Arccotangent

In mathematics, and in particular in trigonometry, the arccotangent is the function defined as the inverse function of the cotangent of an angle in the interval $(0,\pi )$.

Notation

Comparison between arcotangent and arccotangent functions

The mathematical notation of arccotangent is $\mathrm{arccot}$ or $\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{g}$; ${\cot }^{-1}$ is also common.

Properties

The arccotangent is a continuous function and strictly decreasing, defined for all real numbers:

$\mathrm{arccot}:\mathbb{ℝ}\to (0,\pi ).$

The following limits exist:

$\underset{x\to +\mathrm{\infty }}{\lim }\mathrm{arccot}x=0,$

and

$\underset{x\to -\mathrm{\infty }}{\lim }\mathrm{arccot}x=\pi .$

Its graph is symmetric with respect to the point $(0,\frac{\pi }{2})$, being $\mathrm{arccot}x=\pi -\mathrm{arccot}(-x)$.

The derivative of the function arccotangent is:

$\frac{d}{dx}\mathrm{arccot}x=-\frac{1}{1+x^2}=\arctan \frac{1}{x}.$

The corresponding Taylor series is:

$\mathrm{arccot}x=\frac{\pi }{2}-\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k\frac{x^{2k+1}}{2k+1}=\frac{\pi }{2}-x+\frac{x^3}{3}-\frac{x^5}{5}+\frac{x^7}{7}-\cdots .$

Because of the aforementioned symmetry, the relationship holds for negative arguments:

$\mathrm{arccot}(-x)=\pi -\mathrm{arccot}x.$

Applications

In a right triangle the size in radians of an acute angle equals the arccotangent of the ratio between its adjacent cathetus and the opposite cathetus.

Bibliography

• Paolo Baroncini, Roberto Manfredi, Ilaria Fragni. Lineamenti.Math Blu Volume 5. Ghisetti e Corvi, 2012. ISBN 978-88-538-0433-4.CS1 maint: Multiple names: authors list (link) Search this book on
• Paolo Baroncini, Roberto Manfredi, Nella Dodero. Nuovo Corso di Trigonometria. Ghisetti e Corvi, 2010. ISBN 978-88-538-0433-4.CS1 maint: Multiple names: authors list (link) Search this book on

See also

• Trigonometry
• Cotangent
• Arctangent
• Inverse function

Other projects

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