Every triangle has six parts, three angles and three sides.

If you know three of these parts, you can usually find the other
three parts by calculating.

But you must be careful with the capital and small letters
A, B, C, a, b, and c.  Capital and small letters are NOT 
interchangeable.  

The capital letters A, B, C, refer to the three angles only.
The small letters a, b, c, refer to the three sides only.
Side a is across the triangle from angle A.
Side b is across the triangle from angle B.
Side c is across the triangle from angle C.

You should have used small letters, 

a = 27.2 in., b = 33.4 in., c = 44.6 in.

Here is a picture of your triangle:



This is the case SSS, where only the three sides
are given.

In this case we first use the law of cosines to find
each angle.

To find angle A, we use this form of the law of cosines:

a² = b² + c² - 2ab·cos(A)

Solve for cos(A)

2ab·cos(A) = b² + c² - a²

    cos(A) = 

    cos(A) = 

    cos(A) = .7937756773

Now we find the inverse cosine of that, and get

 = 37.45023223°

For the final answer we should round that off
to the nearest tenth of a degree, since the sides are
rounded to three significant digits. So the final answer
should have angle A = 37.5°

To find angle B, we use this form of the law of cosines:

b² = a² + c² - 2ac·cos(B)

Solve for cos(B)

2ac·cos(B) = a² + c² - b²

    cos(B) = 

    cos(B) = 

    cos(B) = .6649960433

Now we find the inverse cosine of that, and get

 = 48.31797816°

For the final answer we should round that off
to the nearest tenth of a degree, since the sides are
rounded to three significant digits. So the final answer
should have angle B = 48.3°

Now all we have to do to find angle C is to
use the fact that the three angles of any
triangle must have sum 180°.

A + B + C = 180°

37.46023223° + 48.31797816° + C = 180°

85.77821039° + C = 180°
               C = 180°- 85.77821039°
               C = 94.22178961°
               C = 94.2°

So the three missing parts, the three
angles, of the triangle are:

A = 37.5°
B = 49.3
C = 94.2°

-------------------------------

Now as a check, let's see if we get the same thing
for angle C if we use the law of cosines as we did to
find angles A and B:

To find angle C, we use this form of the law of cosines:

c² = a² + b² - 2ab·cos(C)

Solve for cos(C)

2ab·cos(C) = a² + b² - c²

    cos(C) = 

    cos(C) = 

    cos(C) = -.0736174709°

Now we find the inverse cosine of that, and get

 = 94.22178961°

That is exactly what we got when we found angle
C the other way.  So we know we are correct.

Edwin