If the two sides of a triangle of a triangle and the included angle are given by `a= sqrt(3)+1, b = 2 and C = 60^(@)`, find the other two angles and the third side.

Answer

To solve the problem, we need to find the other two angles and the third side of the triangle given two sides and the included angle. The given values are:

- Side \( a = \sqrt{3} + 1 \)
- Side \( b = 2 \)
- Included angle \( C = 60^\circ \)

### Step 1: Use the Cosine Rule to find the third side \( c \)

The Cosine Rule states that:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

Substituting the given values:
\[ c^2 = (\sqrt{3} + 1)^2 + 2^2 - 2(\sqrt{3} + 1)(2) \cos(60^\circ) \]

### Step 2: Calculate \( a^2 \) and \( b^2 \)

Calculating \( a^2 \):
\[ a^2 = (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]

Calculating \( b^2 \):
\[ b^2 = 2^2 = 4 \]

### Step 3: Substitute \( a^2 \) and \( b^2 \) into the equation

Now substituting these values into the Cosine Rule:
\[ c^2 = (4 + 2\sqrt{3}) + 4 - 2(\sqrt{3} + 1)(2) \cdot \frac{1}{2} \]

### Step 4: Simplify the equation

Calculating \( -2(\sqrt{3} + 1)(2) \cdot \frac{1}{2} \):
\[ -2(\sqrt{3} + 1)(2) \cdot \frac{1}{2} = -2(\sqrt{3} + 1) = -2\sqrt{3} - 2 \]

Now substituting this back into the equation:
\[ c^2 = (4 + 2\sqrt{3}) + 4 - (2\sqrt{3} + 2) \]
\[ c^2 = 4 + 2\sqrt{3} + 4 - 2\sqrt{3} - 2 \]
\[ c^2 = 6 \]

### Step 5: Solve for \( c \)

Taking the square root:
\[ c = \sqrt{6} \]

### Step 6: Find the other two angles \( A \) and \( B \)

Using the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

From this, we can express:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
\[ \sin A = \frac{a \cdot \sin C}{c} \]

Substituting the known values:
\[ \sin A = \frac{(\sqrt{3} + 1) \cdot \sin(60^\circ)}{\sqrt{6}} \]
\[ \sin A = \frac{(\sqrt{3} + 1) \cdot \frac{\sqrt{3}}{2}}{\sqrt{6}} \]
\[ \sin A = \frac{(\sqrt{3} + 1) \cdot \sqrt{3}}{2\sqrt{6}} \]

### Step 7: Calculate \( \sin A \)

Calculating \( \sin A \):
\[ \sin A = \frac{3 + \sqrt{3}}{2\sqrt{6}} \]

### Step 8: Find angle \( A \)

Using the inverse sine function:
\[ A = \sin^{-1}\left(\frac{3 + \sqrt{3}}{2\sqrt{6}}\right) \]

### Step 9: Find angle \( B \)

Using the triangle angle sum property:
\[ A + B + C = 180^\circ \]
\[ B = 180^\circ - A - 60^\circ \]
\[ B = 120^\circ - A \]

### Final Answers

- The third side \( c = \sqrt{6} \)
- Angle \( A = \sin^{-1}\left(\frac{3 + \sqrt{3}}{2\sqrt{6}}\right) \)
- Angle \( B = 120^\circ - A \)