In `DeltaABC`, given `/_A=45^(@)`, `/_B=105^(@)`, `c=sqrt(2)`, then

Options

A. `b=sqrt(3)`, `/_C=30^(@)`
B. `b=sqrt(3)+1`, `/_C=30^(@)`
C. `b=sqrt(2)`, `/_C=30^(@)`
D. `b=sqrt(3)-1`, `/_C=30^(@)`

Answer

To solve the problem, we need to find the remaining angle and the lengths of the sides of triangle ABC, given that angle A = 45°, angle B = 105°, and side c = √2.

### Step 1: Find Angle C
We know that the sum of the angles in a triangle is 180°.

\[ \text{Angle C} = 180° - \text{Angle A} - \text{Angle B} \]
Substituting the known values:

\[ \text{Angle C} = 180° - 45° - 105° = 30° \]

### Step 2: Use the Sine Rule to Find Side A
The Sine Rule states that:

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

We can use this to find side a. Rearranging the formula gives:

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

Substituting the known values (c = √2, A = 45°, C = 30°):

\[ a = \frac{\sqrt{2} \cdot \sin(45°)}{\sin(30°)} \]

Using the known values of sine:

\[ \sin(45°) = \frac{1}{\sqrt{2}}, \quad \sin(30°) = \frac{1}{2} \]

Substituting these values:

\[ a = \frac{\sqrt{2} \cdot \frac{1}{\sqrt{2}}}{\frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 3: Use the Sine Rule to Find Side B
Now we can find side b using the Sine Rule again:

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

Substituting the known values (c = √2, B = 105°, C = 30°):

\[ b = \frac{\sqrt{2} \cdot \sin(105°)}{\sin(30°)} \]

We need to find \(\sin(105°)\). Using the sine addition formula:

\[ \sin(105°) = \sin(45° + 60°) = \sin(45°)\cos(60°) + \cos(45°)\sin(60° \]
\[ = \frac{1}{\sqrt{2}} \cdot \frac{1}{2} + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2\sqrt{2}} \]

Now substituting back into the equation for b:

\[ b = \frac{\sqrt{2} \cdot \frac{1 + \sqrt{3}}{2\sqrt{2}}}{\frac{1}{2}} = \frac{1 + \sqrt{3}}{2} \cdot 2 = 1 + \sqrt{3} \]

### Final Results
- Angle C = 30°
- Side A = 2
- Side B = 1 + √3