In a `DeltaABC`, if c = 2, `A=120^(@), a=sqrt(6)`, then C=

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. none of these

Answer

To solve the problem, we will use the Law of Sines. Given the values:

- \( c = 2 \)
- \( A = 120^\circ \)
- \( a = \sqrt{6} \)

We need to find angle \( C \).

### Step-by-step Solution:

1. **Write the Law of Sines**:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]

2. **Substitute the known values**:
\[ \frac{\sqrt{6}}{\sin 120^\circ} = \frac{2}{\sin C} \]

3. **Calculate \( \sin 120^\circ \)**:
\[ \sin 120^\circ = \sin (180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \]

4. **Substitute \( \sin 120^\circ \) into the equation**:
\[ \frac{\sqrt{6}}{\frac{\sqrt{3}}{2}} = \frac{2}{\sin C} \]

5. **Simplify the left side**:
\[ \frac{\sqrt{6} \cdot 2}{\sqrt{3}} = \frac{2}{\sin C} \]
This simplifies to:
\[ \frac{2\sqrt{6}}{\sqrt{3}} = \frac{2}{\sin C} \]

6. **Cross-multiply to find \( \sin C \)**:
\[ 2\sqrt{6} \sin C = 2 \sqrt{3} \]

7. **Divide both sides by \( 2\sqrt{6} \)**:
\[ \sin C = \frac{\sqrt{3}}{\sqrt{6}} = \frac{\sqrt{3}}{\sqrt{2} \cdot \sqrt{3}} = \frac{1}{\sqrt{2}} \]

8. **Find angle \( C \)**:
\[ C = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
The value of \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) is \( 45^\circ \).

### Final Answer:
Thus, angle \( C \) is \( 45^\circ \).

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