What kind of algorithm do calculators follow to find values of sine?

Question

If someone gives me an angle of say, 38 degrees, how will I find out the value of the sine function for it value without using actually making a right triangle with 38 degrees and measuring the sides? I know I can use some trigonometric identities to close in on the values but that would be laborious.

I am sure that the algorithms used in calculators and computers won't resort to doing that. Like if I need to find pi, I will use an algorithm like this :

instead of getting a compass and a ruler and drawing circles.

However, I cannot find any such algorithm for a sine function.

Can someone please help me with this?

Answer 1

One of the most common algorithms for this is to take first few terms of Taylor series for sine.

sin x = x - x3/3! + x5/5! - x7/7! + ...,

The more terms you take the better approximation you get. X is in radians here, but you can get radians from degrees quite easily. And then, as you see, only basic operations: +, -, *, / can be used to calculate the sine.

In machines with floating-point coprocessor chips CORDIC algorithm (with a few other modules) is used, as it also can be implemented in hardware.

Answer 2

Improving John's answer:

Create a table of sin(x) for x in different radians from 0 to pi/2.

You can use interpolation as follows: sin(x+dx) = sin(x) + dx*cos(x)

cos(x) = sin(pi/2-x).

Similarly, cos(x+dx) = cos(x) - dx*sin(x).

Answer 3

I don't know what is actually used by any calculators, but a lookup table with linear (or higher order if you want) interpolation should be simple and accurate enough for 3 places of accuracy, or more with a denser table. You would only need a table for one quarter-cycle and could use it for all sine and cosine calculations with appropriate transformations. If you have enough power and precision, you could try iterating a taylor series or something, but rounding errors will accumulate on you.