Hi guys. After a long time of searching, asking, googling, etc, I still cannot find out this answer. How on earth does one find sine, cosine, and tangent without a calculator? Every site I go to does the typical "its complicated so just use your calculator" crap and sadly so does different mathbooks I've purchased. The closest answer I got was that I need a protractor and a slide rule, and that was from a guy who used to have to do it manually but he forgets exactly how it worked. Can anyone please tell me how to do this without a calculator?

You can use a Taylor series to approximate the answer. All you'd have to remember is the one for sine. Cosine should be simple from there, and tangent is just sine divided by cosine.
Computing a Taylor series without a calculator is still troublesome of course, but after about 3 factors, you're usually quite close.
Another way to go about it, is to draw a unit circle, mark out the angle from the center and find the coordinates of the intersection with the unit circle. The X coordinate would be the cosine of the angle, and the Y coordinate would be the sine.

EDIT:
Formula for sine:
Sin(x) =~ x - x^3 / 3! + x^5 / 5! - x^7/ 7! ... etc...

There are iterative solutions, such as Newton-Raphson, which start with an approximation of the answer and refine it. Some quick Googling also lead me to the Taylor series, which I vaguely remember being useful for this kind of thing.

When I was in school we had "log tables", which included a table to look up the sin/cos/tan to certain precisions.

The simple way:

Take out a big piece of paper, a big ruler, a protractor, and something to help you draw a nice big circle. Use your protractor to measure the angle you want to use, and then draw a long line from the center of the circle to the circle's surface. For cos(a), measure the horizontal distance from the origin (not diagonal, just horizontal), and for sin(a), measure the vertical distance from the origin (again, just vertical, not diagonal). Then divide by the circle's radius, and you have your answer [wink]

@ rip-off: I don't see how Newton's method can help here, especially considering you need the derivative the of the function anyway

My apologies, my weak defense is that I'm trying to recall long-forgotten maths. I remember newton raphson being useful for calculating the values of functions, but didn't think about the details.

Quote: Original post by Chrono1081
After a long time of searching, asking, googling, etc, I still cannot find out this answer.

Then you'll have to learn to search better :-)

Just by looking at wikipedia we can find the trigonometric functions page that says:

Quote: Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History below), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2) = 1).

Then, following a "see also" link to the Generating trigonometric tables pages we can find the method used to create the tables that rip-off used in high school as well as some faster approximations.

I assume that Wolfram's MathWorld also explains all this, but I didn't bother to look there.

If you're looking for a way of obtaining sine/cosine values that's faster than using the math lib functions, consider using a lookup table.

Thank you guys sooo much! You have no idea how bad this has been frustrating me :)

I remember asking this question in high school and at three different universities and the closest I got was "Well, before calculators we had look up tables in books". Um..thats nice but how do you get the values in the books!?

I even tried asking friends of mine who are *gasp!* high school math teachers! The reason I gasp is because they both started telling me soh cah toa and didn't understand that I needed to find this without a calculator (or else they simply didn't know and tip-toed around the question).

Anyway I'm off to try this. Hopefully I can scrounge up a protractor from the engineering department. (I work in a remote area so running to walmart is out of the question :P)

Correct me if I am wrong but wouldn't Soh Cah Toa help you? Especially if you are programming it. Just have the computer calculate each one you need instead of a the functions themselves. It should be less math than the Taylor Series, too. nullsquared's method would work just as well. Either way it is just division.

Well, the power series for e, sin and cos has been known since at least Newton's time... given just the power series for e though you can derive the other two (and hence tan) using complex numbers:




Minor elements missing of course (for instance Euler originally published the formula as in 1748 (in his book Introductio in Analysis Infinitorum).