The hypotenuse of a right triangle is 4845. The other two angles are 32.2 and
57.8 degrees. What are the lengths of the other two sides.

Hi ernie1234:

The answer to your question is that the side opposite the 32.2 degree
angle is 2581.8 units long and the side opposite the 57.8 degree angle
is 4099.8 units long.

How did I solve this? I used the Right Angle Triangle Solver at:

http://selectsmart.com/darex/triangle.html

Entered 4845 for the "hypotenuse" and 32.2 for the "small angle" - and
clicked "Solve the Triangle".

If you want to do it "by hand", you can also use the trigonometric
identities:

sin x = Opposite/Hypotenuse
cos x = Adjacent/Hypotenuse

So for the 32.2 angle, you get:

sin(32.2) = Opposite/4845
.53287 = Opposite/4845
Opposite = 2581.8

cos(32.2) = Adjacent/4845
.84619 = Adjacent/4845
Adjacent = 4099.8

Hope this helps! 

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up the good work

I remember from high school math a relationship for triangles that is
relevant to this kind of problem. It is called the sine rule.

(sin x)/X  = (sin y)/Y = (sin z)/Z

Where x is the *angle* that is opposite to *side* X, y is the angle
opposite side Y and so on.

In your case,

sin 90°/4845 = sin 32.2°/Y = sin 57.8°/Z

since sin 90° = 1, cross-multiplying, you get:

Y = 4845*sin 32.2° = 2581.79
Z = 4845*sin 57.8° = 4099.81

The sine rule is applied when you know two sides + one angle or one
side + two angles. There is another rule that is very useful, called
the cosine rule, but I won't post that one here.

Urp.. I meant "the sine rule is applicable" in the above post... where
is the Preview Post feature..? It's not there anymore...

Anyway, if you're interested in the proof, I found one here:
http://www.wikipedia.org/wiki/Trigonometric_function