Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. (This convention is used throughout this article.) 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 often called by the abbreviated forms asin, acos, atan.
The notations sin−1(x), cos−1(x), tan−1(x), etc., as introduced by John Herschel in 1813, are often used as well in English-language sources—conventions consistent with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions such as sin2(x), 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, (cos(x))−1 = sec(x). Nevertheless, certain authors advise against using it for its ambiguity. Another convention used by a few authors is to use an uppercase first letter, along with a −1 superscript: Sin−1(x), Cos−1(x), Tan−1(x), etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by sin−1(x), cos−1(x), etc.
Since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
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 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 (in fact, countably infinite) 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.
(Note: Some authors define the range of arcsecant to be (), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, whereas with the range (), 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 )
If is allowed to be a complex number, then the range of applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.
where denotes set subtraction. In other words, the domain of and is the set of all real numbers that are not of the form for some integer
Similarly, the domain of and is the set
where is the set of all real numbers that do not belong to the set
said differently, the domain of and is the set of all real numbers that are not of the form for some integer
Solutions to elementary trigonometric equationsEdit
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. The expression "LHS RHS" indicates that either (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are both true, or else (b) the left hand side and right hand side are both false; there is no option (c) (e.g. it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to false), because otherwise "LHS RHS" would not have been written (see this footnote[note 1] for an example illustrating this concept).
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 except for those just solved (i.e. except for / and /), the integer in the solution's formula is uniquely determined by (for fixed and ).
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 give interval that satisfies We want to find this The formula for the solution involves:
If (which only happens when ) then and so either way, can only be equal to But if which will now be assumed, then the solution to which is written above as
is shorthand for the following statement: Either
for some integer or else
for some integer
Because and exactly one of these two equalities can hold. Additional information about is needed to determine which one holds. For example, suppose that and that all 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 ).
The equations above can be transformed by using the identities
So for example, by using the equality (found in the table above at row and column ), the equation can be transformed into which allows for the solution to the equation (where ) to be used; that solution being:
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 functionsEdit
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 then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purely algebraic derivations are longer. 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.
Relationships among the inverse trigonometric functionsEdit
The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.
The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.
Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.
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
Similar to the sine and cosine functions, the inverse trigonometric functions can also 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 (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative in a geometric series, and applying the integral definition above (see Leibniz series).
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:
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 functionsEdit
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.
A Riemann surface for the argument of the relation tan z = x. The orange sheet in the middle is the principal sheet representing arctan x. The blue sheet above and green sheet below are displaced by 2π and −2π respectively.
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 Re[x]>0 for the upper branch cut and from Re[x]<0 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;
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.
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:
where is the adjacent side, is the opposite side, and is the hypotenuse. From here, we can solve for .
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.
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:
The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−π, π]. In other words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). 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 (−π/2, π/2), it can be expressed 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 (y, x) 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 (x, y) so some caution is warranted. These variations are detailed at atan2.
For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −π/2 and π/2.