THE

ELEMENTS

OF

PLANE AND SPHERICAL

TRIGONOMETRY.

SECTION I.

TRIGONOMETRY is the science which comprehends all theorems respecting the properties of angles and circular arcs, and the lines which it has been found convenient to consider as belonging to them.

DEFINITIONS.

(1.) The circumference of the circle is considered to be divided into 360 equal parts, called degrees, each of these is subdivided into 60 equal parts, called minutes, each of these into 60, called seconds; the second is sometimes divided into 60 thirds, but it is more usual to divide it decimally. Degrees, minutes, and seconds, are marked thus °,',". The French, however, when they established their decl

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mal system of weights and measures, divided the semicircumfince circumference into 400 equal parts, called grades, each grade into 100 equal parts, called minutes, and each minute into 100, called seconds (a)*.

In the following pages we shall generally consider the radius of the circle as the unit of linear meaThe semicircumference will then

sure.

The value of the semicircumference to radius 1 is generally denoted by ; the value of the quadrant and that of the circumference

will therefore be

2 T.

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(2.) Similar arcs of circles are proportional to

their radii.

Let C (fig. 1.) be the centre of the two similar arcs A B, a b whose radii are R, r, respectively; then, Euclid vi. 33,

* The small letters within parentheses refer to the notes.

+ Vide article (110.)

(3.) Since the angle is proportional to the arc on which it stands, we may take the arc as a relative value of the angle; but, as it is convenient that the radius should be known and constant, the angle is generally measured by the length of the intercepted arc described with the radius unity; hence, by art. (2.), On this supposition, the right

angle=

arc

radius

angle, or 90° =

2

two right angles, or 180°.

(4.) In the higher parts of the science we have frequently to consider arcs as not limited by the circumference, nor even any number of circumferences, and these are still considered to be the measures of angles in this view, the angle is always measured the same way round the circuit, and therefore it may exceed the limit of the angle contained by any two straight lines, or 180°. As the arc has no limit when conceived to be constantly repeated upon the circumference, so the angle may be of any magni

tude whatever, when continually repeated through the circuit.

We proceed now to define the trigonometrical lines belonging to the arc.

(5.) The sine (sin) of an arc is the straight line drawn from the extremity of the arc, perpendicular to the diameter which passes through the origin.

The cosine (cos) is the part of the diameter which passes through the origin intercepted between the centre and the sine.

The tangent (tan) is the straight line touching the arc at the origin, and terminated by the diameter produced, which passes through the end of the arc.

The cotangent (cot) is the straight line which touches the circle at the end of the first quadrant, and terminated by the diameter produced passing through the end of the arc.

The secant (sec) is the straight line drawn from the centre through the end of the arc, and terminated by the tangent.

The cosecant (cosec) is the straight line drawn from the centre through the end of the arc, and terminated by the cotangent.

The versed sine (versin) is the intercepted part of the diameter, between the origin and the sine.

The chord (chd) is the straight line joining the two extremities of the arc.

These definitions are general for all arcs, positive or negative, whatever be their magnitude; but attention must be paid to the signs with which they are affected, according as the arc is terminated in the first, second, third, or fourth quadrant.

(6.) Def. If an arc, or a line measured in one direction, be reckoned positive, it must be considered negative when similarly measured in the opposite direction.

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(7.) Let ABCD (fig. 2.) be a circle, and A C, BD two diameters perpendicular to each other; also suppose A P1, AP, AP, A P, to be positive arcs, commencing at A, and terminating in the first, second, third and fourth quadrants: draw the trigonometrical lines to each arc according to their definitions in (5.); then, of the arc A P1, sin, cos, tan, cot, sec, cosec, are respectively, P, M1, O M1, A T1, Bt, OT, Ot; these lines, belonging to an arc less than the quadrant, it has been agreed to reckon positive.

Of the arc A P2, sin, cos, tan, cot, sec, cosec, are P2 M, O M2, A T1⁄2 B t2, OT, Ot2, respectively;

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