Geometry: what is the answer to this question?

If you're going to use technology to get help, you may as well use technology that can give you the answer immediately. A triangle solver (or your graphing calculator) can tell you
  ∠A ≈ 29.5°
  ∠B ≈ 115.4°
  ∠C ≈ 35.1°

You can rearrange the formula of the Law of Cosines to give you the angles unambiguously. (It often works best to start with the largest angle.) That law tells you
  b² = a² + c²- 2ac·cos(B)
  B = arccos((a² + c² - b²)/(2ac))
  B = arccos((6² + 7² - 11²)/(2·6·7))
  B = arccos(-36/84) ≈ 115.377°
Similarly,
  A = arccos((b²+c²-a²)/(2bc)) = arccos(134/154) ≈ 29.526°
  C = arccos((a²+b²-c²)/(2ab)) = arccos(108/132) ≈ 35.097°
Of course, once you have 2 angles you can find the third by subtracting their sum from 180°.

Once you have one angle, you can also use the Law of Sines to find another.
  a/sin(A) = b/sin(B)
  A = arcsin(a/b·sin(B)) = arcsin(6/11·sin(115.377°))
    ≈ arcsin(.492822) ≈ 29.526°

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Always maintain intermediate results to the full precision your calculator offers. Only round the final answer.

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