Finding the Length of AC in Triangle ABC: A Comprehensive Guide

Understanding how to solve geometry problems involving triangles is a fundamental part of trigonometry. In this article, we will delve into a specific problem: determining the length of side AC in triangle ABC, given that AB 8 cm, BC 10 cm, and angle ACB 42 degrees. This involves applying the sine law effectively and understanding the underlying principles of triangle geometry.

Problem Overview

The problem at hand is to find the length of side AC in triangle ABC, given the following information:

AB 8 cm BC 10 cm Angle ACB 42 degrees

Let's break down the solution step-by-step.

Solution Using the Sine Law

The sine law is a powerful tool in trigonometry. It states that in any triangle with sides a, b, and c, and corresponding angles A, B, and C, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. In our problem, we can use the sine law to find the length of side AC.

Step-by-Step Solution

First, we set up the sine law equation: tequation: 8 cm / sin(42°) 10 cm / sin(∠CAB) Solving for sin(∠CAB): t

t

sin(∠CAB) (10 cm * sin(42°)) / 8 cm

t

Using a calculator, we find: t

t

sin(∠CAB) ≈ 0.8364

t

Next, we find the angle CAB using the inverse sine function: t

t

∠CAB arcsin(0.8364) ≈ 56.76°

t

To find the measure of angle ABC, we use the fact that the sum of the angles in a triangle is 180°: t

t

∠ABC 180° - ∠ACB - ∠CAB

t

t

t

∠ABC 180° - 42° - 56.76° ≈ 81.24°

t

Finally, we use the sine law again to find the length of side AC: t

t

AC / sin(81.24°) 8 cm / sin(42°)

t

AC (8 cm * sin(81.24°)) / sin(42°)

t

t

t

AC ≈ 11.82 cm

t

Therefore, the length of side AC is approximately 11.82 cm.

Key Concepts and Applications

Solving this problem demonstrates the power of the sine law in triangle geometry. Understanding the sine law and its applications can be crucial in various fields, such as architecture, engineering, and physics. By mastering these concepts, you can solve complex real-world problems involving angles and distances.

Further Reading and Practice

For more information on triangle ABC and trigonometry, visit the resources below:

Math Is Fun - Trigonometry Khan Academy - Law of Cosines Soft Schools - Trigonometric Functions

To practice solving more geometric problems, consider working through additional exercises and real-world scenarios that involve angles and distances.

Conclusion

By applying the sine law to find the length of side AC in triangle ABC, we have demonstrated a practical and essential skill in trigonometry. Understanding and mastering geometry problems like this one will greatly enhance your mathematical problem-solving abilities and prepare you for more complex challenges in your chosen field.