Automatic evaluations and transformations
Evaluation for exact, machine-number, and high-precision arguments
For the exact argument , Mathematica returns an exact result.
For a machine‐number argument (a numerical argument with a decimal point and not too many digits), a machine number is also returned.
The next inputs calculate 100‐digit approximations at and .
Within a second, it is possible to calculate thousands of digits for the sine function. The next input calculates 10000 digits for and analyzes the frequency of the digit in the resulting decimal number.
Here is a 50‐digit approximation of the sine function at the complex argument .
Mathematica automatically evaluates mathematical functions with machine precision, if the arguments of the function are machine‐number elements. In this case, only six digits after the decimal point are shown in the results. The remaining digits are suppressed, but can be displayed using the function InputForm.
Simplification of the argument
Mathematica knows the symmetry and periodicity of the sine function. Here are some examples.
Mathematica automatically simplifies the composition of the direct and the inverse sine functions into its argument.
Mathematica also automatically simplifies the composition of the direct and any of the inverse trigonometric functions into algebraic functions of the argument.
If the argument has the structure or , and or with integer , the sine function can be automatically transformed into trigonometric or hyperbolic sine or cosine functions.
Simplification of simple expressions containing the sine function
Sometimes simple arithmetic operations containing the sine function can automatically produce other trigonometric functions.
The sine function arising as special cases from more general functions
The sine function can be treated as a particular case of some more general special functions. For example, can appear automatically from Bessel, Mathieu, Jacobi, hypergeometric, and Meijer functions for appropriate values of their parameters.