If `b=sqrt(3), c=1` and `angleA=30^(@)`, then the measure of `angleB` is

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. `150^(@)`

Answer

To find the measure of angle B in triangle ABC, where \( b = \sqrt{3} \), \( c = 1 \), and \( \angle A = 30^\circ \), we can apply the sine rule. The sine rule states that:

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

### Step-by-Step Solution:

1. **Identify the known values**:
- \( b = \sqrt{3} \)
- \( c = 1 \)
- \( \angle A = 30^\circ \)

2. **Use the sine rule**:
According to the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
We can also express \( c \) in terms of \( \angle C \):
\[ \frac{c}{\sin C} = \frac{b}{\sin B} \]

3. **Calculate \( \sin A \)**:
Since \( \angle A = 30^\circ \):
\[ \sin A = \sin 30^\circ = \frac{1}{2} \]

4. **Set up the equation using the sine rule**:
Substitute the known values into the sine rule:
\[ \frac{a}{\frac{1}{2}} = \frac{\sqrt{3}}{\sin B} \]

5. **Rearranging the equation**:
This gives:
\[ a = \frac{\sqrt{3}}{\sin B} \cdot \frac{1}{2} \]
or:
\[ \sin B = \frac{\sqrt{3}}{2} \cdot \frac{1}{a} \]

6. **Find the relationship with \( \sin B \)**:
We know that \( \sin B = \frac{\sqrt{3}}{2} \) corresponds to an angle of \( 60^\circ \):
\[ \sin B = \frac{\sqrt{3}}{2} \implies B = 60^\circ \]

7. **Conclusion**:
Therefore, the measure of angle B is:
\[ \angle B = 60^\circ \]