how to find x axis distance with cos without using a calculator?

Except for "special" angles, that's normally beyond the scope of trigonometry classes.

There are ways to do it today (Maclaurin series), but it's advanced. If you had to do it in the past, you'd use half-angle and double-angles repeatedly to cover much of the possibilities and write them down in a table. Then you could look in a table when you needed it again.

You could use the half-angle formula twice on 90 degrees to get the answer for 22.5 degrees. That would be pretty close.
http://www.purplemath.com/modules/idents.htm#half

Otherwise, look it up or use a calculator. Doing a Maclaurin expansion is annoying.
http://blogs.ubc.ca/infiniteseriesmodule/units/unit-3-power-series/taylor-series/the-maclaurin-expansion-of-cosx/

I suppose you could use a slide rule, like I had to use when I studied trig a lifetime ago, though I expect that's probably not the answer you're looking for.
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The newly-invented HP-35 cost $395 back then, if I recall correctly. I only saw one, and nobody I knew owned one. It was sticks for us.
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They used to publish trig tables that you could look up. Also, the formula for a circle is:

x² + y² = r²

So that actually relates what you need except angle, you would have to get that from dividing down the perimeter of the circle but between the equation for a circle and the perimeter of a circle, with a little bit of calculus, you could avoid using trig in a rather complicated fashion.

You could also remember the first few terms of the Taylor sequence, that would get you very close.

It would not be easy to do that without a calculator or tables. We memorize sine and cosine of 60, 30, and 45 degree angles. Using trig identities we can calculate the value trig functions of the sums and differences of angles we already know the value of these functions for. We can also calculate the value of trig functions of the half angle if we know the value of trig functions of the angle. For example, if we know sin(30), we can calculate sin(15). Once we can calculate the sin of 15 degrees, we can calculate the sin of 7.5 degrees. Once we know that, we can figure out the sin of 67.5 degrees, to get to exactly 68 degrees from there isn't in the cards.

The trig functions can be expressed as infinite series with the Taylor expansion. We could use Taylor expansion to estimate trig function of any angles, but it's a good bit of computation. Be glad you have a calculator. It saves lots of time. Before calculators we used slide rules and tables. It was more work. I pretty much missed that period. Scientific hand calculators were just starting to become available at a reasonable price when I started studying. This is the one I first used in high school.

http://en.wikipedia.org/wiki/TI-30

Note: Big Daddy mentions the Maclaurin expansion. It is a specialized case of the Taylor expansion. We can use Maclaurin to estimate sine and cosine.