Inverse trigonometric functions
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with 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 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 , , , 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 e.g. to define the multivalued version of each inverse trigonometric function: e.g. formula_1. However, this might appear to conflict logically with the common semantics for expressions such as (although only , without parentheses, is the really common one), 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 precarious convention used by a tiny number of authors is to use an uppercase first letter, along with a superscript: , , , etc. Although its intention is to avoid confusion with the multiplicative inverse, which should be represented by , , etc., or, better, by , , etc., it in turn creates yet another major source of ambiguity: especially in light of the fact that many popular high-level programming languages (e.g. Wolfram's Mathematica, and University  of Sydney's MAGMA) use those very same capitalised representations for the standard trig functions; whereas others (Python (ie SymPy and NumPy), Matlab, MAPLE etc) 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 (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 formula_2 could be defined from formula_3 the function formula_4 is defined so that formula_5 For a given real number formula_6 with formula_7 there are multiple (in fact, countably infinitely many) numbers formula_8 such that formula_9; for example, formula_10 but also formula_11 formula_12 etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each formula_13 in the domain, the expression formula_14 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 formula_15, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, formula_16 whereas with the range formula_17, we would have to write formula_18 since tangent is nonnegative on formula_19 but nonpositive on formula_20 For a similar reason, the same authors define the range of arccosecant to be formula_21 or formula_22 If formula_13 is allowed to be a complex number, then the range of formula_8 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 formula_25: This periodicity is reflected in the general inverses, where formula_27 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 formula_44 formula_45 formula_46 formula_6 and formula_8 all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some formula_49" is just another way of saying "for some integer formula_50  The symbol formula_51 is logical equality. The expression LHS formula_51 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 false), because otherwise "LHS formula_51 RHS" would not have been written (see this footnote for an example illustrating this concept). For example, if formula_54 then formula_55 for some formula_56 While if formula_57 then formula_58 for some  formula_59 where formula_27 will be even if formula_61 and it will be odd if formula_62 The equations formula_63 and formula_64 have the same solutions as formula_54 and formula_66 respectively. In all equations above for those just solved (i.e. except for formula_67/formula_64 and formula_69/formula_70), the integer formula_27 in the solution's formula is uniquely determined by formula_72 (for fixed formula_73 and formula_8). The solutions to formula_76 and formula_77 involve the "plus or minus" symbol formula_78 whose meaning is now clarified. Only the solution to formula_76 will be discussed since the discussion for formula_77 is the same. We are given  formula_13 between formula_82 and we know that there is an angle formula_72 in some give interval that satisfies formula_84 We want to find this formula_85 The formula for the solution involves: formula_86 If formula_87 (which only happens when formula_88) then formula_89 and formula_90 so either way, formula_91 can only be equal to formula_92 But if formula_93 which will now be assumed, then the solution to formula_94 which is written above as formula_95 is shorthand for the following statement: Because formula_100 and formula_101 exactly one of these two equalities can hold. Additional information about formula_72 is needed to determine which one  holds. For example, suppose that formula_103 and that that is known about formula_72 is that formula_105 (and nothing more is known). Then formula_106 and moreover, in this particular case formula_107 (for both the formula_108 case and the formula_109 case) and so consequently, formula_110 This means that formula_72 could be either formula_112 or formula_113 Without additional information it is not possible to determine which of these values formula_72 has. An example of some additional information that could determine the value of formula_72 would be knowing that the angle is above the formula_13-axis(in which case formula_117) or alternatively, knowing that it  is below the formula_13-axis(in which case formula_119). Transforming equations The equations above can be transformed by using the reflection and shift identities: These formulas imply, in particular, that the following hold: formula_120 where swapping formula_121 swapping formula_122 and swapping formula_123 gives the analogous equations for formula_124 respectively. So for example, by using the equality formula_125 the equation formula_76 can be transformed into formula_127 which allows for the solution to the equation formula_128 (where formula_129) to be used; that solution being: formula_130 which becomes: formula_131 where using the fact that formula_132 and substituting formula_133 proves that another solution to formula_134  is: formula_135 The substitution formula_136 may be used express the right hand side of the above formula in terms of formula_137 instead of formula_138 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 formula_6 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 formula_13 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: 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 It is obtained by recognizing that formula_146. From the half-angle formula, formula_147, 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": Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: formula_154 where formula_155 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 (−, ]. 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 (−, ), it can be expressed as follows: It also equals the principal value of the argument of the complex number "x" + i"y". This limited version of the function above 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 ("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. Arctangent function with location parameter. In many applications the solution formula_8 of the equation formula_161 is to come as close as possible to a given value formula_162. The adequate solution is produced by the parameter modified arctangent function The function formula_164 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 (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −/2 and /2.