The smallest angle of the triangle whose sides are `6 + sqrt(12)`, `sqrt(48), sqrt(24)` is

Options

A. `pi//3`
B. `pi//4`
C. `pi//6`
D. none of these

Answer

To find the smallest angle of the triangle with sides \( a = 6 + \sqrt{12} \), \( b = \sqrt{48} \), and \( c = \sqrt{24} \), we will follow these steps:

### Step 1: Identify the sides of the triangle
Let:
- \( a = 6 + \sqrt{12} \)
- \( b = \sqrt{48} \)
- \( c = \sqrt{24} \)

### Step 2: Determine the smallest side
To find the smallest angle, we need to identify the smallest side of the triangle. We will compare the lengths of the sides.

1. Calculate \( a \):
\[ a = 6 + \sqrt{12} = 6 + 2\sqrt{3} \approx 6 + 3.464 = 9.464 \]

2. Calculate \( b \):
\[ b = \sqrt{48} = 4\sqrt{3} \approx 4 \times 1.732 = 6.928 \]

3. Calculate \( c \):
\[ c = \sqrt{24} = 2\sqrt{6} \approx 2 \times 2.449 = 4.898 \]

From the calculations, we find:
- \( a \approx 9.464 \)
- \( b \approx 6.928 \)
- \( c \approx 4.898 \)

Thus, the smallest side is \( c = \sqrt{24} \).

### Step 3: Use the Law of Cosines
The smallest angle \( C \) is opposite the smallest side \( c \). We use the Law of Cosines:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

### Step 4: Calculate \( a^2 \), \( b^2 \), and \( c^2 \)
1. Calculate \( a^2 \):
\[ a^2 = (6 + \sqrt{12})^2 = 36 + 12 + 12\sqrt{3} = 48 + 12\sqrt{3} \]

2. Calculate \( b^2 \):
\[ b^2 = (\sqrt{48})^2 = 48 \]

3. Calculate \( c^2 \):
\[ c^2 = (\sqrt{24})^2 = 24 \]

### Step 5: Substitute values into the cosine formula
Now substitute \( a^2 \), \( b^2 \), and \( c^2 \) into the cosine formula:
\[ \cos C = \frac{(48 + 12\sqrt{3}) + 48 - 24}{2 \cdot (6 + \sqrt{12}) \cdot \sqrt{48}} \]
\[ = \frac{72 + 12\sqrt{3}}{2 \cdot (6 + 2\sqrt{3}) \cdot 4\sqrt{3}} \]

### Step 6: Simplify the expression
Calculate the denominator:
\[ 2 \cdot (6 + 2\sqrt{3}) \cdot 4\sqrt{3} = 8\sqrt{3}(6 + 2\sqrt{3}) = 48\sqrt{3} + 16 \cdot 3 = 48\sqrt{3} + 48 \]

Now, we have:
\[ \cos C = \frac{72 + 12\sqrt{3}}{48 + 48\sqrt{3}} \]

### Step 7: Find \( C \)
To find angle \( C \), we need to compute \( \cos^{-1} \) of the above expression. After simplification, we find:
\[ C = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \text{ or } 60^\circ \]

### Conclusion
Thus, the smallest angle of the triangle is \( C = 60^\circ \).

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