Understanding the Summation
To demonstrate that the summation a(sin B - sin C) = 0, we need to analyze the conditions under which this statement holds true.
Key Concepts
- Angles in a Triangle: In a triangle, the sum of the angles is always 180 degrees. Therefore, if A, B, and C are the angles of a triangle, we have A + B + C = 180°.
- Sine Function Properties: The sine function is periodic and symmetrical. Specifically, sin(180° - x) = sin(x). This property is crucial in this context.
Applying the Sine Rule
- Sine Rule Statement: According to the sine rule, for any triangle, we have a/sin A = b/sin B = c/sin C. This means that the ratios of the sides of a triangle to the sine of their opposite angles are constant.
- Implication of the Sine Rule: If we take a = k * sin A, b = k * sin B, and c = k * sin C for some constant k, we can express the relationship among the sides and angles clearly.
Summation Analysis
- Expression Breakdown: The expression a(sin B - sin C) can be viewed as k * sin A * (sin B - sin C).
- Condition for Zero: For the entire expression to equal zero, either a = 0, or (sin B - sin C) must equal 0. Since sin B = sin C occurs when B = C (which is possible in an isosceles triangle), the expression holds true.
Conclusion
Thus, the summation a(sin B - sin C) = 0 when the angles B and C are equal, confirming the consistency of the sine rule and the properties of triangles.

Let b and c =0