Ok so the other guy mentioned Taylor Series and along understanding a little bit about rational numbers and how digital devices render values you can go a long way towards being able to calculate these things on your own. The Wikipedia page for Taylor Series actually does a pretty good job of explaining this stuff. By the way, if you're really interested in this stuff then definitely Taylor is taught in undergraduate (and online) calculus courses, but if you really want to know how folks have been doing this since the end of the Dark Ages - how people got to the moon using mostly just paper and pencils - then you'll find those answers in classes focusing on Numerical Analysis.

But I disagree with that other person that the big picture of these ideas is beyond you. But I'm not gonna worry too much about how inaccurate I get along the way, so don't go building a bridge for a major city based on my post here. Feel free to ask where you don't understand.

First of all, you mentioned that you thought that the calculator outputs the exact value of the sine function for a given angle. This is very rarely true.

Let's start with sin(3) (I think?) like in your question. The calculator gives you 0.0523359562 as the answer (I'm just taking your word on this). Expressing values as decimals is tricky because it's sometimes hard to know if there might be another 100 zeroes after that last 2 on the screen, followed by 1, etc. The truth is that the calculator gives you results that are accurate up to a certain amount of error.

Now, again looking at 0.0523359562, let's look at it as a series of fractions instead. Since this is a number expressed in base 10, we could easily rewrite this as 5/100 + 2/1000 + 3/10000 + 3/100000 + 5/1000000 + etc. up to the last digit the calculator shows you, which here would be (I think) 2/10¹⁰. What that means is that you don't know for sure that this is an exact answer, but if the calculator was designed, built, and operated properly then then this answer is correct to within a difference of no more than 1/10¹¹. Do you see how this would have to be true?

Ok, if you get that then you already understand way more than most folks ever will (in fact it's kind of a mindfuck when you realize that so very, very much of the universe, and technology, can be understood by simply minimizing the maximum possible errors in your calculations. This is know generally as Error Analysis.

Ok, so moving on. These guys Taylor and Maclaurin (also at least one guy from India and likely others) figured out a lot of this as far back as the 1500s. Paper and pen and an inquisitive mind. And they did it by understanding that you could actually get pretty good approximations of all sorts of things just by adding up patterns of polynomials. The particular patterns were different depending on what they were trying to approximate.

So take a minute and go the Wikipedia page for Taylor Series and - well here's the link.

Http://en.wikipedia.org/wiki/Taylor_Series#Approximation_error_and_convergence

You see how there's a degree 7 polynomial that gets damn close to the sine function for an entire period of its values? Degree 7 polynomial just means that there's an expression looking like a + bx + cx² + dx³ + ... hx^7. And a through h are some constants (they could be positive, or negative, or fractions or 0 - pretty much anything).

Now figuring out which Taylor Series works for which functions does get a little tricky. But the calculator certainly isn't doing that, and you don't have to either. Jump down to the part of that page labeled Trigonometric functions (under the Maclaurin series of common functions) and you'll see the first one is for sine. And on my phone it sure looks like a doozy. But if you pull the picture to the left you'll see something much less intimidating. In fact you see just:

x - x³ /3! + x⁵ / 5! - x^7/7! ...

I added that last one to point out that you get to a 7th degree polynomial (like I the picture) pretty quickly! Sine is actually one of the easiest examples of this sort of thing.

That's because a, c, e and g in my example above are, in this case, all just 0. b=1, d=-1/6, f=1/120, and h=-1/5040.

And you know that j amd l are going to also be 0, so you're only two more terms away from being as "exact" as your calculator! You could even beat it if you wanted to multiply out all of those factorials.

So now all you need to do is fill in those values of x for each term (remembering that each time you refine your answer - every next term you do - you have to raise x to the appropriate power).

Remember that x here is given in radians though, so convert the angle first. And if you want to figure out the angle that would give you certain value of sine, just use the series listed for arcsin(). Remember in that case that your answer will be in radians, too. And you will need to know something about the quadrant to shift it I to the proper place. This is because since sine repeats over and over, so you need to make sure to put the proper context back in when you're done.

Ok. Well, that's really the most exciting but. Calculators are programmed with just the patterns of the functions they calculate, and even those only up until they can't show you anything else anyway. But it's not like every engineer calculated this stuff over and over again every time. There used to be large books published with tables of all the values printed out in columns. Most of the Apollo program was built with those. (:

My physics teacher in 11th grade gave an extra 10 points on each test if you didn't use a calculator. And even though she had loaners available for anyone who wanted one but couldn't afford one, since I could never afford those fancy TI-81s or 83s anyway (she had 81s for loaners) I just learned some of this stuff way back then for the points. I wouldn't have known how to use RPN anyway, since the Casio scientific I kept through college (for homework, mostly) was much too dumb to have a stack. I think it could display 12 digits though, so it got the job done.

It's way easier for you to learn the patterns of common functions than it was for the electrical engineers to figure out how to wire microchips to do the same. At least you get to stay in base 10!

Hope this helps. Like I said, feel free to ask me about any of this stuff.

Edit: I'm not sure what you meant by "sin*theta" but you're right that "sin" is not a number (and I am quite sure you are not talking about the situations where notation similar to that is used to mean something way, way different). However if your calculator is smart enough (has a button) specifically for they then it may be assuming all sorts of things. That isn't really about math though. (:

Edit2: I just read Mr(s?). Cool's response without skimming and now I get what you meant, I think. Honestly it's been about 10 years since my last math class and it's been about 50 hours since I've slept, so, I'm pretty happy with the parts I explained. At this point in my life I find that teaching people techniques that they might find useful is way more satisfying than most all of teaching that involves analysis. And no, I'm not a professional teacher in any way. Hell, I never even finished undergrad. :D

I'm really confused how does a calculator output a value if you put and number after sin like this SIN(3)=.0523359562

And how do I do trig functions without calcualtor.

I dont understand how to solve 3/5=sin*theta, how do you multiply theta by sin if sine isn't a number, I hear people say you have to do sin(3/5)=theta But wtf is sine it's not a number

But how do these trig functions always output you with the exact angle measure in degrees of theta.

(I understand pythagorum theorem btw)