Question

How do you use the Law of Cosines and Sines in various ambiguous cases?

Answer 1

Hint: Here in this question we have been asked to explain how we use the Law of Cosines and Sines in various ambiguous cases. The Law of Cosines and Sines is generally used if at least any three elements of the triangle elements (except the case the three elements are the three angles, in that case the triangle is not unique) are known to us.

Complete step by step solution:
Now considering from the question we have been asked to explain how we use the Law of Cosines and Sines in various ambiguous cases.
From the basic concepts of triangles we know that the Law of Cosines and Sines is generally used if at least any three elements of the triangle elements (except the case the three elements are the three angles, in that case the triangle is not unique) are known to us.

The Law of Cosines is used when we are aware of two sides and their enclosed angle. In all other cases we use the Law of Sines.
The Law of Sines establishes a relationship between the angles and the side lengths of $\Delta ABC$ given as \[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\] .
The Law of Cosines establishes a relationship between the angles and the side lengths of $\Delta ABC$ given as
$\begin{align}
  & {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\cos C \\
 & {{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A \\
 & {{b}^{2}}={{a}^{2}}+{{c}^{2}}-2ac\cos B \\
\end{align}$ .

Note: While answering questions of this type we should be sure with the concepts that we are going to apply. This is a very simple and easy question and can be answered accurately in a short span of time. This is a completely theoretical based question and it can be answered if we have a basic concept. Very few mistakes are possible in questions of this type. If someone has confused and written Law of Cosines for Law of Sines and vice-versa then they will end up having a wrong conclusion.