Solving a Triangle Using the Sine Law: A Step-by-Step Guide

In this article, we will explore how to solve a triangle given two sides and one angle using the Sine Law. The Sine Law is an essential tool in geometry and trigonometry, providing a relationship between the sides and angles of any triangle. Let's dive into the process with a practical example.

Understanding the Sine Law

The Sine Law is a formula that relates the lengths of the sides of a triangle to their opposite angles. It is expressed as:

[frac{a}{sin A} frac{b}{sin B} frac{c}{sin C}]

Where (a), (b), and (c) are the lengths of the sides opposite angles (A), (B), and (C) respectively.

Given Information

Consider a triangle ABC, with the following given information:

(AB c 12 text{ cm})(BC a 5 text{ cm})(C 40^circ)

Applying the Sine Law

According to the Sine Law, we can set up the following equation:

[frac{a}{sin A} frac{c}{sin C}]

Substituting the given values, we get:

[frac{5}{sin A} frac{12}{sin 40^circ}]

Step-by-Step Calculation

Let's break down the calculation step by step:

Substitute the given sine value of (40^circ):[sin 40^circ 0.6428]Thus, the equation becomes:[frac{5}{sin A} frac{12}{0.6428}]Let's solve for (sin A):[sin A frac{5 times 0.6428}{12} frac{3.214}{12} 0.2678]Now, we need to find the angle (A):[sin^{-1}(0.2678) A]Using a calculator, we find:[A approx 15.53^circ]To find angle (B), we use the fact that the sum of the angles in a triangle is (180^circ):[B 180^circ - A - C 180^circ - 15.53^circ - 40^circ 124.47^circ]

Hence, the angles of the triangle are approximately (A 15.53^circ), (B 124.47^circ), and (C 40^circ).

Verification and Application

It is always beneficial to verify the side-clengths with the found angle to ensure consistency with the given information. However, for now, we have successfully solved the triangle using the Sine Law.

Conclusion

Solving a triangle with the Sine Law involves setting up and solving the equation that relates the sides to the sines of the angles. By following these steps, you can solve any triangle given two sides and one included angle. This method is particularly useful in various fields, including engineering, physics, and surveying.

Feel free to practice more problems and use the Sine Law to solve triangles. Understanding these concepts will greatly enhance your problem-solving skills in geometry and trigonometry.