# Basics of Trigonometry

Published on: by Khan Academy

**Table of Contents**

- Introduction
- Definition of a right triangle and its properties
- Introduction to the core trigonometric functions: sine, cosine, tangent
- Explanation of the mnemonic 'soh cah toa' to remember trigonometric ratios
- Calculation of trigonometric ratios for different angles in a right triangle
- Transition to understanding trigonometry for angles beyond 90 degrees
- Highlights
- FAQ

## Introduction

In this article, we will explore the fundamentals of trigonometry, which is the study of the ratios of sides of triangles. Trigonometry simplifies the understanding of geometry by focusing on the relationships between angles and sides in triangles.

## Definition of a right triangle and its properties

Trigonometry is essentially the study of the ratios of sides within triangles, with a focus on right triangles. The name itself, 'trigonometry,' breaks down to 'trigon' meaning triangle and 'metry' meaning measure. One of the defining features of a right triangle is that one of its angles measures 90 degrees, creating a right angle. The Pythagorean theorem is commonly used to verify the relationships between the sides of a right triangle.

## Introduction to the core trigonometric functions: sine, cosine, tangent

The core trigonometric functions in trigonometry include sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively. Each of these functions serves to determine the ratio of specific sides within a right triangle for a given angle. For instance, the sine function calculates the ratio of the opposite side to the hypotenuse, the cosine function calculates the ratio of the adjacent side to the hypotenuse, and the tangent function determines the ratio of the opposite side to the adjacent side.

Understanding these core trigonometric functions is crucial in trigonometry as they form the basis for calculating various trigonometric ratios for different angles within right triangles. These functions enable mathematicians to determine the relative lengths of sides within a triangle based on specific angles, providing a framework for solving various geometric problems involving right triangles.

## Explanation of the mnemonic 'soh cah toa' to remember trigonometric ratios

To aid in remembering the definitions of the core trigonometric functions, a helpful mnemonic known as 'soh cah toa' is often utilized. This mnemonic serves as a memory aid for recalling the definitions of sine, cosine, and tangent functions. 'Soh' stands for sine being equal to opposite over hypotenuse, 'cah' represents cosine as adjacent over hypotenuse, and 'toa' signifies tangent as opposite over adjacent.

Remembering the 'soh cah toa' mnemonic can assist students and individuals studying trigonometry in quickly identifying and applying the correct trigonometric ratios for different angles within right triangles. By associating the mnemonic with the definitions of the trigonometric functions, learners can streamline their problem-solving processes when working with trigonometric concepts and calculations.

## Calculation of trigonometric ratios for different angles in a right triangle

Trigonometry involves calculating trigonometric ratios for various angles within right triangles using the core functions of sine, cosine, and tangent. These ratios provide insights into the relationships between sides in a triangle relative to specific angles. For example, the sine of an angle measures the ratio of the length of the side opposite the angle to the length of the hypotenuse.

By applying the principles of trigonometry and utilizing trigonometric functions, individuals can determine the trigonometric ratios such as sine, cosine, and tangent for any given angle within a right triangle. These ratios play a pivotal role in solving geometric problems, analyzing angles, and understanding the properties of triangles in the realm of trigonometry.

## Transition to understanding trigonometry for angles beyond 90 degrees

While the foundational concepts of trigonometry primarily focus on angles within right triangles, the study of trigonometry extends to angles beyond 90 degrees. As angles approach and exceed the 90-degree mark, traditional trigonometric functions based on 'soh cah toa' may require modifications to accurately calculate ratios for these angles.

Exploring trigonometry for angles beyond 90 degrees introduces advanced concepts such as unit circle trigonometry and the extension of trigonometric functions to accommodate larger angles. By transitioning to angles beyond 90 degrees, mathematicians and students delve into more complex trigonometric calculations and expand their understanding of geometric relationships in various contexts.

## Highlights

- Trigonometry is the study of ratios in triangles, making geometry more accessible
- Core trigonometric functions like sine, cosine, and tangent simplify angle-side relationships
- Mnemonic devices like 'soh cah toa' aid in memorizing trigonometric ratios
- Trigonometric ratios remain constant for specific angles in right triangles
- Exploration of advanced trigonometry for angles greater than 90 degrees

## FAQ

**Q: What is the significance of trigonometry in mathematics?**

A: Trigonometry helps in calculating unknown angles and sides in triangles, making it a vital tool in geometry and real-world applications.

**Q: How do trigonometric functions simplify geometry problems?**

A: Trigonometric functions like sine, cosine, and tangent provide precise ratios that describe angle-side relationships, aiding in problem-solving.

**Q: Why is the Pythagorean theorem relevant in trigonometry?**

A: The Pythagorean theorem establishes the relationship between the sides of a right triangle, laying the foundation for trigonometric calculations.

**Q: How does trigonometry extend beyond right triangles?**

A: Trigonometry can be applied to angles greater than 90 degrees by using advanced definitions that build upon the basic trigonometric functions.

**Q: What are the practical uses of trigonometry in various fields?**

A: Trigonometry is essential in fields like physics, engineering, and astronomy for calculating distances, angles, and trajectories accurately.