In a `DeltaABC`. If a=2,b=3 and `SinA=2//3` then what is value of angleB

Options

A. `45^(@)`
B. `90^(@)`
C. `60^(@)`
D. `30^(@)`

Answer

To solve the problem, we will use the sine rule in triangle ABC. Given the values of sides and angle, we can find angle B.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Side \( a = 2 \)
- Side \( b = 3 \)
- \( \sin A = \frac{2}{3} \)

2. **Apply the Sine Rule:**
The sine rule states that:
\[ \frac{\sin A}{a} = \frac{\sin B}{b} \]
We can rearrange this to find \( \sin B \):
\[ \sin B = \frac{\sin A \cdot b}{a} \]

3. **Substitute the Known Values:**
Substitute \( \sin A = \frac{2}{3} \), \( a = 2 \), and \( b = 3 \) into the equation:
\[ \sin B = \frac{\left(\frac{2}{3}\right) \cdot 3}{2} \]

4. **Simplify the Expression:**
\[ \sin B = \frac{2}{3} \cdot \frac{3}{2} = 1 \]

5. **Find Angle B:**
Since \( \sin B = 1 \), we can find angle B:
\[ B = \sin^{-1}(1) = 90^\circ \]

### Final Answer:
Thus, the value of angle B is \( 90^\circ \).