In `a Delta ABC,` if `angle C =105^(@), angleB=45^(@)` and length of side AC =2 units, then the length of th side AB is equal to

Options

A. `sqrt2`
B. `sqrt3`
C. `sqrt2+1`
D. `sqrt3+1`

Answer

To find the length of side AB in triangle ABC, we will follow these steps:

### Step 1: Find the third angle A
Given:
- Angle C = 105°
- Angle B = 45°

Using the angle sum property of triangles:
\[ \text{Angle A} = 180° - \text{Angle B} - \text{Angle C} \]
Substituting the values:
\[ \text{Angle A} = 180° - 45° - 105° = 30° \]

### Step 2: Use the Sine Rule
The Sine Rule states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Where:
- \( a = BC \)
- \( b = AC = 2 \) units
- \( c = AB \)

We can express the Sine Rule in terms of the known values:
\[ \frac{AB}{\sin C} = \frac{AC}{\sin B} \]

### Step 3: Substitute Known Values
Substituting the known values into the Sine Rule:
\[ \frac{AB}{\sin 105°} = \frac{2}{\sin 45°} \]

### Step 4: Calculate \( K \)
To find \( K \):
\[ K = \frac{\sin 45°}{2} \]
Since \( \sin 45° = \frac{1}{\sqrt{2}} \):
\[ K = \frac{1/\sqrt{2}}{2} = \frac{1}{2\sqrt{2}} \]

### Step 5: Find AB
Now, we can express AB in terms of K:
\[ AB = K \cdot \sin 105° \]
Substituting the value of K:
\[ AB = \frac{1}{2\sqrt{2}} \cdot \sin 105° \]

### Step 6: Simplify \( \sin 105° \)
Using the identity \( \sin(90° + \theta) = \cos \theta \):
\[ \sin 105° = \cos 15° \]
Thus:
\[ AB = \frac{1}{2\sqrt{2}} \cdot \cos 15° \]

### Step 7: Calculate \( \cos 15° \)
Using the cosine subtraction formula:
\[ \cos 15° = \cos(45° - 30°) = \cos 45° \cos 30° + \sin 45° \sin 30° \]
Substituting the known values:
\[ \cos 15° = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) \]
\[ = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 8: Final Calculation of AB
Substituting \( \cos 15° \) back into the equation for AB:
\[ AB = \frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{3} + 1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{4} \]

### Conclusion
Thus, the length of side AB is:
\[ AB = \frac{\sqrt{3} + 1}{4} \text{ units} \]

---