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Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. It's fundamental for understanding the properties of any triangle and has applications in various fields such as physics, engineering, and navigation. The core of trigonometry lies in its functions, such as sine, cosine, and tangent, which relate the angles of a triangle to the ratios of its sides.

In the context of our exercise, trigonometry allows us to determine unknown side lengths or angles using known values. In particular, the cosine function is instrumental as it relates the adjacent side to the hypotenuse in a right-angled triangle and is used in the Law of Cosines for non-right triangles.

The cosine rule, also known as the Law of Cosines, is a crucial theorem in trigonometry that extends the Pythagorean theorem to non-right triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The general form of the cosine rule is given by the equation \( c^2 = a^2 + b^2 - 2ab\cos\gamma \) where \(a\text{, } b\text{, and } c\) are the lengths of the sides of the triangle, and \(\gamma\) is the angle opposite to side \(c\text{.}\)

The cosine rule is especially useful when we have information about two sides and the included angle, permitting the calculation of the third side. As seen in the provided exercise, the rule enables us to derive a relationship between the sides of the triangle and prove the existence of two potential side lengths based on the given parameters.

Triangle side calculation involves determining the length of an unknown side of a triangle when some other aspects of the triangle are known. This can involve applying the Pythagorean theorem in right-angled triangles or the Law of Sines and the Law of Cosines in non-right triangles. Knowledge of trigonometry and the use of trigonometric identities are essential tools in these calculations.

In the case of our exercise, by cleverly substituting the given expressions into the Law of Cosines, we arrive at a quadratic equation in terms of one side of the triangle. Simplifying this equation can show that the third side, \( c \text{,}\) can have two possible values. This demonstrates a profound concept: given certain conditions, a triangle's side can be determined in more than one way, illustrating the versatility and power of triangle side calculations in various geometric scenarios.