Inverse trigonometric functions on arvau

Wikipedia / Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. 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 for the inverse trigonometric functions. The most common uses the prefix : , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of radians will correspond to an arc whose length is , where is the radius of the circle. Thus in the unit circle, the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is " is the same as "the angle whose cosine is ", 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 often called by the abbreviated forms , , .

Another common notation in English sources uses a superscript : , , , etc., as introduced by John Herschel in 1813. This conflicts with the common syntax , which denote exponentiation rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function. The inverse function convention using a superscript in brackets is unambiguous but not commonly used: , , .

The ambiguity is mitigated by the use of special names for reciprocal trigonometric functions — for example, . Nevertheless, certain authors advise against using it, since it is ambiguous. Another problematic convention used by a small number of authors is an uppercase first letter, along with a “” superscript: , , , etc. Although it is intended to avoid confusion with the reciprocal, which should be represented by , , etc., or, better, by , , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA) use capitalised names for the standard trig functions.

Since 2019, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.

Basic concepts

Principal values

Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore, the result rangess of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.

For example, using in the sense of multivalued functions, just as the square root For a given real number there are multiple (in fact, countably infinitely etc. When only one value is desired, the function may be restricted to its principal branch 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.Note: Some authors define the range of arcsecant to be or whereas with the range or

Domains

If is allowed to be a complex number, then the range of applies only to its real part.

Solutions to elementary trigonometric equations

is some integer.

all lie within appropriate ranges so that the relevant expressions below are well-defined" is just another way of saying "for some integer"

is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote for more details and an example illustrating this concept).where the first four solutions can be written in expanded form as: respectively. In all equations above ).

With the help of integer parity symbol:

And similarly for the secant function,

when it's odd.

Detailed example and explanation of the "plus or minus" symbol ±

which is a shorthand way of saying that (at least) one of the following statement is true:

statements (1) and (2) are different and furthermore, exactly one and that ).

Equal identical trigonometric functions

Set of all solutions to elementary trigonometric equations

is always a solution), the set of all solutions to it are:

Transforming equations

The equations above can be transformed by using the reflection and shift identities:These formulas imply, in particular, that the following hold:

respectively.

Relationships between trigonometric functions and inverse trigonometric functions

then applying the Pythagorean theorem is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.

Relationships among the inverse trigonometric functions

Complementary angles:

Negative arguments:

Reciprocal arguments:

are reciprocals

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 (or positive imaginary part if the square was negative real).

A useful form that follows directly from the table above is

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:

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

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as a binomial series in a geometric series, and applying the integral definition above (see Leibniz series).

The Taylor series for arctangent is also known as the arctangent series

, and so on. Another series is given by:

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

Another series for the arctangent function is given by

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 (after the first) are just (nz)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 derivativess 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 functionss:

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

(i.e. integration by parts), set

Then

which by the simple substitution yields the final result:

Extension to the 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 (−i and +i) 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 for the upper branch cut and from for the lower branch cut.

The arcsine function may then be defined as:

where (the square-root function has its cut along the negative real axis and) 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. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

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:

is the hypotenuse.

, 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

above and simplifying.

The particular form of the simplified expression can cause the output to differ from the usual principal branch convention.

, the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.

Example proof

Using the exponential definition of sine

(the positive branch is chosen)

Applications

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 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 meters as it runs out 20 meters. 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 given and , but with a range of . In other words, is the angle between the positive -axis of a plane and the point on it, with positive sign for counter-clockwise angles (upper half-plane, ), and negative sign for clockwise angles (lower half-plane, ). 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 argumentument of the complex number .

This limited version of the function above may also be defined using the tangent half-angle formulae provided that either or . However this fails if given and 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.

Arctangent function with location parameter

In many applications. The adequate solution is produced by the parameter modified arctangent function

rounds to the nearest integer.

Numerical accuracy

For angles near 0 and , arccosine is ill-conditioned, and similarly with arcsine for angles near /2 and /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods.