55.

If all the right- angled triangles shown have the same angle

, they are, by the second similarity rule, similar and the ratios of their sides are invariant. In the preceding section (The tan function), we have selected an arbitrary ratio (a/b) and have shown that it is a function of

. The same considerations can be applied to every other ratio of the sides of a right-angled triangle.

A definite angle

determines also b/c = b₁/c₁ = b₂/c₂= ··· . The same is true for a/c, etc. How many such ratios are there? There are six quotients, that is six functions of which have received special names.

sin =a/c=opposite side/hypotenuse		tan =a/b=opposite/adjacent side
cos =b/c=adjacent side/ hypotenuse		cot =a/b=adjacent/opposite side

In former times, c/a was called secant (sec

), c/b cosecant (cosec

), but they have turned out to be unnecessary, so that now only these four trigonometric functions are used.

In order to find for a given angle

the value of the quotient a/c, that is sin

, you can draw from the infinite number of similar triangles, which you can construct by means of

and the right angle, anyone and measure a and c, to obtain the quotient a/c and thus sin

. The figure yields

= 46º and sin 46º = 0.72.

In order to save time, you can follow the next figure' s design. Draw about A the first quadrant of a unit circle - say, with radius 1 dm - and apply by means of a protractor angles

- say 30º or 60º; since the hypotenuse of all triangles within the circle is 1, you can measure the segments CB = a, that is sin

For sin 30º, a = 0.5 dm, whence sin 30º = 0.5, for sin 60º, it is 0.87, whence sin 60º = 0.87, etc.

Denoting by CB the perpendicular, belonging to the angle

, the definition can be given the form:

sin of an angle equals the length of the associated perpendicular in the unit circle.

This definition allows you to find sin

for angles, which are not possible in a right- angled triangle, like

= 0º and

= 90º. For 0º, the length of CB is 0, whence sin 0º = 0, for 90º, it is 1, whence sin 90º=1.

You can use the same figure to obtain the values of cos

= AC/AB, the denominator again being unity.

For 30º, AC = 0.87, whence cos 30º = 0.87/1 = 0.87; for 60º, AC = 0.5, that is cos 60º = 0.5, etc

If you denote by AC the projection, belonging to the angle , cos can be defined as:

cos of an angle is equal to the length
of the corresponding projection in the unit circle.

As you have seen earlier, you can also arrange for tan that the denominator is always 1. Draw the tangent to the unit circle at the point A, then tan = AT/MA.

For 20º, AT=0.36, whence tan 20º=0.36; for 50º, AT=1.19, whence tan 50=1.19, etc.

If you denote by AT the segment of the tangent belonging to the angle, you can define it by

tan of an angle is the length of the corresponding segment of the tangent to the unit circle.

For 0º, AT = 0, whence tan 0º = 0. As grows, AT increases. For tan 90º, you have no value, and you write tan 90º =

(the symbol for infinity).

Comment: If there appears in a calculation tan 90º =

( cot 0º =

) you always deal with this situation saparately, but do not compute with

like with a number. It is not a number!

Draw the tangent to the unit circle at the point B - the cotangent - and find cot with the aid of the triangles MBC:

as alternate angles on parallels and cot = BC/MB. In this way, the denominator MB will always 1.

For 30º, BC = 1.7, whence cot 30º = 1.7; for 50º BC = 0.84, whence cot 50º = 0.84, etc. BC is the cotan of the angle , whence

cot equals the measure of the
corresponding cotangent segment on the unit circle.

becomes smaller, BC grows and eventually tends to

, whence you write cot 0º =

. For 90º, BC = 0, whence cot 90º = 0.

Task: Draw in the right-angled triangle ABC the height CD = h₀. Determine the trigonometric functions of and

in different ways by quotients of two sides! The figure shows that

sin= a/c	cos = b/c	tan = a/b	cot = b/a	from ABC
sin = b/c	cos = a/c	tan = b/a	cot = a/b	from CDA
sin = h_c/b	cos = p/b	tan = h_c/p	cotan = p/h_c	from CDB
sin = h_c/a	cos = q/a	tan = h_c/q	cotan = q/h_c	from CDB

You must draw a triangle for which a/b = 0.9; for example, you can draw a right-angled triangle with a=0.9 cm, b=1 cm or for a=1.8 cm, b=2 cm, etc.

As simple as the trigonometric functions seem to be now, as difficult were their first definitions. Certainly, the Babylonians must have had already some knowledge of trigonometry, as they executed many astronomical computations, which enabled them, for example, to predict Sun and Moon eclipses. Astronomy advanced further with the Egyptians. Trigonometric knowledge was transferred into other activities. We know, for example, that in their building practice two lengths were important; they introduced for their proportions, which you would call today cos, a special term.

The Greek mathematicians, who predominantly enjoyed a system of faultless conclusions, considered it to be unworthy to deal with practical applications, in which one was forced to handle approximate quantities. Their surveyors were not sufficiently educated in mathematics, so that further development became mainly the task of astronomers, who had to be practically and theoretically equally proficient. Trigonometry remained closely linked to astronomy. Names like that of Aristarch (~310 B.C. - ~ 230 B.C.), Hipparch (~180 B.C. - ~125 B.C.) and Ptolemy (~105 - ~165 P.C.) predominated in trigonometry.

Hipparch discovered that the ratio of a chord to the radius s : r does not change for a certain central angle and created a chord table. The most significant work of the Greeks is Ptolemy's. It reached us through the Arabs under the name Almagest and, like Euclid's Elements has dominated trigonometry through many centuries, indeed for even more than a millennium. His work also contained a table of chords for every half degree. A further significant advance was due to the Indians, who also stepped over from the ratio s : r to the ratio of half the chord to the radius: s/2 : r, and thereby changed in the circle from the isosceles to the right-angled triangle. They gave this ratio a name, which the Arabs adopted as a foreign word. The peculiarity of their characters to write words without vowels, that is only with consonants, caused them to mix up the foreign word (chord) with one word they knew well which means bosom - "sin" is the literal translation of "bosom" into Latin. Also, the ratio d : r, which is now called "cos", was introduced and computed by the Indians. The Indian sin function gradually displaced the chord of Hipparch and in the 10th century of our time the Arabs created the cot and tan functions. It is also their achievement to have freed plane trigonometry from astronomy and have developed it as an autonomous science. For the occident, the large, expert and over several centuries influential work of the German mathematician Regiomontanus (1436 - 1476) yielded a foundation. Later on, trigonometry developed as a separate section goniometry which is only concerned with the definitions and interrelations of the functions of angles.