It would be a lot of work, but you could use a series expansion, such as the simplest, Maclaurin series (see http://mathworld.wolfram.com/MaclaurinSeries.html). For accuracy you could either go to many terms or reduce the argument for say the sine function by dividing by a power of 2, then apply the series for both the sine and cosine, and then apply the double-angle formula as many times as the power of two you used.

I'd think it would be worthwhile to get a calculator that has the trig functions (and thereby does this for you internally).

An alternative to more terms in the Maclaurin series is to use the argument reduction specified. For example, if you want the sin(pi), which we know to be zero, first divide pi by some power of 2, say 1024, getting 0.003067961575771, then use this in two short Maclaurin series: sin(x) = x - x^3 / 6 and cos(x) = 1 - x^2/2 + x^4/24, resulting in sin and cos of 0.003067956762964 and 0.999995293809576 respectively. Then use the double angle formulae, sin 2x = 2 sin x cos x and cos 2x = cos^2 x - sin^2 x, applying it 10 times because 1024 is 2^10. Successive sin, cos pairs are then:

0.003067956762964  0.999995293809576


0.006135884649150  0.999981175282601


0.012271538285711  0.999924701839145


0.024541228522894  0.999698818696205


0.049067674327382  0.998795456205174


0.098017140329489  0.995184726672205


0.195090322015986  0.980785280403260


0.382683432364823  0.923879532511400


0.707106781186140  0.707106781186961


1.000000000000008  0.000000000001160


0.000000000002320 -1.000000000000016

Thanks for your help- that looks like a lot of work. I think i'll take your advice and get a calculator!

And yet there are two other ways to do this. I have one in my desk drawer which I use from time to time, and the other disappeared soon after I left high school. I refer to a slide rule (good for about 3 places accuracy) and a book of log tables (etc) - probably don't print them any more!! What we suffer with the advents of technology!!