If A,B,C,D are angles of a quadrilateral, then sin A + sin (B+ C + D) = (i) 0 (ii) 1 (iii) 2 (iv) None of these

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To solve the problem, we need to analyze the relationship between the angles of a quadrilateral and the sine function.

### Step-by-Step Solution:

1. **Understanding the angles of a quadrilateral**:
The sum of the angles of a quadrilateral is given by the formula:
\[ A + B + C + D = 360^\circ \quad \text{(or } 2\pi \text{ radians)} \]

2. **Rearranging the angles**:
We can express \(B + C + D\) in terms of \(A\):
\[ B + C + D = 360^\circ - A \quad \text{(or } 2\pi - A \text{ radians)} \]

3. **Using the sine function**:
We need to evaluate \( \sin A + \sin(B + C + D) \). Substituting the expression for \(B + C + D\):
\[ \sin A + \sin(360^\circ - A) \quad \text{(or } \sin(2\pi - A) \text{)} \]

4. **Applying the sine identity**:
We know from trigonometric identities that:
\[ \sin(360^\circ - A) = -\sin A \quad \text{(or } \sin(2\pi - A) = -\sin A \text{)} \]

5. **Combining the sine values**:
Now we can combine the sine values:
\[ \sin A + \sin(360^\circ - A) = \sin A - \sin A = 0 \]

6. **Conclusion**:
Therefore, we find that:
\[ \sin A + \sin(B + C + D) = 0 \]
The correct answer is (i) 0.