Finding the Value of a Triginometric Function Without a Calculator

[DCircuit]

How can you find, say, sin(41.3°) without using a calculator? Or maybe a better question is: How does a calculator find that value when you punch it in? Also, what about arcsin, arccos, etc... How does a calculator find those values?

 

[gerben]

They use Taylor series (http://en.wikipedia.org/wiki/Taylor_series) expansions to get an answer that is correct up to a limited number of digits. That way only repeated additions and multiplications are needed.

 

[Gerenuk]

They for sure don't use Taylor because that is much too slow in convergence.
For trigonometric functions in particular some simple systems might use
http://en.wikipedia.org/wiki/CORDIC
In general there are iterative algorithms for each elementary function which use only the basic operation (multiplication,...). I once heard a lecture but can't find the notes anymore :(
Maybe you find some information on
http://www.math.niu.edu/~rusin/known-math/index/65-XX.html
http://www.math.niu.edu/~rusin/known-math/98/special.func.comp

 

[DCircuit]

Actually, Euler's Formula gives the exact values of the Trig functions and is very easy to use.

e^ix = cos(x) + isin(x)

 

[Gerenuk]

:-D

How exactly would you calculate exp(i*41.3°) on paper? (with multiplication and addition only) ;-)

 

[Mark44]

And furthermore, I believe that this formula applies only to real numbers (i.e., radian measure), not angles in degree measure, so you would need to convert 41.3° to radians.

 

[DCircuit]

41.3° = 41.3pi/180 That is easy.

$$sinx\ =\ \frac{1}{2i}\left(e^{ix}\ -\ e^{-ix}\right)$$

 

[Gerenuk]

So go ahead and do the complex exponential :D
How would you think calculators do it?
(Hint: They use the trig functions!)

 

[DCircuit]

They can't use the trig functions to get the trig functions. I guess the only way to evaluate complex exponents is with the Taylor's power series, which will only give you a decimal approximation depending on how far you want to go.

 

[Gerenuk]

Good point. And the conclusion is?
(Hint: Calculators don't use the complex exponential to find trig functions)

If you go back to Taylor series, then you start over at Post #2. And the answer is post #3 which says Taylor series converge too slowly to be useful.