In A `DeltaABC ` , if a = 2 , b = 3 and `sin A = 2/3 ` then `angleB` is (a)`90^(@)` (b)`80^(@)` (c)`110^(@)` (d)`140^(@)`

Options

A. `90^(@)`
B. `80^(@)`
C. `110^(@)`
D. `140^(@)`

Answer

To solve the problem, we will use the sine rule in triangle ABC. The sine rule states that:

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

Given:
- \( a = 2 \)
- \( b = 3 \)
- \( \sin A = \frac{2}{3} \)

We need to find \( \angle B \).

### Step 1: Apply the Sine Rule
Using the sine rule, we can express \( \sin B \) in terms of \( a \), \( b \), and \( \sin A \):

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

Substituting the known values:

\[ \frac{2}{\frac{2}{3}} = \frac{3}{\sin B} \]

### Step 2: Simplify the Left Side
Now, simplify the left side:

\[ \frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 3 \]

So, we have:

\[ 3 = \frac{3}{\sin B} \]

### Step 3: Cross Multiply
Cross multiplying gives us:

\[ 3 \sin B = 3 \]

### Step 4: Solve for \( \sin B \)
Dividing both sides by 3:

\[ \sin B = 1 \]

### Step 5: Determine \( \angle B \)
The sine of an angle is equal to 1 at:

\[ B = 90^\circ \]

### Final Answer
Thus, the angle \( B \) is:

\[ \boxed{90^\circ} \]

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