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Question
Answer the following questions in one word or one sentence or as per exact requirement of the question.
In ∆ABC, if a = 8, b = 10, c = 12 and C = λA, find the value of λ.
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Solution
Using cosine rule, we have
\[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]
\[ \Rightarrow \cos A = \frac{{10}^2 + {12}^2 - 8^2}{2 \times 10 \times 12}\]
\[ \Rightarrow \cos A = \frac{100 + 144 - 64}{240}\]
\[ \Rightarrow \cos A = \frac{180}{240} = \frac{3}{4} . . . . . \left( 1 \right)\]
Now, using sine rule, we have
\[ \Rightarrow \cos A = \frac{{10}^2 + {12}^2 - 8^2}{2 \times 10 \times 12}\]
\[ \Rightarrow \cos A = \frac{100 + 144 - 64}{240}\]
\[ \Rightarrow \cos A = \frac{180}{240} = \frac{3}{4} . . . . . \left( 1 \right)\]
\[\Rightarrow \sin\lambda A = 2\sin A\co sA \left[ \text{ Using }\left( 1 \right) \right]\]
\[ \Rightarrow \sin\lambda A = \sin2A\]
\[ \Rightarrow \lambda = 2\]
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