# How exactly does a calculator calculate trigonometric functions?

With simple arithmetic, I can understand how a calculator works. It's fairly simple... but how does the calculator know what the sine of pi is? Is there an arithmetic process to calculating trig functions?

Relevance

It uses the Taylor expansion approximations. It would probably stop after the tenth term or so, since the error from not including the rest would show up in, say, the 15th digit, which your calculator wouldn't display anyway. For example,

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

• Lisa
Lv 4
4 years ago

To calculate exactly, they should either give side lengths, or stick to 45-45-90 or 30-60-90 triangles. To give a student an obscure angle like 2Pi/7 and expect to get a specific answer isn't right, unless it has been specifically talked about in class. I have included a link to help with the triangles I have listed. Sometimes, the only way to give an exact answer is to leave it in the form sin[2Pi/7]

I believe I have read that calculators often use calculus tools, like a Taylor Polynomial and Taylor Series to find numerical values. However, you can probably find out more by just googling 'how does a calculator work,' and Taylor Series and Taylor Polynomial. You might also try researching the Maclaurin Polynomial and the Maclaurin Series.

I hate the really smart-alecy answers but I just can't help myself sometimes. I think it's magic! :~D