In triangle ABC , if `AB=2,BC=4 and AC=5,` then the value of `(sinA-sinB)/(sinC)` is equal to

Options

A. `1//2`
B. `-1//2`
C. `2//5`
D. `-2//5`

Answer

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

Given:
- \( AB = c = 2 \)
- \( BC = a = 4 \)
- \( AC = b = 5 \)

We need to find the value of \( \frac{\sin A - \sin B}{\sin C} \).

### Step 1: Use the Sine Rule
According to the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

From this, we can express \( \sin A \), \( \sin B \), and \( \sin C \):
\[ \sin A = \frac{a \cdot \sin C}{c} \]
\[ \sin B = \frac{b \cdot \sin C}{c} \]

### Step 2: Substitute Values
Substituting the known values into the equations:
\[ \sin A = \frac{4 \cdot \sin C}{2} = 2 \sin C \]
\[ \sin B = \frac{5 \cdot \sin C}{2} = \frac{5}{2} \sin C \]

### Step 3: Calculate \( \sin A - \sin B \)
Now, we can find \( \sin A - \sin B \):
\[ \sin A - \sin B = 2 \sin C - \frac{5}{2} \sin C \]
\[ = \left(2 - \frac{5}{2}\right) \sin C = \left(\frac{4}{2} - \frac{5}{2}\right) \sin C = -\frac{1}{2} \sin C \]

### Step 4: Substitute into the Original Expression
Now we substitute \( \sin A - \sin B \) into the expression \( \frac{\sin A - \sin B}{\sin C} \):
\[ \frac{\sin A - \sin B}{\sin C} = \frac{-\frac{1}{2} \sin C}{\sin C} \]
\[ = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( \frac{\sin A - \sin B}{\sin C} \) is:
\[ \boxed{-\frac{1}{2}} \]