Tangent and cotangent
In modern notation, cot A = cos A ÷ sin A = shadow ÷ stick, and tan A = sin A ÷ cos A = stick ÷ shadow.
22 / ANTIQUITY TO THE RENAISSANCE
Muhammad ibn Jabir al-Battani
The higher the sun, the shorter a stick’s shadow. Measure the shadow, look it up in a table, and you know the sun’s height.
Explore the mathematics ↓
01 / SHADOWS AND SINES
One stick, every height of the sun.
A stick divided into 12 parts stands in the sun. Drag the sun and read its shadow on the floor.
Fill in a shadow table, then see how a table of sines gives every shadow.
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THE IDEAShadow × sin(sun’s height) = stick × sin(90° − sun’s height). Shadow ÷ stick depends only on the sun’s height: today it is the cotangent, and stick ÷ shadow is the tangent.
The person behind the ideas
- Made observations at Raqqa over several decades.
- Compiled astronomical tables and a catalogue of stars.
- Refined measurements of the year and the seasons.
More of the mathematics
The length of a year
He obtained 365 days, 5 hours, 46 minutes and 24 seconds for the year.