91Ó°ÊÓ

Solving a triangle involves finding the unknown sides and angles when only some of these components are known. In the given problem, we start with two given angles (\( \alpha \) and \( \gamma \)) and one given side (\( c \)). To solve the triangle, we need to find the remaining angle (\( \beta \)) and the other two sides (\( a \) and \( b \)). This step-by-step approach ensures that all elements are determined accurately.
Let's begin by understanding the basic steps:

• Find the missing angle using the angle sum property
• Use the Law of Sines to find the missing sides

By following these steps, we can methodically arrive at the complete solution of the triangle.

The angle sum property of a triangle states that the sum of the interior angles is always 180°. This rule is fundamental for solving triangles since it provides a way to find a missing angle when the other two are known.
Using this property for our problem:
Given \( \alpha = 10.3^{\circ} \) and \( \gamma = 143.7^{\circ} \), we find the missing angle \( \beta \) by applying:
\[ \beta = 180^{\circ} - \alpha - \gamma \] Substituting the given values, we get:
\[ \beta = 180^{\circ} - 10.3^{\circ} - 143.7^{\circ} = 26^{\circ} \] This calculation confirms that the sum of the angles is indeed 180°, emphasizing the usefulness of the angle sum property in solving triangles.

Trigonometric functions, such as sine, cosine, and tangent, are essential tools in solving triangles. In our specific problem, we use the Law of Sines, which relates the ratios of the sides of a triangle to the sines of their opposite angles.
The Law of Sines states: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] This allows us to find an unknown side when the corresponding angle and the other side-angle pairs are known.
For side \( a \): \[ a = \frac{c \cdot \sin(\alpha)}{\sin(\gamma)} \] Substituting the given values, we find:
\[ a = \frac{48.3 \cdot \sin(10.3^{\circ})}{\sin(143.7^{\circ})} \approx 14.71 \] Similarly, for side \( b \): \[ b = \frac{c \cdot \sin(\beta)}{\sin(\gamma)} \] After substituting the known values, we get:
\[ b = \frac{48.3 \cdot 0.4384}{0.5878} \approx 36.04 \] Using trigonometric functions simplifies the process of solving a triangle, making it more accessible and accurate.