In Triangle Abc, Prove the Following: a ( Sin B − Sin C ) + ( Sin C − Sin a ) + C ( Sin a − Sin B ) = 0

प्रश्न

In triangle ABC, prove the following:

\[a \left( \sin B - \sin C \right) + \left( \sin C - \sin A \right) + c \left( \sin A - \sin B \right) = 0\]

उत्तर

Let \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\]

Then,
Consider the LHS of the equation

\[a \left( \sin B - \sin C \right) + \left( \sin C - \sin A \right) + c \left( \sin A - \sin B \right) = 0\]

\[LHS = a\left( \sin B - \sin C \right) + b\left( \sin C - \sin A \right) + c\left( sin A - sin B \right)\]
\[ = k\sin A\left( \sin B - \sin C \right) + k\sin B\left( \sin C - \sin A \right) + k\sin C\left( sin A - sin B \right) \]
\[ = k\sin A\sin B - k\sin A\sin C + k\sin B\sin C - k\sin B\sin A + ksin C\sinA - k\sinC\sinB\]
\[ = 0 = RHS\]
\[\text{ Hence proved } . \]

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Sine and Cosine Formulae and Their Applications

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