Hello GoldenBoy

Welcome to Math Help Forum!You'll have to use a series expansion of arcsin which you'll find here.

This will give the angle in radians, if you know the sine of the angle.

If you look at the attachment, you'll see I've used this series in a spreadsheet to demonstrate how the answer builds up as you add each term.

In column B I've worked out each individual term, and in column C the accumulated totals. You'll see that the total, for z = 0.6, is approaching 0.643501, which is the answer correct to 6 d.p. The more terms you take, the more accurate the answer. But it will be hard work without a calculator!

Grandad