The Origin and Discovery of Trigonometric Functions

Discovering the Problem

Early in human history, people faced the practical problem of measuring heights—buildings, mountains, and tall trees all required clever methods.

Imagine a scene from ancient times: you are a surveyor with no modern instruments and no ready-made formulas. You need to solve a simple-sounding but tricky problem:

How do you measure the height of a tall tree?

You look at the great tree at the edge of the village and think: if only you could fly to the top, you could measure it directly—but you are stuck on the ground.

You try ropes and poles, but none are long enough.

Just as you begin to worry, you notice the tree casting a long shadow in the sunlight.

A thought strikes you: the shadow’s length is easy to measure. Perhaps you can use the shadow to work out the height of the tree.

You decide to run a few experiments.

Experiment 1: The Ratio Between Height and Shadow Length

At the same time and place, measure the shadow lengths of objects of different heights (for example, a 1-meter wooden pole and a 5-meter flagpole).

You discover that the taller the object, the longer the shadow, and there is a fixed ratio between them.

In other words:

This experiment shows: at the same time and place, shadow length is proportional to object height.

Inspired by this experiment, you can already solve the original problem of “how to measure the height of a tall tree.”

For example:

• You measure a 1-meter pole’s shadow as 2 meters.
• You measure the tree’s shadow as 10 meters.

By the ratio from Experiment 1:

Substituting the numbers:

So the tree’s height is $h = 5$ meters.

In other words, as long as you measure the shadows of the pole and the tree at the same moment, you can use the ratio to find the tree’s height.

This simple, practical method was one of the earliest uses of shadows to measure heights.

The Ratio Changes with Time

Below are measurements of the shadow of the same 1-meter pole taken every hour throughout a day. You can clearly see the ratio (shadow/height) changes periodically: smallest at noon, largest in the morning and evening.

You may notice that at different times of day, the ratio is not constant. In the morning and evening the shadow is long and the ratio large; at noon the shadow is short and the ratio small.

The reason is that the sun’s position in the sky—its altitude angle—changes continuously.

• The higher the sun, the shorter the shadow and the smaller the ratio;
• When the sun is near the horizon, the shadow is long and the ratio is large.

So the changing ratio essentially reflects the changing solar altitude angle. This leads to a deeper question: what exactly is the solar altitude angle, and how does it control the shadow’s length and ratio?

The Solar Altitude Angle

The solar altitude angle is the angle between the sun’s rays and the ground.

• When the sun is directly overhead, the altitude angle is 90°.
• When the sun is just rising or setting, the altitude angle is close to 0°.

The solar altitude angle changes throughout the day:

• In the morning and evening, the altitude angle is small and shadows are long.
• At noon, the altitude angle is largest and shadows are shortest.

When measuring the height of a tree, the solar altitude angle is an important parameter. It can be measured with a protractor, a phone app, or by using the shadow of a known object.

Solar Altitude Angle at Different Locations

The solar altitude angle depends not only on the time of day but also on where you are on Earth.

• Latitude: near the equator, the altitude angle is generally large; near the poles, it is small.
• Season: the altitude angle also varies over the year—for example, it is higher at noon in summer than in winter.
• Same time, different places: at the same moment, places at different latitudes have different altitude angles. For example:
  • Near the equator, the sun is almost overhead at noon, with an altitude angle near 90°.
  • In Beijing, the noon altitude angle is smaller than at the equator, and even lower in winter.
  • Inside the Arctic Circle in winter, the sun sometimes never rises, and the altitude angle is 0°.

So when measuring the solar altitude angle, you must consider your geographic location and the date.

Solar Altitude Angle Within One Village

At the east and west ends of the same village, the solar altitude angle is almost identical, because it mainly depends on latitude, date, and time—not on the tiny east–west span (usually a few hundred meters to a few kilometers) of a village.

• Principle: compared with Earth’s radius, the east–west distance in a village is minuscule, so at the same moment the altitude angle is essentially the same everywhere in the village.
• Practical use: whether you stand at the east end or the west end, as long as you measure at the same time, the solar altitude angle can be treated as equal.

Only across a large longitude span (a whole time zone or hundreds of kilometers) does the solar altitude angle differ noticeably.

Experiment 2: The Magic Link Between Sides and Angles in Right Triangles

You plant a 1-meter pole vertically in the ground and measure its shadow. As the sun rises and sets, the shadow length changes. You then measure the tree’s shadow the same way.

Now you begin to wonder:

• What is the real relationship among the solar altitude angle, the object’s height, and the shadow’s length?
• Can you find a “ratio” or “rule” so that, by measuring only the shadow length and the altitude angle, you can compute any object’s height?

You experiment, record, and calculate, finally discovering the magical connection between sides and angles in a right triangle. You and your companions organize these “ratios” into a table for easy lookup and computation.

This was the prototype of what later became the tangent table:

Back to the tree-measuring problem.

As shown above, suppose you have measured the tree’s shadow length $L$ and, through the vertical pole experiment, obtained the solar altitude angle $\theta$.

The tree, the ground, and the end of the shadow form a right triangle:

• The tree’s height $h$ is the opposite side;
• The shadow length $L$ is the adjacent side;
• The solar altitude angle $\theta$ is the angle between the tree and the ground.

By the definition of the tangent function:

So, by measuring the shadow length $L$, and looking up or computing $\theta$, you can find the tree’s height:

For example, if the solar altitude angle is 30°, the tangent is 0.577; if the tree’s shadow is 2 meters, the tree height is about $2 \times 0.577 \approx 1.154$ meters.

This is the basic idea behind how ancient people used trigonometric functions to solve practical measurement problems.

Thinking Further

1. When Did This Research Happen?

Systematizing shadow measurement into trigonometric theory mainly occurred in the 8th–12th centuries.

• From the 8th century: Indian trigonometry spread to the Arab world, and Arab mathematicians began refining trigonometric tables
• 8th–10th centuries: scholars such as Al-Battani (about 858–929) perfected tables of sine, cosine, and tangent
• 12th century: Arabic works were translated into Latin, bringing trigonometry to Europe

Although shadow-based height measurement may have existed in earlier civilizations (such as ancient Egypt and Babylon), the systematic trigonometric theory was mainly developed in the Arab world.

2. Who Was Studying It?

Mainly mathematicians of the Islamic world developed trigonometry.

• Geography: mainly the Middle East of today, including Iraq, Iran, Syria, and nearby regions
• Key scholars:
  • Al-Battani (858–929): perfected trigonometric tables
  • Other Arab mathematicians: built a complete system of trigonometry
• Naming note: in Arabic, the tangent function was called “zill” (shadow), showing the close link between shadow measurement and the tangent

Before that:

• Ancient Babylon (c. 1800 BC): first studied concepts related to trigonometry
• Ancient Greece (2nd century BC–2nd century AD): used chord tables in astronomy
• Ancient India (5th–6th century AD): introduced the first sine table for astronomical computation

3. What Dynasty Was China In Then?

Mathematics in China during the same period:

• Tang dynasty: mathematics continued to develop, mainly in practical computation
• Song dynasty: mathematics flourished, with famous mathematicians such as Qin Jiushao and Li Ye
• Chinese mathematics emphasized practical applications of algebra and geometry, a different emphasis from the Arab world’s systematic study of trigonometry

Summary

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