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Cosine is one of the fundamental trigonometric functions, often abbreviated as 'cos'. It relates the angle of a right-angled triangle to the ratio of the length of the adjacent side to the hypotenuse. In the context of circular functions, it represents the x-coordinate of a point on the unit circle.

The function is periodic with a period of \(2\pi\), meaning that \(\cos(\theta + 2\pi) = \cos(\theta)\). This property is widely used in trigonometric identities and transformations. It's important to understand that cosine, like other trigonometric functions, can be applied beyond acute angles, accommodating negative angles and those exceeding \(360^\circ\) or \(2\pi\) radians.

Cosine plays a crucial role in many formulas, including the sum and double angle formulas. These help break down more complex angles into more manageable computations.

Double angle formulas are a set of trigonometric identities that express functions of double angles (e.g., \(2A\)) in terms of single angles (e.g., \(A\)). In our exercise, we specifically use the double angle formula for cosine and sine:

• \(\cos 2A = 2\cos^2 A - 1\)
• \(\sin 2A = 2\sin A \cos A\)

These formulas are particularly helpful as they simplify trigonometric expressions and equations that involve double angles. By reducing expressions like \(\cos 3A\) into more basic forms, they allow steps to be carried out using simpler algebraic manipulations.

In the given problem, the double angle formula for cosine helps to express \(\cos 2A\) in terms of \(\cos A\), making it easier to find an expression for \(\cos 3A\). This reduces the complexity of working directly with the angle \(3A\) by converting it into combinations of less complex angles, enabling more straightforward algebraic handling.

The sum of angles formula in trigonometry allows one to calculate the cosine or sine of a sum of two angles, such as \(A + B\), in terms of the sine and cosine of the individual angles. This is known as the angle addition formula. For cosine, it is expressed as:

\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]

This formula is crucial in the solution of \(\cos 3A\) because it is used to express \(\cos(2A + A)\). Breaking \(3A\) into \(2A + A\) helps decompose the problem into obtainable parts, which can be individually calculated using known identities and formulas.

In the step-by-step solution, the sum of angles formula allowed for the substitution into components that could make use of the double angle formulas, ultimately leading to the expression of \(\cos 3A\) in terms of \(\cos A\) alone. Understanding sum of angles gives students a powerful tool to tackle complex trigonometric expressions involving angles that are combinations of other angles.