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The area of a triangle can be found using different methods, and one effective technique when two sides and the included angle are known is using the following formula:

\[ Area = \frac{1}{2}ab \sin C \]
In this formula, 'a' and 'b' are the lengths of two sides of the triangle, and 'C' is the included angle between those sides. The sine function (\( \sin \)) connects the angle with the area through trigonometry. For students encountering this, it's crucial to remember that the sine of an angle can be found using a scientific calculator or reference tables.

In the given problem, the triangle is defined by sides 'a' and 'b', and the angle 'C'. By inputting these values into the formula, you can directly calculate the area without needing to know the height. This approach is useful, especially for oblique triangles where finding heights is non-trivial. For instance, with sides \( a = 4 \) and \( b = 6 \), and the angle \( C = 120^\circ \), you would use \( \sin 120^\circ \), which gives a precise area after computation.

Trigonometry is an integral branch of mathematics that focuses on the relationships between angles and sides of triangles. In precalculus, students are introduced to various trigonometric functions, including sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), which are essential for solving a wide range of problems involving triangles.

Understanding the Unit Circle

Most trigonometric values can be understood and memorized using the unit circle. This circle has a radius of one and provides a visual angle-side relationship that is fundamental in solving trigonometric problems. For example, when you are given a specific angle, such as \( 120^\circ \), you can locate it on the unit circle to find the corresponding sine and cosine values.

The Law of Cosines

Another crucial concept in trigonometry used within precalculus is the Law of Cosines. This theorem extends beyond right triangles and provides a method to find a side length when two sides and the included angle are known, or to find an angle when all the side lengths are known.

Solving triangles involves finding the unknown angles and sides of triangles using trigonometric methods like the Law of Cosines and the Law of Sines along with other geometric principles. When it comes to non-right triangles, or oblique triangles, the Law of Cosines is particularly valuable.

In our exercise, to solve for the third side 'c' when sides 'a' and 'b', and angle 'C' are given, we use the Law of Cosines:
\[ c = \sqrt{a^2 + b^2 - 2ab \cos C} \]
The next step involves substituting the known values into this formula. After obtaining the value of 'c', the triangle's sides are all known, and other properties like area can be calculated.
This process is a part of solving the triangle, which not only helps in understanding the geometry but also lays the foundation for more advanced topics in trigonometry and calculus. For students, mastering the skill of solving triangles is a substantial achievement in their study of precalculus.