Question

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Answer 1

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Answer 2

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Answer 3

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Answer 4

I'll assume that this involves trigonometry, in which case we'll want to make use of the sine or cosine ratios that apply to right triangles.
Whether we use sine or cosine in this case is more a matter of preference. Please stop me if I'm getting ahead of myself.

Answer 5

As you may already know, the sine ratio of an angle in a right triangle is equal to the length of the opposite edge divided by the length of the adjacent edge; meanwhile, cosine is the reverse and would be calculated by the length of the adjacent edge divided by the opposite edge.

Answer 6

The diagram above may illustrate what I mean in clearer terms. For the angle of x degrees, sin(x) would be equal to opp/adj, and cos(x) would be equal to adj/opp.

Answer 7

So, how does this help us? Well, if you have a calculator or a table of sin and cos values, you can find the measure of an angle from its sin or cos value.
In case you do not have a calculator, here is a link to a chart you can use. Unfortunately, the chart only gives angle measures to the nearest degree, not tenth of a degree, so you'll be better off using a calculator.

Answer 8

So, to summarize:
Draw a right-triangle with the vector as the hypotenuse;
find the lengths of both legs of that right triangle;
calculate either the sin or cos of the angle using the ratio of the leg lengths
with a calculator, use the inverse sin or inverse cos function to calculate the angle.
I hope this helps!

Answer 9

http://cruzrich.com/sin-cos-tan-chart-uptodate/sin-cos-tan-chart-delux-drawing-table/
Oh, here's the chart I said I posted, if you want to use it as a reference