Inverse trigonometric functions


In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Notation

There are several notations used for the inverse trigonometric functions.
The most common convention is to name inverse trigonometric functions using an arc- prefix:,,, etc. This notation arises from the following geometric relationships:
When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus, in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages the inverse trigonometric functions are usually called by the abbreviated forms asin, acos, atan.
The notations,,, etc., as introduced by John Herschel in 1813, are often used as well in English-language sources, and this convention complies with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions like, which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse or reciprocal and compositional inverse. The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name—for example, =. Nevertheless, certain authors advise against using it for its ambiguity. Another convention used by a few authors is to use a majuscule first letter along with a −1 superscript:,,, etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by,, etc.
Since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.

Basic properties

Principal values

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. Therefore, the ranges of the inverse functions are proper subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions, just as the square root function could be defined from, the function is defined so that. For a given real number, with, there are multiple numbers such that ; for example,, but also,, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each in the domain the expression will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
NameUsual notationDefinitionDomain of for real resultRange of usual principal value
Range of usual principal value
arcsiney = x = −1 ≤ x ≤ 1− ≤ y ≤ −90° ≤ y ≤ 90°
arccosiney = x = −1 ≤ x ≤ 10 ≤ y ≤ 0° ≤ y ≤ 180°
arctangenty = x = all real numbers− < y < −90° < y < 90°
arccotangenty = x = all real numbers0 < y < 0° < y < 180°
arcsecanty = x = x ≤ −1 or 1 ≤ x0 ≤ y < or < y ≤ 0° ≤ y < 90° or 90° < y ≤ 180°
arccosecanty = x = x ≤ −1 or 1 ≤ x− ≤ y < 0 or 0 < y ≤ −90° ≤ y < 0° or 0° < y ≤ 90°

, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range,, whereas with the range
If x is allowed to be a complex number, then the range of y applies only to its real part.

Relationships between trigonometric functions and inverse trigonometric functions

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x, then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purely algebraic derivations are longer.
Diagram

Relationships among the inverse trigonometric functions

Complementary angles:
Negative arguments:
Reciprocal arguments:
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part.
A useful form that follows directly from the table above is
It is obtained by recognizing that.
From the half-angle formula,, we get:

Arctangent addition formula

This is derived from the tangent addition formula
by letting

In calculus

Derivatives of inverse trigonometric functions

The derivatives for complex values of z are as follows:
Only for real values of x:
For a sample derivation: if, we get:

Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,, as a binomial series, and integrating term by term. The series for arctangent can similarly be derived by expanding its derivative in a geometric series and applying the integral definition above.
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example,,, and so on. Another series is given by:
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where is the imaginary unit.

Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:
The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators are just 2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

Indefinite integrals of inverse trigonometric functions

For real and complex values of z:
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example

Using , set
Then
which by the simple substitution yields the final result:

Extension to complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points. One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re>0 for the upper branch cut and from Re<0 for the lower branch cut.
The arcsine function may then be defined as:
where the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.

Logarithmic forms

These functions may also be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion.

Generalization

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where is the adjacent side, is the opposite side, and is the hypotenuse. From here, we can solve for.
or
Simply taking the imaginary part works for any real-valued and, but if or is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of also removes from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input, we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for that result from plugging the values into the equations above and simplifying.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since this definition works for any complex-valued, this definition allows for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.

Example proof

Using the exponential definition of sine, one obtains
Let
Solving for

Applications

General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2. Sine and cosecant begin their period at 2k − , finish it at 2k +, and then reverse themselves over 2k + to 2k +. Cosine and secant begin their period at 2k, finish it at 2k +, and then reverse themselves over 2k + to 2k + 2. Tangent begins its period at 2k −, finishes it at 2k +, and then repeats it over 2k + to 2k +. Cotangent begins its period at 2k, finishes it at 2k +, and then repeats it over 2k + to 2k + 2.
This periodicity is reflected in the general inverses where k is some integer:

Application: finding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: where is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:

In computer science and engineering

Two-argument variant of arctangent

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of is the angle between the positive x-axis of a plane and the point on it, with positive sign for counter-clockwise angles, and negative sign for clockwise angles. It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of, it can be expressed as follows:
It also equals the principal value of the argument of the complex number x + iy.
This function may also be defined using the tangent half-angle formulae as follows:
provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention so some caution is warranted. These variations are detailed at atan2.

Arctangent function with location parameter

In many applications the solution of the equation is to come as close as possible to a given value. The adequate solution is produced by the parameter modified arctangent function
The function rounds to the nearest integer.

Numerical accuracy

For angles near 0 and, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation. Similarly, arcsine is inaccurate for angles near −/2 and /2.