Trigonometry I INTRODUCTION Trigonometry, branch of mathematics that deals with the relationships between the sides and angles of triangles and with the properties and applications of the trigonometric functions of angles.

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Trigonometry
I

INTRODUCTION

Trigonometry, branch of mathematics that deals with the relationships between the sides and angles of triangles and with the properties and applications of the
trigonometric functions of angles. The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical
trigonometry, which deals with triangles that are sections of the surface of a sphere.
The earliest applications of trigonometry were in the fields of navigation, surveying, and astronomy, in which the main problem generally was to determine an
inaccessible distance, such as the distance between the earth and the moon, or of a distance that could not be measured directly, such as the distance across a large
lake. Other applications of trigonometry are found in physics, chemistry, and almost all branches of engineering, particularly in the study of periodic phenomena, such
as vibration studies of sound, a bridge, or a building, or the flow of alternating current.

II

PLANE TRIGONOMETRY

The concept of the trigonometric angle is basic to the study of trigonometry. A trigonometric angle is generated by a rotating ray. The rays OA and OB (Fig. 1a, 1b, and
1c) are considered originally coincident at OA, which is called the initial side. The ray OB then rotates to a final position called the terminal side. An angle and its
measure are considered positive if they are generated by counterclockwise rotation in the plane, and negative if they are generated by clockwise rotation. Two
trigonometric angles are equal if they are congruent and if their rotations are in the same direction and of the same magnitude.

An angular unit of measure usually is defined as an angle with a vertex at the center of a circle and with sides that subtend, or cut off, a certain part of the
circumference (Fig. 2).

If the subtended arc s (AB) is equal to one-fourth of the total circumference C, that is, s = ? so that OA is perpendicular to OB, the angular unit is a right angle. If s =
C,
yC, so that the points A, O, and B are on a straight line, the angular unit is a straight angle. If s = 1/360C, the angular unit is one degree. If s = oC, so that the
subtended arc is equal to the radius of the circle, the angular unit is a radian. By equating the various values of C, it follows that 1 straight angle = 2 right angles = 180
degrees = p radians
Each degree is subdivided into 60 equal parts called minutes, and each minute is subdivided into 60 equal parts called seconds. For finer measurements, decimal parts
of a second may be used. Radian measurements smaller than a radian are expressed in decimals. The symbol for degree is °; for minutes, '; and for seconds, '. For
radian measures either the abbreviation rad or no symbol at all may be used. Thus

The angular unit radian is understood in the last entry. (The notation 42'.14 may be used instead of 42.14' to indicate decimal parts of seconds.)
By convention, a trigonometric angle is labeled with the Greek letter theta (? ). If the angle ? is given in radians, then the formula s = r? may be used to find the length
of the arc s; if ? is given in degrees, then

A

Trigonometric Functions

Trigonometric functions are unitless values that vary with the size of an angle. An angle placed in a rectangular coordinate plane is said to be in standard position if its
vertex coincides with the origin and its initial side coincides with the positive x-axis.
In Fig. 3, let P, with coordinates x and y, be any point other than the vertex on the terminal side of the angle ? , and r be the distance between Pand the origin. Each of
the coordinates x and y may be positive or negative, depending on the quadrant in which the point P lies; x may be zero, if P is on the y- axis, or y may be zero, if P is
on the x-axis. The distance r is necessarily positive and is equal to

in accordance with the Pythagorean theorem (see Geometry).

The six commonly used trigonometric functions are defined as follows:

Since x and y do not change if 2p radians are added to the angle--that is, 360° are added--it is clear that sin (? + 2 p) = sin ?. Similar statements hold for the five
other functions. By definition, three of these functions are reciprocals of the three others, that is,

If point P, in the definition of the general trigonometric function, is on the y-axis, x is 0; therefore, because division by zero is inadmissible in mathematics, the tangent
and secant of such angles as 90°, 270°, and -270° do not exist. If P is on the x-axis, y is 0; in this case, the cotangent and cosecant of such angles as 0°, 180°, and 180° do not exist. All angles have sines and cosines, because r is never equal to 0.
Since r is greater than or equal to x or y, the values of sin ? and cos ? range from -1 to +1; tan ? and cot ? are unlimited, assuming any real value; sec ? and csc ?
may be either equal to or greater than 1, or equal to or less than -1.
It is readily shown that the value of a trigonometric function of an angle does not depend on the particular choice of point P, provided that it is on the terminal side of
the angle, because the ratios depend only on the size of the angle, not on where the point P is located on the side of the angle.
If ? is one of the acute angles of a right triangle, the definitions of the trigonometric functions given above can be applied to ? as follows (Fig. 4). Imagine the vertex A
is placed at the intersection of the x-axis and y-axis in Fig. 3, that AC extends along the positive x-axis, and that B is the point P, so that AB = AP = r. Then sin ? = y/r
= a/c, and so on, as follows:

The numerical values of the trigonometric functions of a few angles can be readily obtained; for example, either acute angle of an isosceles right triangle is 45°, as
shown in Fig. 4. Therefore, it follows that

The numerical values of the trigonometric functions of any angle can be determined approximately by drawing the angle in standard position with a ruler, compass, and
protractor; by measuring x, y, and r; and then by calculating the appropriate ratios. Actually, it is necessary to calculate the values of sin ? and cos ? only for a few
selected angles, because the values for other angles and for the other functions may be found by using one or more of the trigonometric identities that are listed below.

B

Trigonometric Identities

The following formulas, called identities, which show the relationships between the trigonometric functions, hold for all values of the angle ?, or of two angles, ? and ?,
for which the functions involved are:

By repeated use of one or more of the formulas in group V, which are known as reduction formulas, sin ? and cos ? can be expressed for any value of ?, in terms of the
sine and cosine of angles between 0° and 90°. By use of the formulas in groups I and II, the values of tan ? , cot ? , sec ?, and csc ? may be found from the values of
sin ? and cos ? . It is therefore sufficient to tabulate the values of sin ? and cos ? for values of ? between 0° and 90°; in practice, to avoid tedious calculations, the
values of the other four functions also have been made available in tabulations for the same range of ? .
The variation of the values of the trigonometric functions for different angles may be represented by graphs, as in Fig. 5. It is readily ascertained from these curves that
each of the trigonometric functions is periodic, that is, the value of each is repeated at regular intervals called periods. The period of all the functions, except the
tangent and the cotangent, is 360°, or 2 p radians. Tangent and cotangent have a period of 180°, or p radians.
Many other trigonometric identities can be derived from the fundamental identities. All are needed for the applications and further study of trigonometry.

C

Inverse Functions

The statement y is the sine of ? , or y = sin ? is equivalent to the statement ? is an angle, the sine of which is equal to y, written symbolically as ? = arc sin y = sin-1y.
The arc form is preferred. The inverse functions, arc cos y, arc tan y, arc cot y, arc sec y, arc csc y, are similarly defined. In the statement y = sin ? , or ? = arc sin y, a
given value of y will determine infinitely many values of ?. Thus, sin 30° = sin 150° = sin (30° + 360°) = sin (150° + 360°). . .= 1/2; therefore, if ? = arc sin 1/2, then
? = 30° + n 360° or ? = 150° + n 360°, in which n is any integer, positive, negative, or zero. The value 30° is designated the basic or principal value of arc sin 1/2.
When used in this sense, the term arc generally is written with a capital A. Although custom is not uniform, the principal value of Arc sin y, Arc cos y, Arc tan y, Arc cot
y, Arc sec y, or Arc csc y commonly is defined to be the angle between 0° and 90° if y is positive; and, if y is negative, by the inequalities

D

The General Triangle

Practical applications of trigonometry often involve determining distances that cannot be measured directly. Such a problem may be solved by making the required
distance one side of a triangle, measuring othersides or angles of the triangle, and then applying the formulas below.
If A, B, C are the three angles of a triangle, and a, b, c the respective opposite sides, it may be proved that

The cosine and tangent laws can each be given two other forms by rotating the letters a, b, c and A, B, C.
These three relationships can be used to solve any triangle, that is, the unknown sides or angles can be found when one side and two angles, two sides and the
included angle, two sides and an angle opposite one of them (usually there are two triangles in this case), or when three sides are given.

III

SPHERICAL TRIGONOMETRY

Spherical trigonometry, which is used principally in navigation and astronomy, is concerned with spherical triangles, that is, figures that are arcs of great circles (see
Navigation) on the surface of a sphere. The spherical triangle, like the plane triangle, has six elements, the three sides a, b, c and the angles A, B, C. But the three
sides of the spherical triangle are angular as well as linear magnitudes, being arcs of great circles on the surface of a sphere and measured by the angle subtended at
the center. The triangle is completely determined when any three of its six elements are given, since relations exist between the various parts by means of which
unknown elements may be found.
In the right-angled or quadrantal triangle, however, as in the case of the right-angled plane triangle, only two elements are needed to determine all of the remaining
parts. Thus, given c, A in the right-angled triangle, ABC, with C = 90°, the remaining parts are given by the formula as sin a = sin c sin A; tan b = tan c cos A; cot B =
cos c tan A. When any other two parts are given the corresponding formulas may be obtained by Napier's rules concerning the relations of the five circular parts, a, b,
complement of c, complement of A, complement of B. With respect to any particular part, the remaining parts are classified as adjacent and opposite; the sine of any
part is equal to the product of the tangents of the adjacent parts and also to the product of the cosines of the opposite parts.
In the case of oblique triangles no simple rules have been found, but each case depends on the appropriate formula. Thus in the oblique triangle ABC, given a, b, and A,
the formulas for the remaining parts are

In spherical trigonometry, as well as in plane, three elements taken at random may not satisfy the conditions for a triangle, or they may satisfy the conditions for more
than one. The treatment of certain cases in spherical trigonometry is quite formidable, because every line intersects every other line in two points and multiplies the
cases to be considered. The measurement of spherical polygons may be made to depend upon that of the triangle. If, by drawing diagonals, one can divide the
polygons into triangles, each of which contains three known or obtainable elements, then all the parts of the polygon can be determined.
Spherical trigonometry is of great importance in the theory of stereographic projection and in geodesy. It is also the basis of the chief calculations of astronomy; for
example, the solution of the so-called astronomical triangle is involved in finding the latitude and longitude of a place, the time of day, the position of a star, and various
other data.

IV

HISTORY

The history of trigonometry goes back to the earliest recorded mathematics in Egypt and Babylon. The Babylonians established the measurement of angles in degrees,
minutes, and seconds. Not until the time of the Greeks, however, did any considerable amount of trigonometry exist. In the 2nd century

BC

the astronomer Hipparchus

compiled a trigonometric table for solving triangles. Starting with 7y° and going up to 180° by steps of 7y°, the table gave for each angle the length of the chord
subtending that angle in a circle of a fixed radius r. Such a table is equivalent to a sine table. The value that Hipparchus used for r is not certain, but 300 years later the
astronomer Ptolemy used r = 60 because the Hellenistic Greeks had adopted the Babylonian base-60 (sexagesimal) numeration system (see Mathematics).
In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for steps of y°, from 0° to 180°, that is accurate to 1/3600 of a unit. He also
explained his method for constructing his table of chords, and in the course of the book he gave many examples of how to use the table to find unknown parts of
triangles from known parts. Ptolemy provided what is now known as Menelaus's theorem for solving spherical triangles, as well, and for several centuries his
trigonometry was the primary introduction to the subject for any astronomer. At perhaps the same time as Ptolemy, however, Indian astronomers had developed a
trigonometric system based on the sine function rather than the chord function of the Greeks. This sine function, unlike the modern one, was not a ratio but simply the
length of the side opposite the angle in a right triangle of fixed hypotenuse. The Indians used various values for the hypotenuse.
Late in the 8th century, Muslim astronomers inherited both the Greek and the Indian traditions, but they seem to have preferred the sine function. By the end of the
10th century they had completed the sine and the five other functions and had discovered and proved several basic theorems of trigonometry for both plane and
spherical triangles. Several mathematicians suggested using r = 1 instead of r = 60; this exactly produces the modern values of the trigonometric functions. The
Muslims also introduced the polar triangle for spherical triangles. All of these discoveries were applied both for astronomical purposes and as an aid in astronomical timekeeping and in finding the direction of Mecca for the five daily prayers required by Muslim law. Muslim scientists also produced tables of great precision. For example,
their tables of the sine and tangent, constructed for steps of 1/60 of a degree, were accurate for better than one part in 700 million. Finally, the great astronomer Nasir
al-Din al-Tusi wrote the Book of the Transversal Figure, which was the first treatment of plane and spherical trigonometry as independent mathematical sciences.
The Latin West became acquainted with Muslim trigonometry through translations of Arabic astronomy handbooks, beginning in the 12th century. The first major
Western work on the subject was written by the German astronomer and mathematician Johann Müller, known as Regiomontanus. In the next century the German
astronomer Georges Joachim, known as Rheticus introduced the modern conception of trigonometric functions as ratios instead of as the lengths of certain lines. The
French mathematician François Viète introduced the polar triangle into spherical trigonometry, and stated the multiple-angle formulas for sin(nq) and cos(nq) in terms of
the powers of sin(q) and cos(q).
Trigonometric calculations were greatly aided by the Scottish mathematician John Napier, who invented logarithms early in the 17th century. He also invented some
memory aids for ten laws for solving spherical triangles, and some proportions (called Napier's analogies) for solving oblique spherical triangles.
Almost exactly one half century after Napier's publication of his logarithms, Isaac Newton invented the differential and integral calculus. One of the foundations of this
work was Newton's representation of many functions as infinite series in the powers of x (see Sequence and Series). Thus Newton found the series sin(x) and similar
series for cos(x) and tan(x). With the invention of calculus, the trigonometric functions were taken over into analysis, where they still play important roles in both pure
and applied mathematics.
Finally, in the 18th century the Swiss mathematician Leonhard Euler defined the trigonometric functions in terms of complex numbers (see Number). This made the
whole subject of trigonometry just one of the many applications of complex numbers, and showed that the basic laws of trigonometry were simply consequences of the
arithmetic of these numbers.

Contributed By:
J. Lennart Berggren
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