Problem

Solve this triangle.

Answer

Expert–verified
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Answer

\(C = 180^\circ - 30^\circ - 45^\circ = \boxed{105^\circ}\)

Steps

Step 1 ：\(\cos A = \frac{(3 + \sqrt{3})^2 + (2 \sqrt{3})^2 - (\sqrt{6})^2}{2 (3 + \sqrt{3})(2 \sqrt{3})} = \frac{\sqrt{3}}{2}\)

Step 2 ：\(A = \boxed{30^\circ}\)

Step 3 ：\(\cos B = \frac{(3 + \sqrt{3})^2 + (\sqrt{6})^2 - (2 \sqrt{3})^2}{2 (3 + \sqrt{3})(\sqrt{6})} = \frac{1}{\sqrt{2}}\)

Step 4 ：\(B = \boxed{45^\circ}\)

Step 5 ：\(C = 180^\circ - 30^\circ - 45^\circ = \boxed{105^\circ}\)

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