Inverse trigonometric functions

In mathematics, the inverse trigonometric 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

Several notations for the inverse trigonometric functions exist. 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 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,,.
The notations,,, etc., as introduced by John Herschel in 1813, are often used as well in English-language sources, much more than the also established,, – conventions consistent with the notation of an inverse function, that is useful to define the multivalued version of each inverse trigonometric function: However, this might appear to conflict logically with the common semantics for expressions such as , which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal and inverse function.
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, since it is ambiguous. Another precarious convention used by a small number of authors is to use 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 use those very same capitalised representations for the standard trig functions, whereas others use lower-case.
Hence, since 2009, 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 ranges of the inverse functions are proper subsets of the domains of the original functions.
For example, using 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.

Name	Usual notation	Definition	Domain of for real result	Range of usual principal value	Range of usual principal value
arcsine
arccosine
arctangent			all real numbers
arccotangent			all real numbers
arcsecant				or	or
arccosecant				or	or

Note: Some authors define the range of arcsecant to be or because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, whereas with the range or we would have to write since tangent is nonnegative on but nonpositive on For a similar reason, the same authors define the range of arccosecant to be 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

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of

Sine and cosecant begin their period at , finish it at and then reverse themselves over to
Cosine and secant begin their period at finish it at and then reverse themselves over to
Tangent begins its period at finishes it at and then repeats it over to
Cotangent begins its period at finishes it at and then repeats it over to

This periodicity is reflected in the general inverses, where is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values and all lie within appropriate ranges so that the relevant expressions below are well-defined.
Note that "for some " is just another way of saying "for some integer "
The symbol 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.
where the first four solutions can be written in expanded form as:
For example, if then for some While if then for some where will be even if and it will be odd if The equations and have the same solutions as and respectively. In all equations above for those just solved, the integer in the solution's formula is uniquely determined by .
With the help of integer parity
it is possible to write a solution to that doesn't involve the "plus or minus" symbol:
And similarly for the secant function,
where equals when the integer is even, and equals when it's odd.

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

The solutions to and involve the "plus or minus" symbol whose meaning is now clarified. Only the solution to will be discussed since the discussion for is the same.
We are given between and we know that there is an angle in some interval that satisfies We want to find this The table above indicates that the solution is
which is a shorthand way of saying that one of the following statement is true:

for some integer
or
for some integer

As mentioned above, if then both statements and hold, although with different values for the integer : if is the integer from statement, meaning that holds, then the integer for statement is .
However, if then the integer is unique and completely determined by
If then and so the statements and happen to be identical in this particular case.
Having considered the cases and we now focus on the case where and So assume this from now on. The solution to is still
which as before is shorthand for saying that one of statements and is true. However this time, because and statements and are different and furthermore, exactly one of the two equalities holds. Additional information about is needed to determine which one holds. For example, suppose that and that that is known about is that . Then
and moreover, in this particular case and so consequently,
This means that could be either or Without additional information it is not possible to determine which of these values has.
An example of some additional information that could determine the value of would be knowing that the angle is above the -axis or alternatively, knowing that it is below the -axis.

Equal identical trigonometric functions

;Set of all solutions to elementary trigonometric equations
Thus given a single solution to an elementary trigonometric equation, the set of all solutions to it are:

Transforming equations

The equations above can be transformed by using the reflection and shift identities:

Argument:

These formulas imply, in particular, that the following hold:
where swapping swapping and swapping gives the analogous equations for respectively.
So for example, by using the equality the equation can be transformed into which allows for the solution to the equation to be used; that solution being:
which becomes:
where using the fact that and substituting proves that another solution to is:
The substitution may be used express the right hand side of the above formula in terms of instead of

Relationships between trigonometric functions and inverse trigonometric functions

Trigonometric functions of inverse trigonometric functions are tabulated below. They may be derived from the Pythagorean identities. Another way is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that is positive, and thus the result has to be corrected through the use of absolute values and the signum operation.

				Diagram