Trigonometry

Trigonometry is the study of the relationships between the angles and side lengths of triangles — and, more powerfully, the mathematics of anything that rotates or repeats. The name comes from the Greek for "triangle measurement," but that undersells it: the sine and cosine functions that trigonometry produces describe sound waves, alternating current, planetary orbits, tides, and the pixels in the screen you're reading this on.

Learning Objectives

By the end of this subfield, you should be able to:

• Define sine, cosine, and tangent as ratios of a right triangle's sides (SOH-CAH-TOA)
• Use the unit circle to extend these functions to any angle
• Convert between degrees and radians and explain why radians exist
• Apply the Pythagorean and other core identities
• Solve real problems involving heights, distances, and periodic motion

Quick Answer

In a right triangle, the three basic ratios are $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$ — remembered as SOH-CAH-TOA. The unit circle (radius 1) generalizes these to every angle, turning sine and cosine into smooth wave functions that oscillate between $-1$ and $1$. Because those waves repeat, trigonometry becomes the natural language for every periodic phenomenon in science and engineering.

Where It Came From

Trigonometry was born from astronomy and navigation — two problems that share a deep need: measuring distances you cannot walk. Ancient astronomers wanted to know how far away the Moon and Sun were and how large the Earth was; sailors and surveyors needed to fix positions and heights without stretching a tape across the ocean or up a mountain. All of these reduce to the same trick: measure an angle and a known length, and calculate the length you can't reach.

The Greek astronomer Hipparchus (~150 BC) is often called the father of trigonometry for compiling the first table of chord lengths — an early cousin of the sine table. The functions we use now were refined by Indian mathematicians (Aryabhata, ~500 AD, gave us the sine — our word "sine" is a mistranslation of the Sanskrit jya via Arabic) and by Islamic scholars who developed all six trigonometric functions and used them for astronomy and for finding the direction of Mecca. For most of history trigonometry was quite literally applied astronomy; only later did mathematicians realize that sine and cosine, viewed as functions rather than triangle ratios, describe every wave — a shift that made trigonometry indispensable to physics and engineering.

Topics at a Glance

Topic	What You'll Learn	Key Concepts
Right-Triangle Trig	The three ratios and how to use them	SOH-CAH-TOA, solving triangles
The Unit Circle	Extending trig to all angles	Radians, reference angles, signs by quadrant
Identities	Equations true for all angles	Pythagorean identity, angle-sum formulas
Graphs & Periodic Motion	Sine and cosine as waves	Amplitude, period, phase, frequency

Learning Path

Real-World Applications

• Surveying and construction: measuring the height of a building or the width of a river from a distance using one angle and one length.
• Navigation and GPS: positions on a curved Earth are computed with spherical trigonometry.
• Physics and engineering: alternating current, sound, light, and any oscillation are modelled as sine waves.
• Computer graphics and games: rotating and positioning objects on screen uses sine and cosine constantly.

Key Terms

Term	Definition	Related Concept
Sine / Cosine / Tangent	Ratios relating an angle to a triangle's sides	SOH-CAH-TOA
Unit circle	A circle of radius 1 used to define trig for all angles	Radians
Radian	An angle measure based on arc length; $2\pi$ radians = 360°	Circle, period
Period	The length of one full cycle of a wave	Frequency, amplitude

Quick Revision

• SOH-CAH-TOA: sine = opp/hyp, cosine = adj/hyp, tangent = opp/adj.
• Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$.
• Radians: $180° = \pi$ radians; radians make calculus of trig functions clean.
• Sine and cosine are waves oscillating between $-1$ and $1$, repeating every $2\pi$.

Related Topics

Prerequisites: Algebra and Geometry (triangles, the Pythagorean theorem).

Next: Calculus, where the derivatives of sine and cosine reveal deep structure.