Triangle Definition

Similar triangles have been used throughout history to estimate distances that can not be measured directly. Before calculators or computers were used, they used various forms of trigonometric tables that contained the sides of triangles for different angles.

Tables of chords in ancient Greece

Drag the red point to draw the graph!

The first known trigonometric table was made by Hipparchus ( ≈ 150 BC ) who is also considered to be the father of trigonometry. Hipparchus original work has not survived but he is referenced in the book Almagest by Ptolemy ( ≈ 150 AD ). In Almagest there is a trigonometric table of chords. A chord is a segment whose endpoints are on a circle. Ptolemy used base 60 and he measured angles using degrees. The radius of the circle used in the table was therefor 60.

The chord as a function of the angle, which is graphed in the applet above, is an anachronism (just as the applet itself). Graphing functions was first done many centuries after Ptolemy.

Tables of half-chords in India

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The Indian mathematician Aryabhata ( ≈ 500 AD ) was the first to make a table using half-chords instead of chords. When using chords, the triangles are isosceles. When using half-chords, the triangles are right angled.

Etymology of the word sine

Aryabhata called these half-chords ardha-jya (in Sanskrit), which was abbreviated to jya. From this word, the Arabs coined the term jiba which means half-chord. Since vowels were omitted, the word jiba was written jb. When mathematics/science returned to Europe (≈ 1 K years after Ptolemy), the European scholars thought that jb stood for the Arabic word jaib, which means "cave" or "bay". They then translated the wrong word to the Latin word sinus.

Definition of sine and cosine using a right angled triangle

Drag the point A to draw the graphs!

Using the labels in the picture above, the trigonometric functions are defined as

\[\sin \alpha = \frac {\text{opp}}{\text{hyp}} \]

\[\cos \alpha = \frac {\text{adj}}{\text{hyp}} \]

\[\tan \alpha = \frac {\text{opp}}{\text{adj}} \]

Reciprocal trigonometric functions

There are six different ratios between two sides in a triangle. The three most common ratios are sine, cosine and tangent. All ratios have a name, the three other ratios are called cosecant (\(\csc \)), secant (\(\sec \)) och cotangent (\(\cot \)). These ratios are defined like this:

\[\csc \alpha = \frac {\text{hyp}}{\text{opp}} = \frac{1}{\sin \alpha}\]

\[\sec \alpha = \frac {\text{hyp}}{\text{adj}} = \frac{1}{\cos \alpha} \]

\[\cot \alpha = \frac {\text{adj}}{\text{opp}} = \frac{1}{\tan \alpha}\]

Extending the definitions

The trigonometric ratios can be seen as functions of an angle. The graphs of such functions are drawn in the applets above. If the angle is called \(x\), the functions are defined for \(0\lt x \lt 90^\circ\), otherwise \(x\) can not be a non-right angle in a right-angled triangle. We can extend the definition of the trigonometric functions so they are defined for all \(x\). When making such a definition, a unit circle is used instead of a triangle. The unit circle definitions are described at the page Trigonometry - The Unit Circle.

further info:

Wikipedia - Hipparchus

Wikipedia - Ptolemy's table of chords

Wikipedia - Aryabhata's sine table


by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License