. PART I.

PLANE TRIGONOMETRY.

CHAPTER I.

MEASURES OF ANGLES AND ARCS.

1. TRIGONOMETRY is that branch of Mathematics which treats of methods of subjecting angles and triangles to numerical compu

tation.

2. PLANE TRIGONOMETRY treats of methods of computing plane angles and triangles.

It embraces the investigation of the relations of angles in general, a branch of the science not necessarily connected with the elementary solution of triangles, and which has been distinguished as the Angular Analysis.

3. By the solution of a triangle, in trigonometry, is meant the computation of unknown parts of the triangle from given ones.

The triangle has six parts; three angles and three sides. It is shown in geometry, that when any three of these parts are given, provided one of them is a side, the triangle may be constructed, and the unknown parts found by mechanical measurement.

In the same cases, by trigonometry, we compute the unknown parts from the three given ones, without resorting to construction and measurement: a method of inferior accuracy, on account of the unavoidable imperfections of the instruments employed, and the difficulty of distinguishing with the eye the smallest subdivisions of lines and angles.

But here also the case is excluded in which the three angles are given without a side, because there may be an indefinite number of plane triangles, whose angles are equal to the same three given ones, as

B"

B'

Fig. 1.

in Fig. 1. the triangles A B C, A' B' C', &c. In this case, all these triangles are similar, and their sides are proportional; or the ratio of AB to A C is equal to the ratio of A'B' to A' C', C" &c.; so that the ratios of the sides to each other are fixed or determinate, although the absolute lengths of these sides are indeterminate.

B

C

C

4. Now, in order to subject a triangle to computation, we must first express the sides and angles by numbers. For this purpose proper units of measure must be adopted.

The unit of measure for the sides of plane triangles is a straight line, as an inch, a foot, a mile, &c.; and the number expressing a side is the number of units of the adopted kind that the side contains.

5. The units by which angles are expressed are, the degree, minute, and second; distinguished by the characters ° ' "'.

Fig. 2.

A'

A degree is an angle equal to of a right angle; or a degree is 30 of the whole angular space about a point, or aʊ of four right angles. Thus, Fig. 2, if the angular space about O is divided into 360 equal parts, of which A OB is one, then AOB is one degree. The right angle B will be expressed by 90°; two right angles by 180°, and the whole angular space about a point by 360°.

A"

A"

=

A

=

=

A minute is an angle equal to go of a degree. Therefore, 1° 60'; and a right angle 90 x 60' 5400'. A second is an angle equal to of a minute. Therefore, 1'= 60′′; 1° = 60 × 60′′=3600′′; and a right angle =90 × 60 × 60′′ 324000".

=

Angles less than seconds are sometimes expressed by thirds, fourths, fifths, &c., marked "" Iv V, &c. ; a third being of a second; a fourth, go of a third; &c. But the more convenient method is to express them as decimal parts of a second; thus of a right angle will be either

or more conveniently

12° 51′ 25′′ 42′′" 51", &c.

12° 51' 25"-714, &c.

6. The above division of angles is called sexagesimal, from the divisor 60 employed in the subdivision of the degree. The centesimal division, however, would be preferable in all cases, but cannot now be generally introduced without, at the same time, changing the arrangement of all our tables, the graduation of astronomical and

other instruments, charts, &c. Nevertheless, the attempt has been made in France, and several standard works exist in the French language, in which it is employed throughout.

In the centesimal or French division, the right angle is divided into 100 degrees; the degree into 100 minutes; the minute into 100 seconds, &c. The reduction of these denominations from one to the other requires only a change in the position of the decimal point; thus, in this system 60° 75' 84"-8 is the same as 607584"-8 or 60°-75848 or 0.6075848, the symbol q denoting a quadrant or right angle. To convert centesimal into sexagesimal degrees, since 100° dec. 90° sex. deduct one tenth from the number of centesimal degrees.

EXAMPLE. Required the number of sex. degrees in 85° 47′ 43′′ dec.

To convert sexagesimal into centesimal degrees, since we must take 10 of the sex., divide by 9 and move the decimal point one place to the right.

EXAMPLE. Required the number of centesimal degrees in 76° 55' 36"-732 sex. Reducing the minutes and seconds to the decimal of a degree, we have

To distinguish the degrees of the centesimal from those of the sexagesimal division, the former are frequently called grades, and are denoted by the character g instead of °; thus the preceding angle would be 85o 47′ 43′′.

MEASURES OF ARCS.

7. Since the angles at the center of a circle are proportional to the arcs of the circumference intercepted between their sides, these arcs may be taken as the measures of the angles, and we may express both the arc and the angle by the number of units of arc intercepted on the circumference.

Fig. 3.

A'

The units of arc are also the degree, minute, and second. They are the arcs which subtend angles of a degree, a minute, and a second, respectively, at the center. A degree of arc is thus always 3 of the circumference, whatever the radius of the circle may be; and we obtain the same numerical expression of an angle, whether we refer it directly to the angular unit, or to the corresponding unit of arc. The right angle AOA', Fig. 3, and its measure, the quadrant AA', are therefore both expressed by 90°; the semicircumference by 180°, and the whole circumference by 360°.

A"

A"

B

8. The radius of the circle employed in measuring angles is then

arbitrary, and we may assume for it such a value as will most simplify our calculations. This value is unity; that is, the linear unit employed in expressing the sides of our triangles, or other lines considered. This value will be generally used throughout this treatise.

9. To find the length of an arc of a given number of degrees, minutes, &c.

The semi-circumference of a circle whose radius is unity is known to be 3.14159265; or, the radius being R, the semi-circumference is 3.14159265 R. Hence

An arc x therefore, in the circle whose radius is unity, being expressed in degrees, or minutes, or seconds, we find its length by the formulæ

As these factors for finding the length of an arc are often used, it is convenient to have their logarithms prepared.* Thus

in which the rectangular brackets are used to express that the logarithm of the factor is given instead of the factor itself.

EXAMPLE. What is the length of the arc x=38° 17′ 48′′, the radius being 1.

=

log. 5.1394635 Log. factor for seconds 4.6855749

log. x 9.8250384

10. To find the number of degrees, &c. in an arc equal to the radius.

We have, from the preceding article,

The logarithms in the examples of this work will be taken from Stanley's Tables, (published in New Haven, by Durrie and Peck,) the best tables of seven-figure logarithms yet published in this country.

11. The angle at the center measured by an arc equal to the radius, is often taken as the unit of angular measure, as this angle will be of an invariable magnitude, whatever is the length of the radius. If x is the number of such units in a given angle, the number of degrees, &c., in it will be found by multiplying by the value of the radius in degrees, &c., found in the preceding article. Thus,

which is evidently the same as multiplying by the factors of Art. 9.

It appears, then, that an angle is expressed in the unit of this article by the length of the arc which measures the angle in the circle whose radius is unity. Hence, an angle thus expressed is said to be given in arc. If we put (as is usual)

7 is the circular measure of two right angles, or it is the expression of two right angles in arc. In trigonometry it is therefore common to employ to denote an angular magnitude of 180°; a right angle; 2 four right angles, &c.

π

2

12. The complement of an angle or arc is the remainder obtained by subtracting the angle or are from 90°.

The supplement of an angle or arc is the remainder obtained by subtracting the angle or arc from 180°.

Thus the complement of 30° is 60°; the supplement of 30° is 150°.

Two angles or arcs are complements of each other when their sum is 90°. They are supplements of each other when their sum is 180°.

13. According to these definitions, the complement of an arc that exceeds 90° is negative. Thus the complement of 120° is 90° - 120° - 30°. In like manner the supplement of 200° is 180° 200° = == - 20°.

B