510 trigonometry

History of Trigonometry

From very early times, surveyors, architects, navigators,

and astronomers have made use of

TRIANGLE

s to determine

distances that could not be measured directly. This gave

birth to the subject we today know as trigonometry. There

are problems in the ancient Egyptian text, the R

HIND PAPYRUS

of around 1650

B

.

C

.

E

., that call for the determination of the

slope angles of pyramid faces using the equivalent of the

cotangent function we use today, and a Babylonian clay

tablet from around 1700

B

.

C

.

E

. contains a table of secant val-

ues for the 15 angles between 30°and 45°. The Greek

philosopher T

HALES OF

M

ILETUS

(ca. 600

B

.

C

.

E

.) is said to have

made use of similar triangles to determine the height of the

Cheops pyramid by comparing the length of its shadow with

the length of the shadow of a rod inserted in the ground.

E

RATOSTHENES OF

C

YRENE

(ca. 250

B

.

C

.

E

.) computed the cir-

cumference of the E

ARTH

using lengths of shadows and a

simple geometric argument on angles.

Greek astronomers of ancient times believed that the

stars and planets of the night sky moved along circular arcs

of a giant celestial sphere, and they worked to develop

models that would accurately predict the motion of these

objects on the sphere. Rather than phrase matters in terms

of angles, which proved to be difficult, Greek astronomers

chose to work with measures of straight lengths closely

related to angles, namely, the lengths of

CHORD

s of

CIRCLE

s.

Hipparchus of Rhodes (ca. 200

B

.

C

.

E

.) constructed a

table of such chord lengths for a circle of circumference

21,600 = 360×60 units (which corresponds to 1 unit of cir-

cumference for each minute of arc).

In the second century

C

.

E

., the mathematician C

LAUDIUS

P

TOLEMY

wrote the first extensive treatise on the theory of

chords and their use in obtaining information about “spheri-

cal triangles,” that is, triangles made by great circular arcs

on the surface of a sphere. In addition to working out theo-

rems, Ptolemy explained how to construct tables of chord

values, and presented his own list of chord values for all

angles between zero and 180°in half-degree increments.

The next important step in the development of

trigonometry occurred in India. Scholars of the fifth century

C

.

E

. had by this time discovered that working with half-

chords for half-angles greatly simplified the theory of

chords and its applications to astronomy. As shown on the

right figure, this approach is almost the same construct as

the sine function of today. Whereas we think of sine as a

ratio of lengths (the length of the half-chord to the radius),

Indian scholars interpreted sine as the actual length of the

half-chord. They called this length jy –

a-ardha or simply jy –

a,.

Of course, the jy –

a, value of an angle differed for circles of

different sizes. The scholars A

RYABHATA

, B

H

–

ASKHARA II

, and

others developed astonishingly sophisticated techniques

for computing half-chord values.

The Arab scholars of the 10th century took a great

interest in the work from India. Mathematician Abu al-Wafa

(ca. 950) of Baghdad systemized theorems and proofs of

Indian trigonometry and prepared his own comprehensive

table of half-chord values. He is also believed to have

invented the tangent function, which he called the

“shadow,” and possibly the secant and the cosecant func-

tions. (Still, all were thought of as specific lengths, not as

ratios of lengths.) Arabic scholars did not know how to

yp

yp

As any right triangle can be viewed as coming from

a circle with radius equal to the length of the

hypotenuse, one can use these ratios as the definitions

of the sine and cosine of a given angle θ. This is the

approach usually taken in introductory texts on the

subject. Scholars have given names to all six ratios that

appear in a right triangle containing an angle θ:

Here tan stands for tangent, sec for secant, csc for cose-

cant, and cot for cotangent. These names come from

the following observation:

Let Pbe a point on the unit circle located at

an angle θ. Draw a vertical tangent line to the