In a triangle ABC if a = 2, b = 3 and sin A = `(2)/(3)`, then what is angle B equal to?

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `(pi)/(3)`
D. `(pi)/(6)`

Explanation

To find angle B in triangle ABC given that \( a = 2 \), \( b = 3 \), and \( \sin A = \frac{2}{3} \), we can use the Law of Sines. The Law of Sines states that:

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

### Step 1: Substitute the known values into the Law of Sines

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

Substituting these values into the formula gives us:

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

### Step 2: Simplify the left side of the equation

Calculating the left side:

\[ \frac{\frac{2}{3}}{2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \]

So we have:

\[ \frac{1}{3} = \frac{\sin B}{3} \]

### Step 3: Cross-multiply to solve for \( \sin B \)

Cross-multiplying gives:

\[ 1 \cdot 3 = \sin B \cdot 3 \]

This simplifies to:

\[ \sin B = 1 \]

### Step 4: Determine angle B

The sine of an angle is equal to 1 at:

\[ B = \frac{\pi}{2} \text{ (or 90 degrees)} \]

### Conclusion

Thus, angle B is:

\[ B = \frac{\pi}{2} \]