Trigonometry

Overview

Trigonometry is the part of mathematics that studies how angles and lengths relate to each other, especially in triangles and in circular motion. The tools you learn in trigonometry let you:

• Measure heights and distances that are hard to reach directly.
• Describe oscillations and waves (like sound and light).
• Analyze rotation and circular motion.
• Work with periodic (repeating) patterns in many fields: physics, engineering, computer graphics, and more.

Later chapters inside this trigonometry unit will go into specific topics: how we measure angles (degrees and radians), define the trigonometric ratios (sine, cosine, tangent), use the unit circle, work with trigonometric functions and their graphs, and use identities to simplify and solve equations. This chapter gives the big picture, focusing on what makes trigonometry its own subject and how its key ideas fit together.

Trigonometry’s Core Idea: Angles and Ratios

At the heart of trigonometry is a simple but powerful question:

If you know one angle of a triangle and one side length, what can you say about the other side lengths?

For right triangles, trigonometry answers this using ratios of side lengths. For example (later studied in detail):

• The ratio of the length of the side opposite an angle to the hypotenuse is the sine of that angle.
• The ratio of the adjacent side to the hypotenuse is the cosine of that angle.
• The ratio of the opposite side to the adjacent side is the tangent of that angle.

In this chapter we will not define these precisely yet; that belongs to later sections. Here, what matters is the pattern:

• Each angle in a right triangle corresponds to certain fixed ratios between the sides.
• If you know an angle and one side, you can determine the others (up to scale).

This is already more than geometry alone: geometry describes shapes; trigonometry links shapes with numbers and functions in a systematic way.

From Triangles to Circles

At first, trigonometry begins with right triangles. But its ideas extend far beyond triangles, especially through the concept of the unit circle (a circle of radius 1 centered at the origin in the coordinate plane).

The important conceptual shift is:

• A geometric angle at the center of a circle determines a point on the circle.
• The coordinates $(x, y)$ of that point are closely related to cosine and sine of that angle.
• As the angle changes, the point moves around the circle, and $x$ and $y$ trace out the cosine and sine functions.

This connection to circles turns trigonometry into a powerful language for any phenomenon that repeats in cycles:

• The motion of a pendulum or a planet.
• The alternating current in electrical circuits.
• The up-and-down motion of a vibrating string.

Later sections on the unit circle and trigonometric functions will formalize this, but the essential idea for now is:

Trigonometry transforms questions about angles and circles into questions about numbers and functions.

Trigonometric Functions as Periodic Functions

Unlike most functions you first meet in algebra, trigonometric functions repeat their values over regular intervals. This is called periodicity.

Conceptually:

• As you increase an angle around a circle, you eventually return to where you started.
• Therefore, the associated sine and cosine values also repeat.

This periodic nature makes trigonometric functions the natural tools for modeling:

• Day–night cycles.
• Seasons.
• Sound waves (vibrations in air).
• Light waves and other electromagnetic waves.

You will study graphs and periodicity of trigonometric functions in a later chapter, but here it is important to recognize that repetition is a central theme of trigonometry.

What Trigonometry Lets You Do

Below are some characteristic kinds of problems and applications that trigonometry is designed to handle. The later subchapters (angles, ratios, unit circle, functions, identities) provide the techniques to solve them.

Measuring the Inaccessible

Suppose you want to find:

• The height of a tall building.
• The width of a river you cannot cross easily.
• The distance to a ship offshore.

Direct measurement might be impossible or unsafe. Trigonometry offers an indirect method:

1. Measure an accessible distance on the ground (a baseline).
2. Measure angles (for example, using an instrument that measures the angle of elevation).
3. Use trigonometric relationships to compute the unknown length.

The key idea:

By combining one distance you can measure and one or more angles, you can deduce other distances you cannot directly reach.

Describing Oscillations and Waves

Trigonometric functions describe smooth, repeating motion, such as:

• A mass on a spring moving up and down.
• A sound wave compressing and rarefying air.
• An alternating electrical signal swinging between positive and negative voltages.

Mathematically, these are often modeled by expressions like
$$
A \sin(\omega t + \phi),
$$
where $A$, $\omega$, and $\phi$ are constants representing amplitude, frequency, and phase shift. The specific meanings and how to work with such formulas come later; at this stage, notice:

• These expressions use trigonometric functions to encode key physical characteristics: how big, how often, and how shifted in time the oscillation is.

Navigating and Locating Positions

Trigonometry is essential for:

• Navigation using bearings and headings.
• GPS and satellite positioning.
• Surveying land and making maps.

Triangles created by lines of sight and distances are analyzed using trigonometric methods (right triangles and, at more advanced levels, general triangles with laws of sines and cosines).

The conceptual theme is:

Positions on Earth or in space are converted into angles and distances relative to known points, then trigonometry reconstructs the actual locations.

Trigonometry and Other Areas of Mathematics

Trigonometry does not stand alone; it connects strongly to other branches.

With Geometry

Geometry provides the basic objects: points, lines, angles, triangles, and circles. Trigonometry adds numerical tools:

• Geometry: “This triangle is right-angled, this angle is acute.”
• Trigonometry: “For this angle, the opposite side is $0.6$ times the hypotenuse; the adjacent side is $0.8$ times the hypotenuse.”

Trigonometry makes geometric relationships quantitatively precise.

With Algebra and Functions

In algebra, you learn about functions, equations, and how to manipulate formulas. Trigonometry introduces new specific functions:

• $\sin(x)$, $\cos(x)$, $\tan(x)$, and others.

You will:

• Combine them with algebraic expressions.
• Solve equations that involve angles and trigonometric functions.
• Use identities (equations that are always true for all allowed values of the variables) to simplify expressions.

Thus, trigonometry extends algebra’s toolkit with a new family of important functions.

With Calculus

In calculus, trigonometric functions appear very frequently:

• Differentiation and integration of $\sin(x)$, $\cos(x)$, and related functions.
• Solving differential equations that describe oscillations or wave motion.

You will later see that:

• The derivatives of sine and cosine involve each other.
• Many integrals in physics and engineering feature trigonometric functions.

So trigonometry prepares you to understand a large class of problems in calculus and its applications.

Big Structural Ideas in Trigonometry

The detailed chapters on angles, ratios, the unit circle, functions, and identities will introduce formal definitions and computation techniques. Here, it is helpful to see what kinds of structures you will encounter and why they matter.

Multiple Ways to Measure Angles

You will work with two main angle measures:

• Degrees: a full turn is $360^\circ$.
• Radians: a full turn is $2\pi$ radians.

Radians are built directly from the geometry of a circle (arc length and radius), making them very natural in more advanced mathematics and physics.

Conceptually, you should be prepared for:

• Converting between these measures.
• Recognizing that some trigonometric formulas are simpler in radians.

Defining Trigonometric Functions in Different Ways

Trigonometry introduces functions that can be understood in several equivalent ways:

• As ratios in a right triangle.
• As coordinates of a point on the unit circle.
• As periodic functions defined on the real line.

These views will appear across later subchapters. Understanding that they all describe the same underlying functions helps you move flexibly between geometry, algebra, and analysis.

Identities and Transformations

Trigonometric identities are equations that relate trigonometric functions of the same angle, like:

• Pythagorean-type relationships involving $\sin$ and $\cos$.
• Formulas that involve sums and differences of angles.

You will use identities to:

• Rewrite expressions into more convenient forms.
• Solve trigonometric equations.
• Simplify integrals or other expressions in later mathematics.

Learning trigonometry involves not just computing values, but transforming trigonometric expressions using these identities.

Skills You Will Develop in the Trigonometry Unit

As you work through the chapters under trigonometry, you will build the ability to:

• Interpret and measure angles in degrees and radians.
• Understand and use the basic trigonometric ratios for right triangles.
• Connect angles with points on the unit circle.
• Work with trigonometric functions as functions: read their graphs, describe their periodic behavior, and interpret their parameters.
• Use basic trigonometric identities to simplify expressions and solve equations.
• Apply these ideas to model real-world phenomena involving cycles, rotations, and waves.

Trigonometry thus serves as a bridge between pure geometric thinking and the broader use of functions to model and solve a wide variety of problems.