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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 a $\triangle \mathrm{ABC}$, if $b=\sqrt{3}, c=1$ and $\angle A=30^{\circ}$, find $a$.

Solution:

In $\triangle \mathrm{ABC}, b=\sqrt{3}, c=1$ and $\angle A=30^{\circ}$.

Using cosine formula, we have

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

$\Rightarrow \cos 30^{\circ}=\frac{(\sqrt{3})^{2}+(1)^{2}-a^{2}}{2 \times \sqrt{3} \times 1}$

$\Rightarrow \frac{\sqrt{3}}{2}=\frac{4-a^{2}}{2 \sqrt{3}}$

$\Rightarrow 3=4-a^{2}$

$\Rightarrow a^{2}=4-3=1$

$\Rightarrow a=1$

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