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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!