The value of the expression `sinx+ siny + sinz` where x, y, z are real numbers satisfying `x+y+z= 180^@` is

Options

A. (a)Positive
B. (b)Zero
C. (c)`-3`
D. (d)Negative

Answer

To find the value of the expression \( \sin x + \sin y + \sin z \) where \( x + y + z = 180^\circ \), we can use the properties of sine and the given constraint.

### Step-by-Step Solution:

1. **Use the given constraint**:
Since \( x + y + z = 180^\circ \), we can express \( z \) in terms of \( x \) and \( y \):
\[ z = 180^\circ - (x + y) \]

2. **Apply the sine identity**:
We know that \( \sin(180^\circ - \theta) = \sin \theta \). Therefore, we can rewrite \( \sin z \):
\[ \sin z = \sin(180^\circ - (x + y)) = \sin(x + y) \]

3. **Substitute back into the expression**:
Now we can substitute \( \sin z \) back into the original expression:
\[ \sin x + \sin y + \sin z = \sin x + \sin y + \sin(x + y) \]

4. **Use the sine addition formula**:
The sine addition formula states that:
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]
Thus, we can rewrite our expression:
\[ \sin x + \sin y + \sin(x + y) = \sin x + \sin y + (\sin x \cos y + \cos x \sin y) \]

5. **Combine like terms**:
Rearranging gives us:
\[ \sin x (1 + \cos y) + \sin y (1 + \cos x) \]

6. **Find the maximum value**:
The maximum value of \( \sin x + \sin y + \sin z \) occurs when \( x = y = z = 60^\circ \), leading to:
\[ \sin 60^\circ + \sin 60^\circ + \sin 60^\circ = 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \]

### Conclusion:
The maximum value of the expression \( \sin x + \sin y + \sin z \) given the constraint \( x + y + z = 180^\circ \) is:
\[ \frac{3\sqrt{3}}{2} \]