \cup [1, \infty)	[- \pi/2, 0) \cup (0, \pi/2]

^[13]^[12]

Relationships with Trigonometric Functions

The compositional relationships between inverse trigonometric functions and their trigonometric counterparts form the foundation for simplifying expressions and solving equations. For the inverse sine function, defined with principal range [- \pi/2, \pi/2], the identity \sin(\arcsin x) = x holds for all x in the domain [-1, 1].^[14] Similarly, \arcsin(\sin \theta) = \theta when \theta lies within the principal range [- \pi/2, \pi/2].^[14] Analogous identities apply to the other primary inverse functions: \cos(\arccos x) = x for x \in [-1, 1], with principal range [0, \pi]; and \tan(\arctan x) = x for all real x, with principal range (-\pi/2, \pi/2).^[14] These relations reflect the one-to-one correspondence established by restricting the domains of the trigonometric functions to ensure invertibility.^[15] Outside the principal ranges, these compositions do not generally yield the original angle due to the periodic and non-injective nature of trigonometric functions. For instance, \arcsin(\sin \theta) \neq \theta if \theta \notin [-\pi/2, \pi/2]; instead, the result is the angle within the principal range that has the same sine value.^[14] To adjust for this, one common expression maps \theta back to the principal branch: for \theta in intervals determined by integer k, \arcsin(\sin \theta) = (-1)^k (\theta - k \pi), where k is chosen such that the result falls in [- \pi/2, \pi/2].^[16] This adjustment accounts for the symmetry and periodicity of the sine function, producing a sawtooth-like graph for \arcsin(\sin \theta). Similar restrictions apply to \arccos(\cos \theta) = \theta only for \theta \in [0, \pi], and \arctan(\tan \theta) = \theta only for \theta \in (-\pi/2, \pi/2).^[15] Reciprocal relationships link certain inverse functions, facilitating simplifications. Notably, \arctan(1/x) = \arccot x for x > 0, reflecting the complementary nature of tangent and cotangent.^[14] For x < 0, a sign adjustment is required: \arctan(1/x) = \arccot x - \pi (assuming the principal range for arccot is (0, \pi)), to align with the respective principal branches.^[15] These identities stem from the definition \cot \phi = 1/\tan \phi and the need to preserve the output within defined ranges. In solving trigonometric equations, these relationships underpin the general solutions, often involving a \pm term to capture multiple branches. For \sin \theta = x where x \in [-1, 1], the solutions are \theta = \arcsin x + 2k\pi or \theta = \pi - \arcsin x + 2k\pi, with k \in \mathbb{Z}.^[14] The second form arises from the symmetry \sin(\pi - \phi) = \sin \phi, ensuring all solutions are accounted for across periods of $2\pi. This \pm structure (equivalently expressed via the co-function) highlights the multi-valued inverse nature outside principal branches.^[15]

Identities Among Inverse Functions

One of the fundamental identities among inverse trigonometric functions is the complementary angle relation between arcsine and arccosine. For all x \in [-1, 1], it holds that

\arcsin x + \arccos x = \frac{\pi}{2}.

This identity arises from the principal ranges of the functions: the range of \arcsin x is [- \pi/2, \pi/2], while that of \arccos x is [0, \pi]. To derive it, let \theta = \arcsin x, so \sin \theta = x and \theta \in [- \pi/2, \pi/2]. Then \arccos x = \arccos(\sin \theta). Since \cos(\pi/2 - \theta) = \sin \theta = x and \pi/2 - \theta \in [0, \pi], it follows that \arccos x = \pi/2 - \theta = \pi/2 - \arcsin x.^[17] Similar complementary identities exist for the other pairs of inverse trigonometric functions, reflecting their reciprocal relationships and principal branches. For all x \in \mathbb{R},

\arctan x + \arccot x = \frac{\pi}{2},

where the range of \arctan x is (- \pi/2, \pi/2) and that of \arccot x is (0, \pi). The derivation follows analogously: let \phi = \arctan x, so \tan \phi = x and \phi \in (- \pi/2, \pi/2). Then \arccot x = \arccot(\tan \phi) = \pi/2 - \phi, since \cot(\pi/2 - \phi) = \tan \phi = x and \pi/2 - \phi \in (0, \pi). For the remaining pair, the identity is

\arcsec x + \arccsc x = \frac{\pi}{2}

for all |x| \geq 1, with the principal range of \arcsec x being [0, \pi] \setminus \{\pi/2\} and that of \arccsc x being [- \pi/2, 0) \cup (0, \pi/2]. Let \psi = \arcsec x, so \sec \psi = x and \psi \in [0, \pi] \setminus \{\pi/2\}. Then \arccsc x = \arccsc(\sec \psi) = \pi/2 - \psi, because \csc(\pi/2 - \psi) = \sec \psi = x and \pi/2 - \psi lies in the appropriate range for \arccsc. These identities facilitate simplification by expressing one inverse function in terms of its complement.^[18]

Solving Trigonometric Equations

Basic Equations and Solutions

Inverse trigonometric functions are essential for solving basic trigonometric equations of the form \sin \theta = x, \cos \theta = x, and \tan \theta = x, where x lies within the appropriate domain for each function. These equations typically have infinitely many solutions due to the periodic nature of the trigonometric functions, and the inverse functions provide the principal values from which the general solutions are derived by adding multiples of the period. For the equation \sin \theta = x where -1 \leq x \leq 1, the general solution is given by

\theta = \arcsin x + 2k\pi \quad \text{or} \quad \theta = \pi - \arcsin x + 2k\pi, \quad k \in \mathbb{Z}.

This accounts for the two solutions within one period of $2\pi, one in the first or second quadrant, and then extends periodically.^[19] Similarly, for \cos \theta = x where -1 \leq x \leq 1, the general solution is

\theta = \pm \arccos x + 2k\pi, \quad k \in \mathbb{Z}.

The positive and negative signs capture the symmetric solutions about the x-axis within the period $2\pi. Equivalently, this can be expressed as \theta = \arccos x + 2k\pi or \theta = -\arccos x + 2k\pi.^[20] For \tan \theta = x where x \in \mathbb{R}, the general solution simplifies to

\theta = \arctan x + k\pi, \quad k \in \mathbb{Z},

reflecting the period of \pi for the tangent function, with a single solution per period.^[20] To illustrate the process, consider solving \tan \theta = 1. First, apply the inverse tangent: the principal value is \theta = \arctan 1 = \pi/4, since \tan(\pi/4) = 1 and \pi/4 lies in the principal range (-\pi/2, \pi/2). Given the periodicity of tangent with period \pi, add multiples of \pi to obtain the general solution \theta = \pi/4 + k\pi, k \in \mathbb{Z}. For example, when k=0, \theta = \pi/4; when k=1, \theta = 5\pi/4; and so on, verifying that \tan(5\pi/4) = 1. This step-by-step approach—finding the principal angle and adding the period—applies broadly to these equations.^[21] These solutions arise from fundamental relationships between trigonometric functions, such as \sin \theta = \cos(\pi/2 - \theta). Setting \sin \theta = x implies \cos(\pi/2 - \theta) = x, so \pi/2 - \theta = \arccos x (principal value), yielding \theta = \pi/2 - \arccos x. But \theta = \arcsin x, thus \arcsin x = \pi/2 - \arccos x for x \in [-1, 1]. This identity connects the solutions of sine and cosine equations directly.^[22]

Multi-Valued Nature

The trigonometric functions are periodic, which implies that their inverses are inherently multi-valued. For instance, the equation \sin \theta = x with x \in [-1, 1] has infinitely many solutions \theta, reflecting the $2\pi-periodicity of sine combined with its symmetry about multiples of \pi. The complete set of solutions is given by \theta = (-1)^k \arcsin x + k\pi for integers k.^[9] Similarly, for \cos \theta = x with x \in [-1, 1], the solutions are \theta = \pm \arccos x + 2k\pi.^[23] To make these multi-valued functions single-valued, principal branches are defined by restricting the range of \theta (e.g., [-\pi/2, \pi/2] for \arcsin x), selecting one representative value from each set of solutions. However, the full inverse relation encompasses all branches, arising from the periodic replicas. In the complex plane, the multi-valued nature of inverse trigonometric functions like \Arcsin z is visualized using a Riemann surface, which consists of infinitely many sheets connected along branch cuts at the branch points z = \pm 1. This structure allows analytic continuation across branches without discontinuities, providing a geometric representation of the function's periodicity and symmetry.^[24] For example, solving \cos \theta = 1/2 yields the principal value \arccos(1/2) = \pi/3, but the complete solutions are \theta = \pm \pi/3 + 2k\pi for integers k, capturing both the even symmetry of cosine and its $2\pi-periodicity.^[23] A common pitfall is assuming \arcsin(\sin \theta) = \theta for arbitrary \theta; this equality holds only when \theta lies in the principal interval [-\pi/2, \pi/2]. Outside this range, the result is the equivalent angle within the interval, as in \arcsin(\sin(2\pi/3)) = \pi/3.

Algebraic Manipulations

Expression Transformations

Expression transformations involve rewriting expressions containing inverse trigonometric functions into equivalent forms, often to eliminate the inverse or convert between different inverse functions. One common transformation equates the inverse sine to the inverse tangent: for |x| < 1,

\arcsin x = \arctan \left( \frac{x}{\sqrt{1 - x^2}} \right).

This identity arises from considering a right triangle where the opposite side to the angle \theta = \arcsin x is x and the hypotenuse is 1, so the adjacent side is \sqrt{1 - x^2}; thus, \tan \theta = x / \sqrt{1 - x^2}, yielding \theta = \arctan(x / \sqrt{1 - x^2}).^[16] Similar transformations express the inverse cosine in terms of the inverse sine using half-angle formulas. For -1 \leq x \leq 1,

\arccos x = 2 \arcsin \left( \sqrt{\frac{1 - x}{2}} \right).

To derive this, let \phi = \arccos x, so \cos \phi = x and \sin(\phi/2) = \sqrt{(1 - \cos \phi)/2} = \sqrt{(1 - x)/2} (taking the positive root since $0 \leq \phi/2 \leq \pi/2); therefore, \phi/2 = \arcsin \sqrt{(1 - x)/2}, or \phi = 2 \arcsin \sqrt{(1 - x)/2}. This form is useful for relating inverse functions to half-angle identities in trigonometric simplifications. A practical example of transforming a composition is simplifying \sin(2 \arcsin x) for |x| \leq 1. Let \theta = \arcsin x, so \sin \theta = x and \cos \theta = \sqrt{1 - x^2} (positive since -\pi/2 \leq \theta \leq \pi/2). Apply the double-angle formula:

\sin(2\theta) = 2 \sin \theta \cos \theta = 2 x \sqrt{1 - x^2}.

Thus, \sin(2 \arcsin x) = 2 x \sqrt{1 - x^2}. This step-by-step process demonstrates how substituting the inverse definition and applying standard trigonometric identities removes the inverse function. In general, to eliminate inverse trigonometric functions from equations or expressions, first isolate the inverse (if needed), then apply the corresponding trigonometric function to both sides, and use identities like the Pythagorean theorem or angle formulas to simplify the resulting trigonometric expression. For instance, in solving equations like \arcsin x = \alpha, take \sin of both sides to get x = \sin \alpha, but verify domain restrictions to avoid extraneous solutions. This strategy relies on the fundamental relationships between inverse and direct trigonometric functions, often visualized via right triangles for conceptual clarity. These core transformation techniques are primary for algebraic manipulations.

Calculus of Inverse Functions

Derivatives

The derivatives of inverse trigonometric functions are fundamental in calculus, providing expressions for the rates of change of these functions with respect to their arguments. These derivatives are typically derived using implicit differentiation and are defined on the principal domains of the functions, excluding points where the expressions are undefined.^[25] The standard derivative formulas are as follows:

\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 - x^2}}, \quad \frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1 - x^2}},

\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2}, \quad \frac{d}{dx} \arccot(x) = -\frac{1}{1 + x^2},

\frac{d}{dx} \arcsec(x) = \frac{1}{|x| \sqrt{x^2 - 1}}, \quad \frac{d}{dx} \arccsc(x) = -\frac{1}{|x| \sqrt{x^2 - 1}}.

These formulas hold for x in the respective domains where the denominators are positive and the functions are differentiable.^[25]^[26] To derive these, implicit differentiation is employed by expressing the inverse function in terms of its trigonometric counterpart. For example, let y = \arcsin(x), so x = \sin(y). Differentiating both sides with respect to x gives $1 = \cos(y) \frac{dy}{dx}, hence \frac{dy}{dx} = \frac{1}{\cos(y)}. Since \cos(y) = \sqrt{1 - \sin^2(y)} = \sqrt{1 - x^2} (taking the positive root due to the range of \arcsin), the derivative is \frac{1}{\sqrt{1 - x^2}}. Similar implicit differentiation applies to the other functions, adjusting for their ranges and the signs of the trigonometric derivatives.^[25]^[27]^[26] For composite functions, the chain rule extends these derivatives. Specifically, \frac{d}{dx} \arcsin(u(x)) = \frac{1}{\sqrt{1 - u(x)^2}} \cdot u'(x), where u(x) is differentiable and lies within the domain of \arcsin. Analogous forms hold for the other inverse functions by replacing the specific derivative with the appropriate formula.^[25]^[28] Geometrically, these derivatives interpret the rate of change of the angle defined by the inverse function. For \arctan(x), which gives the angle \theta such that \tan(\theta) = x in a right triangle with opposite side x and adjacent side 1, the derivative \frac{1}{1 + x^2} equals \cos^2(\theta), representing the squared cosine of the angle and thus the rate at which \theta increases as x varies along the x-axis in the unit circle context. This measures how quickly the angle adjusts with input changes, decreasing as |x| grows since angles approach \pm \pi/2 asymptotically.^[25]^[29]

Integrals

Inverse trigonometric functions frequently appear as antiderivatives in standard integrals of rational functions involving square roots. One fundamental example is the indefinite integral \int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C, valid for |x| < 1.^[30] Similarly, \int \frac{dx}{1 + x^2} = \arctan x + C, which holds for all real x.^[30] For expressions involving hyperbolic inverses, which are closely related to trigonometric ones via analytic continuation, \int \frac{dx}{\sqrt{x^2 - 1}} = \arccosh x + C = \ln \left| x + \sqrt{x^2 - 1} \right| + C, applicable for x > 1.^[30] These functions also admit definite integral representations that reflect their geometric interpretations as accumulated angles. Specifically, \arcsin x = \int_0^x \frac{dt}{\sqrt{1 - t^2}}, for |x| \leq 1, derived from the fundamental theorem of calculus applied to the derivative of \arcsin x.^[31] Likewise, \arctan x = \int_0^x \frac{dt}{1 + t^2}, for all real x, again following directly from differentiation under the integral sign.^[32] Integrating the inverse functions themselves often requires integration by parts. For instance, to evaluate \int \arcsin x \, dx, set u = \arcsin x so du = \frac{dx}{\sqrt{1 - x^2}} and dv = dx so v = x. Then, \int \arcsin x \, dx = x \arcsin x - \int \frac{x \, dx}{\sqrt{1 - x^2}} = x \arcsin x + \sqrt{1 - x^2} + C, where the remaining integral is resolved via the substitution w = 1 - x^2.^[33] In practice, inverse trigonometric functions emerge naturally through trigonometric substitutions in integrals of rational functions with quadratic denominators under square roots. For example, to integrate \int \frac{dx}{\sqrt{a^2 - x^2}} for a > 0, substitute x = a \sin \theta with \theta = \arcsin(x/a), yielding \arcsin(x/a) + C after simplification.^[34] This technique simplifies forms like \sqrt{x^2 + a^2} using x = a \tan \theta, leading to arctangent results, and is essential for handling non-elementary integrals in applied contexts./07%3A_Techniques_of_Integration/7.03%3A_Trigonometric_Substitution)

Series Representations

Inverse trigonometric functions admit power series expansions around zero, which are useful for computational approximations within their domains of convergence. These series are typically derived by integrating the geometric series expansions of their derivatives, such as \frac{d}{dx} \arctan x = \frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^n x^{2n} for |x| < 1. The Taylor series for \arctan x is given by

\arctan x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1}, \quad |x| \leq 1.

This series converges for |x| < 1 and also at the endpoints x = \pm 1, where it evaluates to \pm \pi/4, as established by Abel's theorem on the continuity of power series on the boundary of the disk of convergence. The radius of convergence is 1, and the error after N terms is bounded by the remainder term R_N(x) = \frac{(-1)^N x^{2N+3}}{(2N+3)(1 + \xi^{2N+2})} for some \xi between 0 and x, with |R_N(x)| < \frac{|x|^{2N+3}}{(2N+3)(1 - |x|^2)} for |x| < 1. For \arcsin x, the Taylor series expansion is

\arcsin x = \sum_{n=0}^\infty \frac{\binom{2n}{n}}{4^n (2n+1)} x^{2n+1}, \quad |x| < 1,

where \binom{2n}{n} = \frac{(2n)!}{(n!)^2}. This form arises from the binomial expansion of (1 - t^2)^{-1/2} integrated term by term. The radius of convergence is 1, with divergence at the endpoints x = \pm 1 due to the growth of the coefficients, though the partial sums approach \pm \pi/2 asymptotically. Error estimates follow the Lagrange remainder, with |R_N(x)| \leq \frac{|x|^{2N+3}}{(2N+3) (1 - x^2)^{(2N+3)/2}} for |x| < 1. The series for \arccos x is obtained via the identity \arccos x = \frac{\pi}{2} - \arcsin x, yielding

\arccos x = \frac{\pi}{2} - \sum_{n=0}^\infty \frac{\binom{2n}{n}}{4^n (2n+1)} x^{2n+1}, \quad |x| < 1.

This shares the same radius of convergence as the arcsin series. An alternative representation for \arctan x is the continued fraction

\arctan x = \cfrac{x}{1 + \cfrac{x^2}{3 + \cfrac{4x^2}{5 + \cfrac{9x^2}{7 + \cfrac{16x^2}{9 + \ddots}}}}}

with partial quotients a_0 = 0, b_n = 1, a_n = (n x)^2 for n \geq 1. This fraction converges for all real x, with rapid convergence for |x| < 1 where the terms decrease monotonically, and slower but still convergent behavior for |x| > 1. Truncation after k levels provides approximations with error less than the next term, often superior to the power series for larger |x| due to better asymptotic properties.

Complex Extensions

Logarithmic Forms

In the complex plane, inverse trigonometric functions can be expressed using the complex logarithm, providing an exact representation that accounts for their multi-valued nature. These logarithmic forms arise from solving the defining equations of the trigonometric functions, such as \sin w = z or \tan w = z, through exponential substitutions and quadratic resolutions.^[9] The principal branch of the inverse sine is given by

\operatorname{arcsin} z = -i \ln \left( i z + \sqrt{1 - z^2} \right),

where the square root is the principal branch with branch cut along the negative real axis, and the logarithm is the principal branch with argument in (-\pi, \pi].^[9] Similarly, the principal branch of the inverse tangent is

\operatorname{arctan} z = \frac{i}{2} \ln \left( \frac{1 - i z}{1 + i z} \right),

with the logarithm using the principal branch and appropriate branch cuts along the imaginary axis for | \Im z | > 1.^[9] Analogous expressions exist for the other inverse trigonometric functions, such as arccos, arccsc, arccot, and arcsec, derived from compositions or identities with arcsin and arctan.^[35] The choice of branches for the square root and logarithm is critical to ensure continuity and define the principal values. For arcsin, the square root \sqrt{1 - z^2} selects the branch where the real part is non-negative on the principal sheet, avoiding discontinuities except along the cuts (-\infty, -1] \cup [1, \infty). The principal logarithm then yields imaginary parts between -\pi/2 and \pi/2 for real arguments within the domain.^[9] These branch specifications ensure the functions are analytic in the complex plane minus the branch cuts.^[35] When z is real and |z| \leq 1, these complex logarithmic forms reduce to the standard real principal values. For instance, \operatorname{arcsin} x for x \in [-1, 1] produces a real output in [-\pi/2, \pi/2], matching the conventional definition, as the argument of the logarithm becomes purely imaginary and the expression simplifies accordingly.^[9] The same holds for \operatorname{arctan} x yielding real values in (-\pi/2, \pi/2).^[35] As an example, consider \operatorname{arcsin}(i). Substituting into the formula gives \operatorname{arcsin}(i) = -i \ln(\sqrt{2} - 1), which simplifies to i \operatorname{arcsinh}(1) since \operatorname{arcsinh}(1) = \ln(1 + \sqrt{2}) and \ln(\sqrt{2} - 1) = -\ln(1 + \sqrt{2}). Numerically, this evaluates to approximately i \times 0.8814.^[9]^[35]

Generalizations and Proofs

In the complex plane, inverse trigonometric functions such as arcsin(z) and arctan(z) are multi-valued due to the periodic nature of the sine and tangent functions and the multi-valued complex logarithm involved in their definitions.^[9] The general solution for w such that sin(w) = z includes branches that differ by multiples of 2π in the real part, reflecting the 2π-periodicity of sine, while the logarithmic representation introduces additional branching from the 2πi periods of the complex logarithm.^[35] Similarly, for arctan(z), the branches arise from the π-periodicity of tangent and the logarithmic multi-valuedness.^[9] To derive the logarithmic form of arcsin(z), begin with sin(w) = z, where w is complex. Express sine in exponential form:

\sin w = \frac{e^{iw} - e^{-iw}}{2i} = z.

Let v = e^{iw}, yielding the quadratic equation v² - 2i z v - 1 = 0. The solutions are v = i z ± √(1 - z²), and taking the logarithm gives

w = \frac{1}{i} \ln \left( i z + \sqrt{1 - z^2} \right),

where the square root and logarithm are multi-valued, leading to the principal branch Arcsin(z) = -i Ln( i z + √(1 - z²) ) using the principal logarithm Ln.^[35] For arctan(z), start from tan(w) = z, or equivalently sin(w)/cos(w) = z. Using the exponential form, this simplifies to

\tan w = \frac{1}{i} \frac{e^{iw} - e^{-iw}}{e^{iw} + e^{-iw}} = z,

leading to e^{2iw} = (i - z)/(i + z) after algebraic manipulation. Taking the logarithm yields

w = \frac{1}{2i} \ln \left( \frac{i - z}{i + z} \right),

with multi-valuedness from the logarithm, and the principal branch Arctan(z) = (i/2) Ln( (i + z)/(i - z) ).^[35] Branch cuts are necessary to define single-valued principal branches. For arcsin(z) and arccos(z), standard cuts are placed along the real axis from -∞ to -1 and from 1 to ∞, avoiding the branch points at z = ±1.^[9] For arctan(z), cuts run from -i∞ to -i and from i to i∞ along the imaginary axis.^[9] A key relation connects inverse trigonometric and hyperbolic functions in the complex domain: arcsin(i y) = i arcsinh(y) for real y, arising from the identity sinh(w) = -i sin(i w) and consistent branching.^[36]

Applications

Geometry and Triangles

Inverse trigonometric functions are fundamental in geometry for determining unknown angles in right triangles when side lengths are known. In a right triangle, if the ratio of the length of the side opposite an acute angle θ to the hypotenuse is x (where 0 ≤ x ≤ 1), then θ = arcsin(x).^[37] This application leverages the definitions of sine, cosine, and tangent, allowing the inverse functions to recover the angle from these ratios. Similarly, for the adjacent side over hypotenuse y, θ = arccos(y), and for opposite over adjacent z, θ = arctan(z).^[37] The principal values of these functions, restricted to appropriate ranges, ensure the angles obtained are acute, aligning with the geometry of right triangles.^[37] A classic example is the 3-4-5 right triangle, where the sides are 3, 4, and 5 units, satisfying the Pythagorean theorem. The acute angle θ opposite the side of length 3 has tangent θ = 3/4, so θ = arctan(3/4) ≈ 0.6435 radians or 36.87°. For non-right triangles, inverse trigonometric functions extend through the law of sines, which states that in any triangle with sides a, b, c opposite angles A, B, C respectively, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.^[38] Thus, the principal value of the angle A opposite side a can be found as A = arcsin(a / (2R)), though the actual angle may be π - arcsin(a / (2R)) if obtuse, requiring additional context such as another side or angle to resolve the ambiguity.^[38]^[39] This relation allows determination of possible angles when the circumradius and a side length are known, bridging right and oblique triangle geometry. Historically, inverse trigonometric functions played a vital role in surveying and navigation before the advent of electronic calculators, where trigonometric tables were used to compute angles from side ratios for mapping land and plotting sea routes.^[40] Advances in plane and spherical trigonometry, including inverse computations via logarithms and tables from the 16th century onward, enabled accurate distance measurements and celestial positioning essential for exploration and cartography.^[41]

Computing and Engineering

In computing and engineering, the two-argument arctangent function, commonly denoted as \operatorname{atan2}(y, x), computes the angle \theta between the positive x-axis and the point (x, y) in the Cartesian plane, returning values in the interval (-\pi, \pi] to correctly determine the quadrant based on the signs of y and x. This distinguishes it from the single-argument \operatorname{atan}(y/x), which cannot resolve quadrant ambiguity and is limited to (-\pi/2, \pi/2).^[42] The function handles the edge case where x = 0 by returning \pi/2 if y > 0, -\pi/2 if y < 0, and undefined (or 0 in some implementations) if both are zero, ensuring robust angle computation without division by zero errors. Generalizations of \operatorname{atan2} extend its utility for non-origin-centered coordinates, such as \operatorname{atan2}(y - y_0, x - x_0) to find the angle of a point (x, y) relative to a center (x_0, y_0).^[43] This form is prevalent in applications requiring offset calculations, like orienting objects around arbitrary pivots. Numerical implementations of inverse trigonometric functions must address floating-point precision challenges, particularly for \operatorname{asin}(x) and \operatorname{acos}(x) near x = \pm 1, where small input perturbations can lead to significant output errors due to the functions' derivatives approaching infinity at the boundaries.^[44] For small angles, approximations like \operatorname{atan}(x) \approx x (valid for |x| \ll 1) improve efficiency and accuracy in low-precision regimes by avoiding full series or table lookups.^[45] Modern math libraries, such as those in C99 and beyond, strive for IEEE 754 compliance by providing correctly rounded results within one unit in the last place (ULP) for most inputs, though edge cases may require specialized handling like argument reduction or fused operations.^[46] In signal processing, \operatorname{atan2} computes the phase (argument) of a complex number z = x + iy as \arg(z) = \operatorname{atan2}(y, x), enabling accurate frequency-domain analysis and demodulation in systems like FFT-based spectrum estimation.^[47] In robotics, it resolves joint angles during inverse kinematics for manipulators, such as calculating base rotation from end-effector coordinates to avoid singularity issues in multi-link arms.^[48] Computer graphics leverages \operatorname{atan2} for rotation matrices and orientation, converting vector directions to Euler angles for rendering scenes or animating transformations around arbitrary centers.^[49]