In a triangle ABC, if `c=2`, `A=120^(@)` and `a=c` then what is angle C equal to

Options

A. A. `75^(@)`
B. B. `60^(@)`
C. C. `45^(@)`
D. D. `30^(@)`

Answer

To solve the problem, we will use the Law of Sines, which states that in any triangle ABC:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Given:
- \( c = 2 \)
- \( A = 120^\circ \)
- \( a = c = 2 \)

We need to find angle \( C \).

### Step-by-Step Solution:

1. **Set up the Law of Sines**:
We can write the Law of Sines for our triangle as:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]

2. **Substitute the known values**:
Since \( a = c = 2 \) and \( A = 120^\circ \), we can substitute these values into the equation:
\[ \frac{2}{\sin 120^\circ} = \frac{2}{\sin C} \]

3. **Simplify the equation**:
Since both sides have a numerator of 2, we can cancel them out:
\[ \frac{1}{\sin 120^\circ} = \frac{1}{\sin C} \]

4. **Cross-multiply**:
This gives us:
\[ \sin C = \sin 120^\circ \]

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

6. **Find angle \( C \)**:
Since \( \sin C = \frac{\sqrt{3}}{2} \), we can find angle \( C \):
\[ C = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

The angles that satisfy this equation are:
\[ C = 60^\circ \quad \text{or} \quad C = 120^\circ \]

7. **Determine the valid angle**:
In triangle ABC, the sum of angles must equal \( 180^\circ \). Since \( A = 120^\circ \), if \( C = 120^\circ \), then:
\[ A + C = 120^\circ + 120^\circ = 240^\circ \]
which is not possible. Thus, the only valid solution is:
\[ C = 60^\circ \]

### Final Answer:
\[ C = 60^\circ \]