How to Determine the Lengths of All Sides in a Right-Angled Triangle with One Side and an Included Angle

Knowing how to determine the lengths of all sides in a right-angled triangle when you have only one side and the included angle is an invaluable skill for many practical applications. Whether you are a math student, a professional engineer, or someone solving real-world problems, this guide will walk you through the process step by step using trigonometric principles.

Understanding the Components of a Right-Angled Triangle

A right-angled triangle consists of:

The right angle (90°) The hypotenuse, which is the longest side and the one opposite the right angle Two legs, labeled as the opposite and adjacent sides relative to an angle

Key Concepts in Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In a right-angled triangle, the three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

Sine (sin)

The sine of an angle is the ratio of the length of the opposite side to the hypotenuse:

sin(α) opposite / hypotenuse

Cosine (cos)

The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse:

cos(α) adjacent / hypotenuse

Tangent (tan)

The tangent of an angle is the ratio of the length of the opposite side to the adjacent side:

tan(α) opposite / adjacent

Determining Side Lengths Given One Side and an Included Angle

Let's consider a scenario where you know the length of one side (s) and the measure of the included angle (α) in a right-angled triangle. The goal is to find the lengths of the other two sides: the unknown side and the hypotenuse.

Example: Finding the Unknown Side (x) when the Opposite Side is Known

Suppose you know the length of the opposite side (s) and the measure of the included angle (α). To find the length of the unknown side (x) adjacent to the angle, use the tangent function.

x s · tan(α)

Example: Finding the Hypotenuse (h) when the Adjacent Side is Known

Suppose you know the length of the adjacent side (s) and the measure of the included angle (α). To find the length of the hypotenuse (h), use the cosine function.

h s / cos(α)

Step-by-Step Guide

Identify the known side and angle in the right-angled triangle. Choose the appropriate trigonometric function based on the given information: If the opposite side and angle are known, use the tangent function to find the unknown side. If the adjacent side and angle are known, use the cosine function to find the hypotenuse. Substitute the known values into the equation. Solve the equation for the unknown side or hypotenuse.

Additional Tips

Remember the inverse trigonometric functions: If you are given two side lengths and need to find the angle, use the inverse trigonometric functions (arcsin, arccos, arctan). Learn the trigonometric ratios: Familiarity with the sine, cosine, and tangent ratios will help you quickly identify which function to use in different scenarios. Use a calculator: For angles that are not standard (0°, 30°, 45°, 60°, 90°), use a calculator to find the values of the trigonometric functions.

By mastering these techniques, you can confidently determine the lengths of all sides in a right-angled triangle when only one side and its included angle are known. Whether you are working on a mathematical problem or a real-world application, this skill will be a valuable asset.