In a `DeltaABC`, if a 100, c = 100 `sqrt(2)` and `A=30^(@)`, then B equals to

Options

A. `45^(@) or 135^(@)`
B. `105^(@) or 15^(@)`
C. `60^(@) or 120^(@)`
D. none of these

Explanation

To solve the problem, we will use the sine rule in triangle ABC. Given the values:

- Side \( a = 100 \)
- Side \( c = 100\sqrt{2} \)
- Angle \( A = 30^\circ \)

We need to find angle \( B \).

### Step 1: Apply the Sine Rule
The sine rule states that:

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

Substituting the known values into the sine rule:

\[ \frac{100}{\sin 30^\circ} = \frac{100\sqrt{2}}{\sin C} \]

### Step 2: Calculate \(\sin 30^\circ\)
We know that:

\[ \sin 30^\circ = \frac{1}{2} \]

### Step 3: Substitute \(\sin 30^\circ\) into the equation
Now substituting \(\sin 30^\circ\) into the equation:

\[ \frac{100}{\frac{1}{2}} = \frac{100\sqrt{2}}{\sin C} \]

This simplifies to:

\[ 200 = \frac{100\sqrt{2}}{\sin C} \]

### Step 4: Rearrange to find \(\sin C\)
Now, we can rearrange this equation to solve for \(\sin C\):

\[ \sin C = \frac{100\sqrt{2}}{200} \]

This simplifies to:

\[ \sin C = \frac{\sqrt{2}}{2} \]

### Step 5: Find angle \( C \)
Now we need to find the angle corresponding to \(\sin C = \frac{\sqrt{2}}{2}\). The angles that satisfy this equation are:

\[ C = 45^\circ \quad \text{or} \quad C = 135^\circ \]

### Step 6: Calculate angle \( B \) for both cases
1. **Case 1: \( C = 45^\circ \)**

Using the triangle angle sum property:

\[ A + B + C = 180^\circ \]

Substituting the known values:

\[ 30^\circ + B + 45^\circ = 180^\circ \]

Solving for \( B \):

\[ B = 180^\circ - 30^\circ - 45^\circ = 105^\circ \]

2. **Case 2: \( C = 135^\circ \)**

Again using the triangle angle sum property:

\[ 30^\circ + B + 135^\circ = 180^\circ \]

Solving for \( B \):

\[ B = 180^\circ - 30^\circ - 135^\circ = 15^\circ \]

### Conclusion
Thus, angle \( B \) can be either \( 105^\circ \) or \( 15^\circ \).

### Final Answer
The possible values for angle \( B \) are \( 105^\circ \) or \( 15^\circ \).