91Ó°ÊÓ

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles.
Specifically, it helps us understand and find these relationships in various triangle formations.
It's divided into several functions, with the sine, cosine, and tangent being the most common.
In our problem, we particularly use the sine function to find the area of a triangle when we know two sides and the included angle. Understanding the basic functions in trigonometry can greatly simplify solving geometry problems.

One of the core results from trigonometry is the triangle area formula when provided with two sides and the included angle.
The formula is: \[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(\gamma) \] This is particularly useful because it bypasses the more complicated Heron's formula and gives a direct computation based on the given measurements.
For our specific example, the sides are 12.9 and 6.4, and the included angle is 13.7°. The formula eases the calculation by allowing us to directly plug in these values and solve using basic multiplication and the sine function.

The sine function is a primary trigonometric function with widespread applications in geometry and physics.
It's typically defined on a right-angled triangle for an angle \( \theta \) as: \[ \text{Sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] For angles in general, especially in non-right-angled triangles, we use the unit circle definition of sine.
In our problem, calculating \( \sin(13.7^{\circ})\) involves using a calculator to get an approximation (≈0.237).
This value helps us in receiving the final area value when included in the area formula.
Understanding how to derive and use the sine value is crucial for efficiently solving triangle area problems involving angles.