How are trigonometric and logarithmic functions calculated?

Asked Aug 26 '24 at 06:40

Active Aug 26 '24 at 06:41

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How are complex (yet fundamental) functions calculated without a graph? In even simpler words, how do graphing calculators actually compute them?

Trigonometric functions and the like such as logarithms seem to be really useful in defining piecewise functions and the like using an equation, however is it possible to define them over all their domains using the arithmetic basic operators we have?

Since these trigonometric often don't show a progression, how do we go on to define them using something using our arithmetic operators? Do we need to create a generating-piecewise function first?

Sorry if a similar question has already been asked, I didn't seem to find one with a satisfactory answer.

edited Aug 26 '24 at 06:41

Frederik vom Ende

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asked Aug 26 '24 at 06:40

Sanskar Anand

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Taylor series may be useful. – Átila Correia Aug 26 '24 at 06:45
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@ÁtilaCorreia but it still is a series that goes on till infinity. Can't these functions be computed in a finite number of steps? – Sanskar Anand Aug 26 '24 at 06:46
You can stop to add the highest order terms based on some approximation criteria. – Átila Correia Aug 26 '24 at 06:48
you can start with $\sin(x)\sim x$ for small $x$ and then use trigonometric identities such as $\sin(2x)=2\sin(x)\cos(x)$ – user619894 Aug 26 '24 at 07:43
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See https://math.stackexchange.com/questions/395600/how-does-a-calculator-calculate-the-sine-cosine-tangent-using-just-a-number and several questions listed there under "Linked" and "Related". – Gerry Myerson Aug 26 '24 at 07:45
You need to realise that Taylor (or McLaurin) series are created in such a way that the result gets more and more accurate while the last entry becomes smaller and smaller. Hence, no need to go up to infinity is you're just interested in an approximation. – Dominique Aug 26 '24 at 09:08
Find anything useful at those links, Sanskar? – Gerry Myerson Aug 27 '24 at 10:34
@GerryMyerson Yeah. Thanks. – Sanskar Anand Aug 27 '24 at 15:50

How are trigonometric and logarithmic functions calculated?

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