Try this:
79.5° = 60° + 15° + 4.5°
The first two are easy because the sin and cos of 60 are well known.
The sin and cos of 15° can be obtained using the sin and cos of 30° and the half-angle formulae.
Then sin and cos of 75° can be obtained from the addition formulae.
Now the hard part
In isosceles triangle with vertex angle of 72° and base angles of 36° the side is the "golden ratio" times the base. This can be shown using Ptolemy's Theorem to prove that the ratio of the diagonal of a pentagon is the "golden ratio" times the side of the pentagon. This allows calculation of the sin and cos of 36°
The "golden ratio" is [1 + sqrt(5)]/2
Using the half-angle formula 3 times, starting with 36° gives you the sin and cos of 4.5°.
Now use the addition formula to get the cos of 79.5°.
There may be an easier way to do this and I will try to think of one, but for now this answer is correct.
Of course, if you already know the Taylor series expansion of cos you can use it to get the cos of 4.5°...and not too many terms will be required.
