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 ranges 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 function could be defined from the function is defined so that For a given real number with there are multiple (in fact, countably infinitely many) 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 (radians) Range of usual principal value (degrees) 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 For example: 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 (where is an integer), 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 (forward) over to Cotangent begins its period at finishes it at and then repeats it (forward) 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 (see this footnoteThe expression "LHS RHS" indicates that (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are true, or else (b) the left hand side and right hand side are false; there is option (c) (e.g. it is possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS RHS" would not have been written.
To clarify, suppose that it is written "LHS RHS" where LHS (which abbreviates left hand side) and RHS are both statements that can individually be either be true or false. For example, if and are some given and fixed numbers and if the following is written:
then LHS is the statement "". Depending on what specific values and have, this LHS statement can either be true or false. For instance, LHS is true if and (because in this case ) but LHS is false if and (because in this case which is not equal to ); more generally, LHS is false if and Similarly, RHS is the statement " for some ". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values and have). The logical equality symbol means that (a) if the LHS statement is true then the RHS statement is also true, and moreover (b) if the LHS statement is false then the RHS statement is also false. Similarly, means that (c) if the RHS statement is true then the LHS statement is also true, and moreover (d) if the RHS statement is false then the LHS statement is also false. for more details and an example illustrating this concept).
+ Equation if and only if Solution for some for some for some for some for some for some
where the first four solutions can be written in expanded form as:
+ Equation if and only if Solution or for some or for some or for some or for some
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 (i.e. except for / and /), the integer in the solution's formula is uniquely determined by (for fixed and ).
With the help of integer parity
it is possible to write a solution to that doesn't involve the "plus or minus" symbol: if and only if for some And similarly for the secant function, if and only if for some 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 (at least) one of the following statement is true: for some integer or for some integer As mentioned above, if (which by definition only happens when ) then both statements (1) and (2) hold, although with different values for the integer : if is the integer from statement (1), meaning that holds, then the integer for statement (2) is (because ). However, if then the integer is unique and completely determined by If (which by definition only happens when ) then (because and so in both cases is equal to ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). 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 (1) and (2) is true. However this time, because and statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about is needed to determine which one holds. For example, suppose that and that that is known about is that (and nothing more is known). Then
and moreover, in this particular case (for both the case and the 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 (in which case ) or alternatively, knowing that it is below the -axis (in which case ).
Equal identical trigonometric functions
Set of all solutions to elementary trigonometric equations
Thus given a single solution to an elementary trigonometric equation ( is such an equation, for instance, and because always holds, is always a solution), the set of all solutions to it are:
+ If solves then Set of all solutions (in terms of ) then then then then then then
Transforming equations
The equations above can be transformed by using the reflection and shift identities:
+ Transforming equations by shifts and reflections 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 (where ) 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 (sgn) operation.
Diagram
Relationships among the inverse trigonometric functions
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that and are reciprocals (i.e. ), as are and and and
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).