Exact Sin Calculus: Solving for Trigon Funcs Without Triangles

[vjacheslav]

Hi!
While computer programming Us encountered problem of exact calculus of trigonometry funcs.
As is well known, all calculators and comp progs do x for Sin
and x^2/2 for Cos on 0..Pi/2 and so on. It seems insufficient.
While solving - next problem:
trigon func definition without triangle.

 

[SteamKing]

Hi!
While computer programming Us encountered problem of exact calculus of trigonometry funcs.
As is well known, all calculators and comp progs do x for Sin
and x^2/2 for Cos on 0..Pi/2 and so on. It seems insufficient.
While solving - next problem:
trigon func definition without triangle.

I'm sorry, but this doesn't make a lot of sense. Can you explain it better?

 

[bhillyard]

I think you may be thinking of how a calculator/computer generates the sin/cosine of a number.
They will use several terms to calculate the value.
I don't think it will be the first few terms of a Taylor series for the function - the convergence is likely to be too slow. The only case I recall is the sine function in the Fortran library for a Univac 1108. This used a fifth degree expression in x² - I don't remember the coefficients.

 

[SteamKing]

This article describes one basic algorithm which can be used to calculate trigonometric and hyperbolic functions on relatively primitive computers:

http://en.wikipedia.org/wiki/CORDIC

Many of the processors nowadays have microcode for calculating trig functions built into the CPU itself.

 

[CellCoree]

Hello! Thank you for bringing up this interesting topic. I understand the importance of finding exact solutions in mathematics and its applications in fields such as computer programming.

In terms of solving for trigonometric functions without using triangles, there are various methods that have been developed over the years. One approach is through the use of power series expansions, which allow for the calculation of trigonometric functions for any angle without the need for triangles. Another method is through the use of complex numbers and Euler's formula, which relates trigonometric functions to exponential functions.

Additionally, advancements in technology have also allowed for more precise and efficient calculations of trigonometric functions, making it possible to obtain more accurate results without the use of triangles.

Overall, while the traditional methods of using triangles to define trigonometric functions are still valid and useful, it is important to continue exploring and developing alternative approaches to solve for these functions in a more precise and efficient manner.