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In the points, the values of are algebraic. Students usually learn the following basic table of sine function values for special points of the circle:įor real values of argument, the values of are real. The best-known properties and formulas for the sine function Here are two graphics showing the real and imaginary parts of the sine function over the complex plane. Taking the difference of the preceding formulas and dividing by 2ⅈ gives the following result: The key role in this definition of belongs to the famous Euler formula connecting the exponential, the sine, and the cosine functions: 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 sine 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 sine function for a complex argument Other Bessel functions can also be expressed through sine 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, multiplied by : The function is a particular case of more complicated mathematical functions. Representation through more general functions Here is a graphic of the sine function for real values of its argument.
SINE FUNCTION EQUATION MAKER SERIES
This series converges for all finite numbers. The following formula can also be used as a definition of the sine function: This approach to sine 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 sine function for real arguments is: "the sine of an angle in a right‐angle triangle is the ratio of the length of the opposite leg to the length of the hypotenuse." This description of is valid for when the triangle is nondegenerate. de Chesters translated Abu Ja'far Muhammed ibn Musa al‐Khwarizme's works and used the word "sine" (in Latin, "sinus"). It was used in ancient Greece and India, and in 1140, R. The sine function is one of the oldest mathematical functions.