What is the easiest way to find the sine of any number?
Answers
Very interesting question! A similar question is, how does the calculator figure out the value of sin, cos, etc.? Or you could ask, what did people do before the calculator was invented, i.e. before ca. 1970 ? These are all very similar questions, and the answers are closely related.
But I assume you're asking for what would be a practical method today of calculating sin, cos, etc in case you don't have access to any electronic devices.
The answers given are all good. You see, it is really a big bag of different tricks. It depends on how accurate you want your answer. So you must first of all accept, that whatever you do, you'll only get an approximate result. You can get any desired accuracy, but a more accurate result will require more calculations. Each calculation "improves" the accuracy of the previous result - so to speak.
If you want to learn more about this question, then the whole subject falls under . The general method is to approximate the function, e.g. sin(x), by some polynomial. It is usually possible to find a polynomial whose function values are very close to that of sin(x), provided that x is very close to 0.
Looking specifically at the function sin(x) we have some additional options. For instance, we may use the special property that:
sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
Of course, this only works for sin(x) . But for e.g. ln(x) we have something similar:
ln(x⋅y)=ln(x)+ln(y)
These special relations may be used in various ingenious ways to add to the bag of tricks.
For another method not mentioned in the other answers, some computers today use the method.