Modern French writers, instead of using the sexagesimal division, use the centesimal; and it is to be regretted that the latter is not universally used, on account of the great ease with which all calculations are made in that division.

If E denote the number of English degrees in a given angle, and F denote the number of French grades in the same angle, we shall obviously have

E=% of FF of F; and F of EE+ ¿ of E. ' F= →2 ¦

7. When the sum of two angles is equal to a right angle, those angles are mutually complements of each other. In such

case, the sum of their measuring arcs is

and the sum of the

When the sum of two

degrees which these arcs contain is 90°. angles is equal to two right angles, these angles are mutually supplement of each other: the sum of the arcs which measure

these angles is equal to, and the sum of the degrees which they contain is 180°. Thus the

angles CAC, CAC, are mutually complements of each other. Also if we denote any arc by a,

its complement will be

T

2-a,

and its supplement will be -a. If an angle is expressed in degrees, as A°, where A° denotes

Cs

C3

C2

C4

C8

Co

B

the number of degrees in the arc which measures the given angle, then will 90°-A° denote the number of degrees in the arc which measures its complement. For brevity, we say that a anda are complementary arcs, and that A° and 90°-A°

T

2

are complementary degrees. For a like reason, a and -a are supplementary arcs, and A° and 180°-A° are supplementary degrees.

Definition of Sines, Tangents, Secants, &c.

§ 8. In a right triangle the sine of either of the acute angles

is the quotient obtained by divi

ding the side opposite the given angle by the hypotenuse.

Thus, in the right triangle ABC, if we represent the base, perpendicular, and hypotenuse, by b, p, and respectively, we shall have

A

B

The tangent of either of the acute angles is the quotient obtained by dividing the side opposite the angle by the adjacent side. Thus,

The secant of either of the acute angles is the quotient obtained by dividing the hypotenuse by the side adjacent the given angle. Thus,

The cosine, cotangent, and cosecant of any angle are respectively the sine, tangent, and secant of its complement. The two acute angles of a right triangle being complements of each other, it follows that we shall have these relations :

The exponent 2, equation (5) of (B), which is used to denote the square of the sine and cosine of the angle A, is placed immediately after sin. and cos., so that sin. A is the same as (sin. A), cos. A is the same as (cos. A).

From the above relations, we see that the sine and cosecant are reciprocals of each other; cosine and secant are also reciprocals; so also are the tangent and cotangent.

C3 Cz

§ 9. If from one extremity of the arc which measures an angle, a perpendicular be drawn to the diameter which passes through the other extremity, such perpendicular will be the sine of that angle. Thus, C,D, is the sine of the angle СAC1. For we have (§8) defined the sine of this angle to be the quotient obtained by dividing C,D, by AC1, which quotient becomes C,D,, since the divisor is the radius of the circle, and therefore ($5) equal to unity. In the same. way it may be shown that CD is the sine of the angle CAC, that CD, is the sine of the angle CAC. As the arc is the measure of angles, we say that C1D1, CD2, C,D,, are respectively the sines of the arcs CoC1, CoC2 CoC1

C4

D4

D1

C8

C7

Ci

Passing from the angle to the arc, we also say, when the arc exceeds the semi-circumference, that CD, CDs, are respectively the sines of the arcs CCC, CCC,Сg

In all these cases we suppose the arc to be estimated from the point Co, at the extreme right of the circle, upwards and around towards the left. When angles are estimated from C, downwards around towards the left, they are considered as neg ative arcs. Thus, CAC is considered as a negative angle, measured by the negative arc CoCs. By reference to the diagram, it will be readily seen that so long as a positive arc does not exceed or 180°, the sines are all situated above the horizontal diameter, and are considered as positive; for arcs greater than 180° and less than 360°, their sines are situated below the horizontal diameter, and are considered as negative.

§ 10. If from one extremity of an arc a tangent line be drawn, and limited by its intersection with the radius drawn through the other extremity of the arc, then will this portion of the tangent line thus in- C4 cluded between the radius and the radius produced, be the trigonometrical tangent of the arc. And the radius thus produced, or simply the produced part, will be the secant of the arc. Thus, C,D, is the tangent of the arc CC, and C,D, is the tan

Cs

gent of the arc C,C3, in the same way CD, CD, are respectively the tangents of the arcs CCC, CCCC. For, by the former definition (§ 8), the tangent of the angle CAD1, which is measured by the arc CoC1, is the quotient obtained by dividing C,D, by AC, which quotient becomes CoD1, since AC, the radius, is a unit. Also the secant of this angle is the quotient obtained by dividing AD1 by AC, which quotient becomes AD1. When the arc terminates in the first and third quadrants, the tangents are counted on the line D,D, from C, upwards, and are considered as positive. But in the second and fourth quadrants, the tangents are counted from C, downwards, and are considered as negative.

The secants of the arcs CoC, CoC3, CoC2C5, CCC,C,, are respectively AD, AD3, AD, AD,. By inspecting the diagram it will be seen that the secants represented by the dotted lines

consist of the produced part of the radius, while those represented by the full lines consist of the radius together with the produced part; the latter are considered as positive and the former as negative. Hence in the first and fourth quadrants the secants are positive, and in the second and third quadrants they are negative.

§ 11. These different lines may be clearly exhibited in the four quadrants as follows:

In the above diagrams the dotted lines are considered as negative and the full lines as positive.

The letters in the four diagrams are so arranged that in each case BC counted in the positive direction (§9) is the arc, CD the sine, CH the cosine, BE the tangent, GF the cotangent, AE the secant, and AF the cosecant.