In a `triangleABC, s/R=`

Options

A. sin A+sin B+ sin C
B. cos A +cos B+ cos C
C. 2
D. none of these

Answer

To solve the problem of finding the value of \( \frac{s}{R} \) in triangle \( ABC \), we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Terms**:
- Let \( a, b, c \) be the lengths of the sides opposite to angles \( A, B, C \) respectively.
- \( s \) is the semi-perimeter of the triangle, defined as \( s = \frac{a + b + c}{2} \).
- \( R \) is the circumradius of the triangle.

2. **Using the Sine Rule**:
- According to the sine rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
- From this, we can express the sides in terms of the circumradius \( R \):
\[ a = 2R \sin A, \quad b = 2R \sin B, \quad c = 2R \sin C \]

3. **Finding the Semi-Perimeter**:
- The semi-perimeter \( s \) can be expressed as:
\[ s = \frac{a + b + c}{2} = \frac{2R \sin A + 2R \sin B + 2R \sin C}{2} \]
- Simplifying this gives:
\[ s = R (\sin A + \sin B + \sin C) \]

4. **Calculating \( \frac{s}{R} \)**:
- Now, we can find \( \frac{s}{R} \):
\[ \frac{s}{R} = \frac{R (\sin A + \sin B + \sin C)}{R} \]
- This simplifies to:
\[ \frac{s}{R} = \sin A + \sin B + \sin C \]

5. **Final Result**:
- Therefore, the value of \( \frac{s}{R} \) is:
\[ \frac{s}{R} = \sin A + \sin B + \sin C \]

### Conclusion:
The final answer is:
\[ \frac{s}{R} = \sin A + \sin B + \sin C \]