In theory you could just use the series expansion:
cos(theta) = 1 - theta^2/(2!) + theta^4/(4!) - theta^6/(6!) +...
This will converge faster for small values of theta, so it's better to use it in conjunction with some angle sum formulae:
cos(alpha + beta) = cos(alpha)cos(beta)-sin(alpha)sin(beta)
sin(alpha + beta) = sin(alpha)cos(beta) + sin(beta)cos(alpha)
and
sin(theta) = theta - theta^3/(3!) + theta^5/(5!) - theta^7/(7!) +...
So if you can find an angle alpha for which you know some trig values that is close to theta, then you can use these to construct a good approximation for cos(theta)
For your example, I am not sure whether 5.22 is in radians or degrees. I will assume radians and just try to find an answer to a few significant digits.
I will use pi ~= 355/113
5.22 / pi ~= 113 * 5.22 / 355 = 113 * 522 / 35500 = 58986 / 35500 ~= 5/3
cos((5pi)/3) = cos(-pi/3) = cos(pi/3) = 0.5
sin((5pi)/3) = sin(-pi/3) = -sin(pi/3) = -sqrt(3)/2
~= -1.7320508/2 = -0.8660254
Let alpha = (5pi)/3 and beta = 5.22 - (5pi)/3
beta = 5.22 - (5pi)/3 ~= 5.22 - (5*355)/(113*3) ~= 5.22 - 5.235988
=-0.0159882
cos(5.22) = cos(alpha+beta)
=cos(alpha)cos(beta) - sin(alpha)sin(beta)
~=0.5 cos(beta)+0.8660254 sin(beta)
cos(beta) ~= 1 - beta^2/(2!)
~=1 - 0.0002556/2 = 1 - 0.0001278 = 0.9998722
sin(beta) ~= beta - beta^3/(3!)
~=-0.0159882 + 0.0000007
=-0.0159875
So
cos(5.22) ~= 0.5 * 0.9998722 - 0.8660254 * 0.0159875
~= 0.49993610 - 0.01384558
=0.48609052
Actually cos(5.22) ~= 0.486090886
so we were good to about 6 significant digits.
As you see, the process is a little tedious and fraught with possibilities for miscalculation.
