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Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, [10] 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 .
What links here Related changes Upload file Special pages Permanent link Page information Cite this page Get shortened URL Download QR code In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that ...
- Characterizing Properties
- Useful Relations
- Second Derivatives
- Standard Integrals
- Taylor Series Expressions
- Infinite Products and Continued Fractions
- Comparison with Circular Functions
- Relationship to The Exponential Function
- Hyperbolic Functions For Complex Numbers
- See Also
Hyperbolic cosine
It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc lengthcorresponding to that interval:
Hyperbolic tangent
The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f 2, with f (0) = 0.
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for θ {\displaystyle \theta } , 2 θ {\displaystyle 2\theta } , 3 θ {\displaystyle 3\theta } or θ {...
Each of the functions sinh and cosh is equal to its second derivative, that is: All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x {\displaystyle e^{x}} and e − x {\displaystyle e^{-x}} .
The following integrals can be proved using hyperbolic substitution: where C is the constant of integration.
It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The following series are followed by a description of a subset of their domain of convergence, where...
The following expansions are valid in the whole complex plane: 1. sinh x = x ∏ n = 1 ∞ ( 1 + x 2 n 2 π 2 ) = x 1 − x 2 2 ⋅ 3 + x 2 − 2 ⋅ 3 x 2 4 ⋅ 5 + x 2 − 4 ⋅ 5 x 2 6 ⋅ 7 + x 2 − ⋱ {\displaystyle \sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)={\cfrac {x}{1-{\cfrac {x^{2}}{2\cdot 3+x^{2}-{\cfrac {2\cdot 3x^{2}}{4\cd...
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle. Since the area of a circular sector with radius r and angle u (in radians) is r2u/2, it will be equal to u when r = √2. In the diagram, such a circle is tangent to the hyperbola ...
The decomposition of the exponential function in its even and odd partsgives the identities Additionally,
Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh z and cosh z are then holomorphic. Relationships to ordinary trigonometric functions are given by Euler's formulafor complex numbers: Thus, hyperbolic functions are periodic ...
Euler. Fourier. v. t. e. In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.
False sharing. In computer science, false sharing is a performance-degrading usage pattern that can arise in systems with distributed, coherent caches at the size of the smallest resource block managed by the caching mechanism.
This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. [1] Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering.
Cosine similarity. In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine ...
