If `a=sqrt3 +1, B=30^@, C=45^@`, then `c=`

Options

A. 3
B. 2
C. 4
D. none of these

Explanation

To solve the problem where \( a = \sqrt{3} + 1 \), \( B = 30^\circ \), and \( C = 45^\circ \), we need to find the side \( c \) using the sine rule.

### Step-by-Step Solution:

**Step 1: Identify the angles and sides of the triangle.**
- We have:
- \( a \) opposite angle \( A \)
- \( b \) opposite angle \( B \) (which is \( 30^\circ \))
- \( c \) opposite angle \( C \) (which is \( 45^\circ \))

**Step 2: Use the sine rule.**
- The sine rule states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

**Step 3: Write the equation for \( c \) using the sine rule.**
- We can express \( c \) in terms of \( a \) and the angles:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
Rearranging gives:
\[ c = a \cdot \frac{\sin C}{\sin A} \]

**Step 4: Find angle \( A \).**
- Since the sum of angles in a triangle is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Substituting the known angles:
\[ A + 30^\circ + 45^\circ = 180^\circ \]
Therefore:
\[ A = 180^\circ - 75^\circ = 105^\circ \]

**Step 5: Substitute values into the equation for \( c \).**
- Now we need to find \( \sin A \) and \( \sin C \):
- \( \sin C = \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin A = \sin 105^\circ \)

**Step 6: Calculate \( \sin 105^\circ \).**
- Using the angle addition formula:
\[ \sin 105^\circ = \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \]
Substituting values:
\[ = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

**Step 7: Substitute back into the equation for \( c \).**
- Now substituting \( a \) and the sine values:
\[ c = (\sqrt{3} + 1) \cdot \frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3} + 1}{2\sqrt{2}}} \]

**Step 8: Simplify the expression for \( c \).**
- The \( \sqrt{2} \) cancels out:
\[ c = (\sqrt{3} + 1) \cdot \frac{2}{\sqrt{3} + 1} = 2 \]

### Final Answer:
Thus, the value of \( c \) is \( 2 \).

---