If `a=1+sqrt(3), b=2` and `c=sqrt(2)`, then the measure of `angleB` is

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. `15^(@)`

Answer

To find the measure of angle \( B \) given the values \( a = 1 + \sqrt{3} \), \( b = 2 \), and \( c = \sqrt{2} \), we will use the cosine rule. The cosine rule states that:

\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]

**Step 1: Calculate \( a^2 \), \( b^2 \), and \( c^2 \)**

- \( a^2 = (1 + \sqrt{3})^2 = 1^2 + 2 \cdot 1 \cdot \sqrt{3} + (\sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \)
- \( b^2 = 2^2 = 4 \)
- \( c^2 = (\sqrt{2})^2 = 2 \)

**Step 2: Substitute the values into the cosine formula**

Now substitute \( a^2 \), \( b^2 \), and \( c^2 \) into the cosine formula:

\[ \cos B = \frac{(4 + 2\sqrt{3}) + 2 - 4}{2 \cdot (1 + \sqrt{3}) \cdot \sqrt{2}} \]

**Step 3: Simplify the numerator**

The numerator simplifies as follows:

\[ 4 + 2\sqrt{3} + 2 - 4 = 2 + 2\sqrt{3} \]

So we have:

\[ \cos B = \frac{2 + 2\sqrt{3}}{2(1 + \sqrt{3})\sqrt{2}} \]

**Step 4: Factor out the common terms**

Factor out 2 from the numerator:

\[ \cos B = \frac{2(1 + \sqrt{3})}{2(1 + \sqrt{3})\sqrt{2}} \]

**Step 5: Cancel the common terms**

The \( 2(1 + \sqrt{3}) \) in the numerator and denominator cancels out:

\[ \cos B = \frac{1}{\sqrt{2}} \]

**Step 6: Find the angle \( B \)**

We know that:

\[ \cos B = \frac{1}{\sqrt{2}} \implies B = \frac{\pi}{4} \text{ or } 45^\circ \]

Thus, the measure of angle \( B \) is \( 45^\circ \).

### Summary of Steps:
1. Calculate \( a^2 \), \( b^2 \), and \( c^2 \).
2. Substitute these values into the cosine formula.
3. Simplify the numerator.
4. Factor out common terms.
5. Cancel the common terms.
6. Determine the angle using the cosine value.