Inverse trigonometric function
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In mathematics, the inverse trigonometric functions are a set of relationships closely related to the trigonometric functions. The principal inverses are listed in the following table.
| Name | Usual notation | Definition | Range 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 |
| arccosecant | y = arccsc(x) | x = cosec(y), y = arcsin(1/x) | −∞ to −1 and 1 to ∞ | −π/2 ≤ y < 0, or 0 < y ≤ π/2 |
| arcsecant | y = arcsec(x) | x = sec(y), y = arccos(1/x) | −∞ to −1 and 1 to ∞ | 0 ≤ y < π/2, or π/2 < y ≤ π |
| arccotangent | y = arccot(x) | x = cot(y), y = arctan(1/x) | all | −π/2 ≤ y < 0, or 0 < y ≤ π/2 |
The notations sin−1, cos−1, etc are often used for arcsin, arccos, etc, but this notation sometimes causes confusion between (e.g.) arcsin(x) and 1/sin(x).
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].
Infinite series
Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series.- [\arcsin z = z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots]
- :::[= \sum_^\infty \left( \frac (n!)^2} \right) \frac } \ , \quad \left| z \right| < 1 ]
- [\begin\arccos z & = & \frac - \arcsin z \\ \\& = & \frac - (z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots ) \\& = & \frac - \sum_^\infty \left( \frac (n!)^2} \right) \frac } \end\ , \quad \left| z \right| < 1 ]
- [\begin\arctan z & = & z - \frac +\frac -\frac +\cdots \\ \\& = & \sum_^\infty \frac } \end\ , \quad \left| z \right| < 1 ]
- [\begin\arccsc z & = & \arcsin\left(z^\right) \\ \\& = & z^ + \left( \frac \right) \frac } + \left( \frac \right) \frac } + \left( \frac \right) \frac } +\cdots \\& = & \sum_^\infty \left( \frac (n!)^2} \right) \frac } \end\ , \quad \left| z \right| > 1 ]
- [\begin\arcsec z & = & \arccos\left(z^\right) \\ \\& = & \frac - (z^ + \left( \frac \right) \frac } + \left( \frac \right) \frac } + \left( \frac \right) \frac} + \cdots ) \\& = & \frac - \sum_^\infty \left( \frac (n!)^2} \right) \frac } \end\ , \quad \left| z \right| > 1 ]
- [\begin\arccot z & = & \frac - \arctan z \\ \\& = & \frac - ( z - \frac +\frac -\frac +\cdots ) \\ \\& = & \frac - \sum_^\infty \frac } \end\ , \quad \left| z \right| < 1 ]
Definitions as integrals
These functions may also be defined by proving that they are antiderivatives of other functions.- [\arccos\left(x\right) =\int_x^1 \frac }\,\mathrmz,\quad |x| < 1]
- [\arctan\left(x\right) =\int_0^x \frac 1 \,\mathrmz,\quad \forall x \in \mathbb]
- [\arccot\left(x\right) =\int_x^\infty \frac \,\mathrmz,\quad z > 0]
- [\arcsec\left(x\right) =\int_x^1 \frac 1 }\,\mathrmz, \quad x > 1]
- [\arccsc\left(x\right) =\int_x^\infty \frac }\,\mathrmz, \quad x > 1]
External links
- , [Inverse Trigonometric Functions] at MathWorld.
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