In `Delta ABC`, if a=2, b=3 and `sin A =(2)/(3)`. Then, cos C is equal to

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(2)/(sqrt(13))`
D. `(1)/(sqrt(13))`

Answer

To solve the problem step by step, we will use the sine rule and properties of triangles.

### Step-by-Step Solution:

1. **Given Information**:
- In triangle ABC, we have:
- \( a = 2 \)
- \( b = 3 \)
- \( \sin A = \frac{2}{3} \)

2. **Using the Sine Rule**:
The sine rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
We can express \( \sin B \) in terms of \( \sin A \):
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Rearranging gives:
\[ \sin B = \frac{b \cdot \sin A}{a} \]

3. **Substituting the Values**:
Now, substituting the known values:
\[ \sin B = \frac{3 \cdot \frac{2}{3}}{2} \]
Simplifying this:
\[ \sin B = \frac{3 \cdot 2}{3 \cdot 2} = 1 \]

4. **Finding Angle B**:
Since \( \sin B = 1 \), we have:
\[ B = \sin^{-1}(1) = 90^\circ \]

5. **Finding Angle C**:
Using the property of triangles, the sum of angles in a triangle is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Substituting the known values:
\[ A + 90^\circ + C = 180^\circ \]
Rearranging gives:
\[ C = 180^\circ - 90^\circ - A = 90^\circ - A \]

6. **Finding \( \cos C \)**:
We know that:
\[ C = 90^\circ - A \]
Therefore, using the co-function identity:
\[ \cos C = \sin A \]

7. **Substituting the Value of \( \sin A \)**:
From the given information:
\[ \sin A = \frac{2}{3} \]
Thus:
\[ \cos C = \frac{2}{3} \]

### Final Answer:
\[ \cos C = \frac{2}{3} \]