If in `Delta ABC, /_ C=pi/8, a= sqrt2 and b= sqrt(2+sqrt2)` then find the measure of angle `A` (in degree).

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
We have:
- Angle \( C = \frac{\pi}{8} \)
- Side \( a = \sqrt{2} \)
- Side \( b = \sqrt{2 + \sqrt{2}} \)

### Step 2: Use the cosine and sine formulas
We will use the following formulas:
1. \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \)
2. \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \)

### Step 3: Calculate \( \cos C \) and \( \sin C \)
Substituting \( \theta = \frac{\pi}{8} \):
- For \( \cos C \):
\[ \cos^2\left(\frac{\pi}{8}\right) = \frac{1 + \cos\left(\frac{\pi}{4}\right)}{2} = \frac{1 + \frac{1}{\sqrt{2}}}{2} = \frac{\sqrt{2} + 1}{2\sqrt{2}} \]

- For \( \sin C \):
\[ \sin^2\left(\frac{\pi}{8}\right) = \frac{1 - \cos\left(\frac{\pi}{4}\right)}{2} = \frac{1 - \frac{1}{\sqrt{2}}}{2} = \frac{\sqrt{2} - 1}{2\sqrt{2}} \]

### Step 4: Use the cosine rule
Using the cosine rule:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the known values:
\[ \frac{\sqrt{2} + 1}{2\sqrt{2}} = \frac{(\sqrt{2})^2 + (\sqrt{2 + \sqrt{2}})^2 - c^2}{2 \cdot \sqrt{2} \cdot \sqrt{2 + \sqrt{2}}} \]
Calculating \( a^2 \) and \( b^2 \):
\[ a^2 = 2, \quad b^2 = 2 + \sqrt{2} \]
Thus,
\[ \frac{\sqrt{2} + 1}{2\sqrt{2}} = \frac{2 + (2 + \sqrt{2}) - c^2}{2 \cdot \sqrt{2} \cdot \sqrt{2 + \sqrt{2}}} \]

### Step 5: Simplify and solve for \( c^2 \)
Cross-multiplying and simplifying:
\[ (\sqrt{2} + 1) \cdot 2 \cdot \sqrt{2} \cdot \sqrt{2 + \sqrt{2}} = 2 + 2 + \sqrt{2} - c^2 \]
This leads to:
\[ c^2 = 4 + \sqrt{2} - (2\sqrt{2} + 2) \]
Thus,
\[ c^2 = 2 - \sqrt{2} \]
Taking the square root gives:
\[ c = \sqrt{2 - \sqrt{2}} \]

### Step 6: Use the sine rule to find \( A \)
Using the sine rule:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
Substituting the known values:
\[ \frac{\sqrt{2}}{\sin A} = \frac{\sqrt{2 - \sqrt{2}}}{\sin\left(\frac{\pi}{8}\right)} \]
From earlier calculations, we know:
\[ \sin\left(\frac{\pi}{8}\right) = \frac{1}{2\sqrt{2} - \sqrt{2}} = \frac{1}{2\sqrt{2} - \sqrt{2}} \]
Thus,
\[ \sin A = \frac{\sqrt{2} \cdot \sin\left(\frac{\pi}{8}\right)}{\sqrt{2 - \sqrt{2}}} \]

### Step 7: Calculate the angle \( A \)
From the sine value, we find:
\[ \sin A = \frac{1}{\sqrt{2}} \implies A = 45^\circ \text{ or } 135^\circ \]

### Final Answer
The measure of angle \( A \) is \( 45^\circ \) or \( 135^\circ \).

---