In a `DeltaABC`, if `A=30^(@)`, `C=105^(@)` and `b=3sqrt(2)`, then a =

Options

A. `sqrt(2)`
B. 3
C. `3sqrt(2)`
D. 2

Explanation

To solve the problem, we will use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Given:
- Angle A = 30°
- Angle C = 105°
- Side b = 3√2

We need to find the length of side a.

### Step-by-Step Solution:

1. **Find Angle B**:
Since the sum of angles in a triangle is 180°, we can find angle B:
\[ B = 180° - A - C = 180° - 30° - 105° = 45° \]

**Hint**: Remember that the sum of angles in a triangle is always 180°.

2. **Apply the Law of Sines**:
According to the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Substituting the known values:
\[ \frac{a}{\sin 30°} = \frac{3\sqrt{2}}{\sin 45°} \]

**Hint**: Make sure to use the correct sine values for the angles.

3. **Calculate Sine Values**:
We know:
\[ \sin 30° = \frac{1}{2} \quad \text{and} \quad \sin 45° = \frac{1}{\sqrt{2}} \]
Substitute these values into the equation:
\[ \frac{a}{\frac{1}{2}} = \frac{3\sqrt{2}}{\frac{1}{\sqrt{2}}} \]

**Hint**: Sine values for common angles can be found in trigonometric tables or unit circles.

4. **Simplify the Right Side**:
Simplifying the right side:
\[ \frac{3\sqrt{2}}{\frac{1}{\sqrt{2}}} = 3\sqrt{2} \times \sqrt{2} = 3 \times 2 = 6 \]
So now we have:
\[ \frac{a}{\frac{1}{2}} = 6 \]

**Hint**: Multiplying by the reciprocal can help simplify the equation.

5. **Solve for a**:
Now, multiply both sides by \(\frac{1}{2}\):
\[ a = 6 \times \frac{1}{2} = 3 \]

**Hint**: Isolate the variable to find its value.

### Final Answer:
Thus, the length of side \( a \) is \( 3 \).