In the complex ‐plane, the function is defined using the exponential function in the points and through the formula: Struve functions can also degenerate into the cosine function for similar values of the parameter:īut the function is also a degenerate case of the doubly periodic Jacobi elliptic functions when their second parameter is equal to or :įinally, the function is the particular case of another class of functions-the Mathieu functions:ĭefinition of the cosine function for a complex argument Other Bessel functions can also be expressed through cosine functions for similar values of the parameter: It is also a particular case of the Bessel function with the parameter, multiplied by : For example, it is a special case of the generalized hypergeometric function with the parameter at : The function is a particular case of more complicated mathematical functions. Representation through more general functions Here is a graphic of the cosine function for real values of its argument.
Cosine geometry definition series#
This series converges for all finite numbers. The following formula can also be used as a definition of the cosine function:
This approach to the cosine can be expanded to arbitrary real values of if consideration is given to the arbitrary point in the, ‐Cartesian plane and is defined as the ratio, assuming that α is the value of the angle between the positive direction of the ‐axis and the direction from the origin to the point. The classical definition of the cosine function for real arguments is: "the cosine of an angle in a right‐angle triangle is the ratio of the length of the adjacent leg to the length of the hypotenuse." This description of is valid for when the triangle is nondegenerate. Newton (1658, 1665) found the series expansion for. Gunter (1636) introduced the notation "Cosi" and the word "cosinus" (replacing "complementi sinus"), and I. Viète (1590) evaluated some values of, E. It was first used in ancient Egypt in the book of Ahmes (c. The cosine function is one of the oldest mathematical functions.
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