Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question:

Answer each of 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.

Solution:

Using cosine rule, we have

$\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$

$\Rightarrow \cos A=\frac{10^{2}+12^{2}-8^{2}}{2 \times 10 \times 12}$

 

$\Rightarrow \cos A=\frac{100+144-64}{240}$

$\Rightarrow \cos A=\frac{180}{240}=\frac{3}{4} \quad \ldots .(1)$

Now, using sine rule, we have

$\frac{a}{\sin A}=\frac{c}{\sin C}$

$\Rightarrow \frac{8}{\sin A}=\frac{12}{\sin \lambda A}$

$\Rightarrow \sin \lambda A=\frac{3}{2} \sin A$

$\Rightarrow \sin \lambda A=2 \times \frac{3}{4} \sin A$

$\Rightarrow \sin \lambda A=2 \sin A \cos A$   [Using (1)]

$\Rightarrow \sin \lambda A=\sin 2 A$

$\Rightarrow \lambda=2$

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