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Encyclopedia > Inverse trigonometric functions

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. The principal inverses are listed in the following table. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

Name	Usual notation	Definition	Domain of x for real result	Range of usual principal value
arcsine	y = arcsin(x)	x = sin(y)	−1 to +1	−π/2 ≤ y ≤ π/2
arccosine	y = arccos(x)	x = cos(y)	−1 to +1	0 ≤ y ≤ π
arctangent	y = arctan(x)	x = tan(y)	all	−π/2 < y < π/2
arccotangent	y = arccot(x)	x = cot(y)	all	0 < y < π
arcsecant	y = arcsec(x)	x = sec(y)	−∞ to −1 or 1 to ∞	0 ≤ y < π/2 or π/2 < y ≤ π
arccosecant	y = arccsc(x)	x = csc(y)	−∞ to −1 or 1 to ∞	−π/2 ≤ y < 0 or 0 < y ≤ π/2

If x is allowed to be a complex number, then the range of y applies only to its real part. See also Cauchy principal value for its use in describing improper integrals In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ... Secant is a term in mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

The notations sin⁻¹, cos⁻¹, etc are often used for arcsin, arccos, etc, but this notation causes confusion, e.g. between arcsin(x) and 1/sin(x).

The usual principal values of the f(x) = arcsin(x) and f(x) = arccos(x) functions graphed on the cartesian plane.

The usual principal values of the f(x) = arctan(x) and f(x) = arccot(x) functions graphed on the cartesian plane.

In computer programming languages the functions arcsin, arccos, arctan, are usually called asin, acos, atan. Many programming languages also provide the two-argument atan2 function, which computes the arctangent of y/x given y and x, but with a range of [−π, π]. Image File history File links Asin_acos_plot. ... Image File history File links Asin_acos_plot. ... Image File history File links Atan_acot_plot. ... Image File history File links Atan_acot_plot. ... Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...

1 Relationships among the inverse trigonometric functions
2 General solutions
3 Derivatives of inverse trigonometric functions
4 Expression as definite integrals
5 Infinite series
6 Continued fraction for arctangent
7 Indefinite integrals of inverse trigonometric functions
8 Recommended method of calculation
9 Two argument variant of arctangent
10 Logarithmic forms
- 10.1 Example proof
11 Practical usage
12 See also
13 External links

Relationships among the inverse trigonometric functions

Complementary angles:

$arccos x = frac{pi}{2} - arcsin x$

$arccot x = frac{pi}{2} - arctan x$

$arccsc x = frac{pi}{2} - arcsec x$

Negative arguments:

$arcsin (-x) = - arcsin x !$

$arccos (-x) = pi - arccos x !$

$arctan (-x) = - arctan x !$

$arccot (-x) = pi - arccot x !$

$arcsec (-x) = pi - arcsec x !$

$arccsc (-x) = - arccsc x !$

Reciprocal arguments:

$arccos frac{1}{x} ,= arcsec x$

$arcsin frac{1}{x} ,= arccsc x$

$arctan frac{1}{x} = frac{pi}{2} - arctan x =arccot x,$ if $x > 0$

$arctan frac{1}{x} = -frac{pi}{2} - arctan x = -pi + arccot x,$ if $x < 0$

$arccot frac{1}{x} = frac{pi}{2} - arccot x =arctan x,$ if $x > 0$

$arccot frac{1}{x} = frac{3pi}{2} - arccot x = pi + arctan x,$ if $x < 0$

$arcsec frac{1}{x} = arccos x$

$arccsc frac{1}{x} = arcsin x$

If you only have a fragment of a sine table:

$arccos x = arcsin sqrt{1-x^2},$ if $0 leq x leq 1$

$arctan x = arcsin frac{x}{sqrt{x^2+1}}$

Notice that 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).

From the half-angle formula $tan frac{theta}{2} = frac{sin theta}{1+cos theta}$ , we get: In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. ...

$arcsin x = 2 arctan frac{x}{1+sqrt{1-x^2}}$

$arccos x = 2 arctan frac{sqrt{1-x^2}}{1+x},$ if $-1 < x leq +1$

$arctan x = 2 arctan frac{x}{1+sqrt{1+x^2}}$

General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2π. Sine and cosecant begin their period at 2πk - π/2 (where k is an integer), finish it at 2πk + π/2, and then reverse themselves over 2πk + π/2 to 2πk + 3π/2. Cosine and secant begin their period at 2πk, finish it at 2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk - π/2, finishes it at 2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its period at 2πk, finishes it at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.

This periodicity is reflected in the general inverses:

sin y = x if and only if y = arcsin x + 2kπ or y = π − arcsin x + 2kπ for some integer k.

cos y = x if and only if y = arccos x + 2kπ or y = 2π − arccos x + 2kπ for some integer k.

tan y = x if and only if y = arctan x + kπ for some integer k.

cot y = x if and only if y = arccot x + kπ for some integer k.

sec y = x if and only if y = arcsec x + 2kπ or y = 2π − arcsec x + 2kπ for some integer k.

csc y = x if and only if y = arccsc x + 2kπ or y = π − arccsc x + 2kπ for some integer k.

Derivatives of inverse trigonometric functions

Simple derivatives for real values of $x$ are as follows: For a non-technical overview of the subject, see Calculus. ...

$begin{align} frac{d}{dx} arcsin x & {}= frac{1}{sqrt{1-x^2}}; qquad |x| < 1 frac{d}{dx} arccos x & {}= frac{-1}{sqrt{1-x^2}}; qquad |x| < 1 frac{d}{dx} arctan x & {}= frac{1}{1+x^2} frac{d}{dx} arccot x & {}= frac{-1}{1+x^2} frac{d}{dx} arcsec x & {}= frac{1}{|x|,sqrt{x^2-1}}; qquad |x| > 1 frac{d}{dx} arccsc x & {}= frac{-1}{|x|,sqrt{x^2-1}}; qquad |x| > 1 end{align}$

For an example derivation, letting $theta = arcsin x !$ , we get:

$frac{d arcsin x}{dx} = frac{d theta}{d sin theta} = frac{1} {cos theta} = frac{1} {sqrt{1-sin^2 theta}} = frac{1}{sqrt{1-x^2}}$

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:

$begin{align} arcsin x &{}= int_0^x frac {1} {sqrt{1 - z^2}},dz,qquad |x| leq 1 arccos x &{}= int_x^1 frac {1} {sqrt{1 - z^2}},dz,qquad |x| leq 1 arctan x &{}= int_0^x frac 1 {z^2 + 1},dz, arccot x &{}= int_x^infty frac {1} {z^2 + 1},dz, arcsec x &{}= int_1^x frac 1 {z sqrt{z^2 - 1}},dz, qquad x geq 1 arccsc x &{}= int_x^infty frac {1} {z sqrt{z^2 - 1}},dz, qquad x geq 1 end{align}$

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined. It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ...

Infinite series

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using infinite series, as follows: In mathematics, a series is a sum of a sequence of terms. ...

$begin{align} arcsin z & {}= z + left( frac {1} {2} right) frac {z^3} {3} + left( frac {1 cdot 3} {2 cdot 4} right) frac {z^5} {5} + left( frac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } right) frac{z^7} {7} + cdots & {}= sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{2n+1}} {(2n+1)} ; qquad | z | le 1 end{align}$

$begin{align} arccos z & {}= frac {pi} {2} - arcsin z & {}= frac {pi} {2} - (z + left( frac {1} {2} right) frac {z^3} {3} + left( frac {1 cdot 3} {2 cdot 4} right) frac {z^5} {5} + left( frac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } right) frac{z^7} {7} + cdots ) & {}= frac {pi} {2} - sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{2n+1}} {(2n+1)} ; qquad | z | le 1 end{align}$

$begin{align} arctan z & {}= z - frac {z^3} {3} +frac {z^5} {5} -frac {z^7} {7} +cdots & {}= sum_{n=0}^infty frac {(-1)^n z^{2n+1}} {2n+1} ; qquad | z | le 1 qquad z neq i,-i end{align}$

$begin{align} arccot z & {}= frac {pi} {2} - arctan z & {}= frac {pi} {2} - ( z - frac {z^3} {3} +frac {z^5} {5} -frac {z^7} {7} +cdots ) & {}= frac {pi} {2} - sum_{n=0}^infty frac {(-1)^n z^{2n+1}} {2n+1} ; qquad | z | le 1 qquad z neq i,-i end{align}$

$begin{align} arcsec z & {}= arccosleft(z^{-1}right) & {}= frac {pi} {2} - (z^{-1} + left( frac {1} {2} right) frac {z^{-3}} {3} + left( frac {1 cdot 3} {2 cdot 4} right) frac {z^{-5}} {5} + left( frac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } right) frac{z^{-7}} {7} + cdots ) & {}= frac {pi} {2} - sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{-(2n+1)}} {(2n+1)} ; qquad left| z right| ge 1 end{align}$

$begin{align} arccsc z & {}= arcsinleft(z^{-1}right) & {}= z^{-1} + left( frac {1} {2} right) frac {z^{-3}} {3} + left( frac {1 cdot 3} {2 cdot 4 } right) frac {z^{-5}} {5} + left( frac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} right) frac {z^{-7}} {7} +cdots & {}= sum_{n=0}^infty left( frac {(2n)!} {2^{2n}(n!)^2} right) frac {z^{-(2n+1)}} {2n+1} ; qquad left| z right| ge 1 end{align}$

Leonhard Euler found a more efficient series for the arctangent, which is: Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

$arctan x = frac{x}{1+x^2} sum_{n=0}^infty prod_{k=1}^n frac{2k x^2}{(2k+1)(1+x^2)}.$

(Notice that the term in the sum for n= 0 is the empty product which is 1.) In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

Continued fraction for arctangent

An alternative to the power series for arctangent is its generalized continued fraction: In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity. ...

$arctan(z)=cfrac{z}{1 + cfrac{z^2}{3 + cfrac{4 z^2}{5 + cfrac{9 z^2}{7 + cfrac{16 z^2}{9 + cfrac{25 z^2}{ddots,}}}}}},$

This 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. It was developed by Carl Friedrich Gauss, utilizing the hypergeometric series. Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

Indefinite integrals of inverse trigonometric functions

$begin{align} int arcsin x,dx &{}= x,arcsin x + sqrt{1-x^2} + C int arccos x,dx &{}= x,arccos x - sqrt{1-x^2} + C int arctan x,dx &{}= x,arctan x - frac{1}{2}lnleft(1+x^2right) + C int arccot x,dx &{}= x,arccot x + frac{1}{2}lnleft(1+x^2right) + C int arcsec x,dx &{}= x,arcsec x - lnleft(x+sqrt{x^2-1}right) + C int arccsc x,dx &{}= x,arccsc x + lnleft(x+sqrt{x^2-1}right) + C end{align}$

These are all easily derived using integration by parts and the simple derivative forms shown above. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

Recommended method of calculation

To calculate arcsine, use:

$arcsin x = 2 arctan frac{x}{1+sqrt{1-x^2}}.$

To calculate arccosine, use:

$arccos x = frac{pi}{2} - arcsin x.$

To calculate arctangent for x near zero, use the continued fraction above. To calculate arctangent for other values of x, use:

$arctan x = 2 arctan frac{x}{1+sqrt{1+x^2}}.$

To calculate arccotangent, use:

$arccot x = frac{pi}{2} - arctan x.$

To calculate arcsecant, use:

$arcsec x = frac{pi}{2} - arcsin frac{1}{x}.$

To calculate arccosecant, use:

$arccsc x = arcsin frac{1}{x}.$

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 $( - π,π]$ . It was introduced first in many computer programming languages but is now common in all fields of science and engineering too. Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...

It's defined using the standard arctan function (that is with range of (−π/2, π/2)) as follows:

$operatorname{atan2}(y, x) = begin{cases} arctan(frac y x) & qquad x > 0 pi + arctan(frac y x) & qquad y ge 0 , x < 0 -pi + arctan(frac y x) & qquad y < 0 , x < 0 frac{pi}{2} & qquad y > 0 , x = 0 -frac{pi}{2} & qquad y < 0 , x = 0 text{undefined} & qquad y = 0, x = 0 end{cases}$

This function may be computed using the tangent half-angle formulae as follows: In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. ...

$operatorname{atan2}(y, x)=2arctan frac{y}{sqrt{x^2 + y^2} + x}$

provided that either x > 0 or y ≠ 0. However, in practical implementations it is cheaper and more robust to use the signs of x and y to choose the correct range. Assuming arctan(z) returns a value between −^π⁄₂ and ^π⁄₂ for all real z, we have

$operatorname{atan2}(y, x) = begin{cases} -operatorname{atan2}(-y, x) & qquad y < 0 pi - arctan(-frac y x) & qquad y ge 0 , x < 0 arctan(frac y x) & qquad y ge 0 , x > 0 frac{pi}{2} & qquad y > 0 , x = 0 text{undefined} & qquad y = 0, x = 0 end{cases}$

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. Also, IEEE floating point implementations must handle exceptional (non-numeric) argument values; FDLIBM (available through netlib) shows how this may be done reliably. This is an incomplete list of ISO standards. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... Netlib, www. ...

The function atan2 can be implemented in a numerically reliable manner by the CORDIC method. Thus implementations of atan(y) will probably choose to compute actually atan2(y,1). CORDIC (digit-by-digit method, Volder`s algorithm) (for COordinate Rotation DIgital Computer) is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions. ...

Logarithmic forms

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain to the whole of the complex plane. The natural logarithm is the logarithm to the base e, where e is equal to 2. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

$begin{align} arcsin x &{}= -i,logleft(i,x+sqrt{1-x^2}right) &{}= arccsc frac{1}{x} arccos x &{}= -i,logleft(x+sqrt{x^2-1}right) = frac{pi}{2},+ilogleft(i,x+sqrt{1-x^2}right) = frac{pi}{2}-arcsin x &{}= arcsec frac{1}{x} arctan x &{}= frac{i}{2}left(logleft(1-i,xright)-logleft(1+i,xright)right) &{}= arccot frac{1}{x} arccot x &{}= frac{i}{2}left(logleft(1-frac{i}{x}right)-logleft(1+frac{i}{x}right)right) &{}= arctan frac{1}{x} arcsec x &{}= -i,logleft(sqrt{frac{1}{x^2}-1}+frac{1}{x}right) = i,logleft(sqrt{1-frac{1}{x^2}}+frac{i}{x}right)+frac{pi}{2} = frac{pi}{2}-arccsc x &{}= arccos frac{1}{x} arccsc x &{}= -i,logleft(sqrt{1-frac{1}{x^2}}+frac{i}{x}right) &{}= arcsin frac{1}{x} end{align}$

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof

$arcsin x,=,theta$

$frac{e^{i,theta}-e^{-i,theta}}{2i},=,x$ (exponential definition of sine)

Let

$k=e^{i,theta}.$

Then

$frac{k-frac{1}{k}}{2i},=,x$

$k^2-2,i,k,x-1,=,0$ (solve for

k

)

$k,=,i,xpmsqrt{1-x^2},=,e^{i,theta}$ (the positive branch is chosen)

$theta,=,arcsin,x,=,-ilogleft(i,x+sqrt{1-x^2}right)$ Q.E.D.

Practical usage

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when you already know the length of the sides of the triangle. Remember the acronym SOHCAHTOA. Using inverse trigonometric functions For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigÅnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

$theta = arcsin left( frac{text{opposite}}{text{hypotenuse}} right)$

Often, the hypotenuse is unknown and would need to be calculated before using arcsin or arccos. Arctan comes in handy in this situation. You can compute the angle of the triangles without knowing the length of the hypotenuse.

$theta = arctan left( frac{text{opposite}}{text{adjacent}} right)$

For example, you can calculate the slope of your roof line if you know the rise and run of the roof. If your roof drops 8 feet as it runs out 20 feet then your roof is angled θ degrees up from horizontal, where θ may be computed as follows. Image File history File links Triangle. ...

$begin{align} theta &{}= arctan left( frac{text{opposite}}{text{adjacent}} right) &{}= arctan left( frac{text{rise}}{text{run}} right) &{}= arctan left( frac{8}{20} right) &{}= 21.8^{circ} end{align}$

External links

Categories: Trigonometry | Elementary special functions Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

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	The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin.
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Inverse function - Wikipedia, the free encyclopedia (844 words)
	In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function.
	is also used for the (set valued) function associating to an element or a subset of the codomain, the inverse image of this subset (or element, seen as a singleton).
	Because the range of a left inverse is not restricted, we can adjoin to the domain of this injection an element "undefined", which we then assign to every element of the codomain which is not in the range.

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