About Trigonometry

What is Trigonometry?

Trigonometry is the study of relationships between the angles and sides of triangles. The word comes from the Greek trigonon (triangle) and metron (measure). That might sound narrow, but it turns out those relationships describe an enormous range of things: sound waves, light, planetary orbits, tides, electrical signals, music, architecture, and navigation.

At its core, trigonometry gives you three functions - sine, cosine, and tangent - that connect angles to ratios of sides in a right triangle.

Once you understand those three ratios, you can solve triangles, model periodic behavior, and describe anything that repeats in a cycle. That’s a lot of mileage from three simple ideas.

A classic way to remember them: SOH CAH TOA.

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

Say it out loud a few times. It sticks.

A Brief History

Trigonometry has one of the most international histories in all of mathematics. It wasn’t invented in one place. It was built up over centuries by civilizations that needed to solve practical problems.

The ancient Babylonians (around 1800 BCE) had tables relating angles to ratios, though they didn’t call it trigonometry. The Egyptians used angle-based calculations for building pyramids and surveying land. But the formal study began with the Greeks. Hipparchus of Nicaea, working around 150 BCE, is often called the “father of trigonometry.” He created the first known table of chords, which is essentially a table of sine values. He needed it for astronomy - to predict the positions of stars and planets.

The Indian mathematician Aryabhata (around 500 CE) was the first to define the sine function as we know it today, using the Sanskrit word jya (meaning “bowstring,” because a chord of a circle looks like a drawn bow). When Arab scholars translated Indian texts, they transliterated jya as jiba, which later got confused with the Arabic word jayb (meaning “pocket” or “fold”). When European scholars translated the Arabic into Latin, they used sinus, meaning “curve” or “fold.” And that’s how we got the word “sine.” The whole thing is basically a 1,500-year game of telephone.

Islamic mathematicians, particularly al-Battani and Abu al-Wafa in the 9th and 10th centuries, developed all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) and created highly accurate trigonometric tables. European mathematicians later built on this work during the Renaissance, and in the 17th century trigonometry became essential for navigation, physics, and the development of calculus.

The connection between trigonometry and circles (the unit circle) was formalized by Leonhard Euler in the 18th century. That transformed trigonometry from a tool for solving triangles into a fundamental part of calculus and physics.

Why We’re Learning It

Trigonometry sits at a crossroads. It connects geometry (shapes and angles) with algebra (equations and functions) and opens the door to calculus (rates of change and accumulation). If you want to go further in math, science, or engineering, trigonometry is not optional. It’s the bridge.

But even on its own, trigonometry is remarkably useful. It’s the math behind anything that involves angles, rotation, or waves. And the world is full of all three.

Why It Matters in Real Life

Trigonometry shows up in more places than most people realize:

• Navigation and GPS (calculating positions from angles and distances)
• Architecture and construction (roof pitch, ramp angles, structural loads)
• Music and sound engineering (sound waves are sine waves)
• Physics (projectile motion, forces at angles, wave mechanics)
• Electrical engineering (alternating current is modeled with sine and cosine)
• Computer graphics and game development (rotation, lighting, 3D rendering)
• Astronomy (calculating distances to stars, predicting eclipses)
• Surveying and cartography (measuring land and making maps)
• Medical imaging (CT scans use trigonometric reconstruction)

When you hear that a speaker produces a “440 Hz sine wave,” that’s trigonometry. When an architect calculates the angle of a roof, that’s trigonometry. When your phone’s GPS figures out where you are, that’s trigonometry. Even everyday tasks like measuring the height of a tree or calculating the angle of a ramp use trigonometric ideas.

What You’ll Learn in This Section

Starting from right triangles and building up to the unit circle and beyond:

• Right triangle trigonometry: sine, cosine, tangent, and solving for missing sides and angles
• Applications of right triangle trig (heights, distances, angles of elevation and depression)
• Radians and degree-radian conversion
• The unit circle and trigonometric functions defined on it
• Graphing sine, cosine, tangent, and reciprocal functions
• Transformations of trig graphs (amplitude, period, phase shift, vertical shift)
• Fundamental trigonometric identities
• Verifying identities and sum/difference formulas
• Solving trigonometric equations
• Law of Sines and Law of Cosines
• Applications of triangle solving in real-world contexts

Each lesson includes worked examples, real-world connections, and a quiz.

How to Get the Most Out of This Section

Trigonometry has a lot of new vocabulary and notation. Don’t try to memorize everything at once. Focus on understanding the core idea (ratios of sides in a triangle), and the rest will build naturally from there.

Drawing diagrams helps enormously. Label the sides, mark the angles, and write out the ratios. A quick sketch can turn a confusing problem into a straightforward one.

You should be comfortable with Algebra 2 before starting here, especially functions, graphing, and the Pythagorean theorem from Geometry.

You’ve Got This

Trigonometry might feel new at first, but it builds directly on the algebra and geometry you’ve already learned. Many people find it surprisingly satisfying once the pieces start clicking. Take it one ratio at a time.