Computing the sines of few angles is easy.

The only thing which I recommend to memorize is pi.

To memorize pi, I recommend this:

http://www.math.purdue.edu/~eremenko/dvi/pi.pdf(for English,
French or Russian speakers:-)

Then you proceed as follows: if x is your angle in degrees,

then y=2.pi.x/360 is your angle in radians.

Then

sin(y)=y-y^3/6+y^5/120-y^7/(1.2.3.4.5.6.7)+ etc.

This is better to use for small angles. If your angle is not small

use the doubling formulas like sin(2t)=2sin(t)cos(t),

cos(t)=1-t^2/2+t^4/24-t^6/1.2.3.4.5.6+ etc.

Dividing a 1-foot ruler into 1/10 of an inch divisions,

without any instruments, is a much

harder task:-)

Alex.

P.S. This is a late XVII century math. Ptolemy had to use a different

method, much more sophisticated, starting from few angles (30, 45, 60, 90)

for which sine and cosine can be found precisely,

and then using division and addition formulas. To be more precise, he

did not use our modern functions, sine and cos, but used the chord instead,

chd(x)=2sin(x/2). His book contains the first table of chords that

survived to out time. He
also had to compute without having our decimal

positional system, poor guy:-)