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.

Answer 1

Using cosine rule, we have

$$\begin{gathered} \cos A=\frac{b^2+c^2-a^2}{2bc} \\ \Rightarrow \cos A=\frac{{10}^2+{12}^2-8^2}{2\times 10\times 12} \\ \Rightarrow \cos A=\frac{100+144-64}{240} \\ \Rightarrow \cos A=\frac{180}{240}=\frac{3}{4}.....\left(1\right) \end{gathered}$$

Now, using sine rule, we have

$$\begin{gathered} \frac{a}{\sin A}=\frac{c}{\sin C} \\ \Rightarrow \frac{8}{\sin A}=\frac{12}{\sin \lambda A} \\ \Rightarrow \sin \lambda A=\frac{3}{2}\sin A \\ \Rightarrow \sin \lambda A=2\times \frac{3}{4}\sin A \end{gathered}$$
$$\begin{gathered} \Rightarrow \sin \lambda A=2\sin A\cos A\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow \sin \lambda A=\sin 2A \\ \Rightarrow \lambda =2 \end{gathered}$$