formulas which will serve to express the sines and cosines of negative arcs.

From 0° to 180° the sines are always positive, because they always lie on the same side of the diameter AB; from 180° to 360°, the sines are negative, because they lie on the opposite side of their diameter. Suppose ABN =x an arc greater than 180°; its sine P'N is equal to PM, the sine of the arc AM= x-180°. Hence we have in general

sin x=—sin (x-180°)

This formula will give us the sines between 180° and 360°, by means of the sines between 0° and 180°: in particular it gives sin 360°-sin 180=0; and accordingly, if an arc is equal to the whole circumference, its two extremities will evidently be confounded together at the same point, and the sine be reduced to zero.

It is no less evident, that if one or several circumferences were added to any arc AM, it would still terminate exactly at the point M, and the arc thus increased would have the same sine as the arc AM; hence if C represent a whole circumference or 360°, we shall have

sin x=sin (C+x)=sin (2C+x)=sin (3C+x,) &c. The same observation is applicable to the cosine, tangent, &c.

Hence it appears, that whatever be the magnitude of x the proposed arc, its sine may always be expressed, with a proper sign, by the sine of an arc less than 180°. For, in the first place, we may subtract 360° from the arc x as often as they are contained in it; and y being the remainder, we shall have sin x=sin y. Then if y is greater than 180° make y=180°+ z, and we have sin y——sin z. Thus all the cases are reduced to that in which the proposed arc is less than 180°; and since we farther have sin (90°+x)=sin (90°—x), they are likewise ultimately reducible to the case, in which the proposed arc is between zero and 90o.

XIV. The cosines are always reducible to sines, by means of the formula cos A=sin(90°-A); or if we require it, by means of the formula cos A=sin (90°+A): and thus, if we can find the value of the sines in all possible cases, we can also find that of the cosines. Besides, the figures will easily shew us that the negative cosines are separated from the posi

tive cosines by the diameter DE; all the arcs whose extremities fall on the left side of DE, having a positive cosine, while those whose extremities fall on the right have a negative cosine.

Thus from 0° to 90° the cosines are positive; from 90° to 270° they are negative; from 270° to 360° they again become positive; and after a whole revolution, they assume the same values as in the preceding revolution, for cos (360°+x)

=cos x.

From these explanations, it will evidently appear, that the sines and cosines of the various arcs which are multiples of the quadrant have the following values:

And generally, k designating any whole number we shall

What we have just said concerning the sines and cosines renders it unnecessary for us to enter into any particular detail respecting the tangents, cotangents, &c. of arcs greater than 180°; the value of these quantities are always easily deduced from those of the sines and cosines of the same arcs: as we shall see by the formulas, which we now proceed to explain.

THEOREMS AND FORMULAS RELATING TO SINES, COSINES,

TANGENTS, &c.

XV. The sine of an arc is half the chord which subtends a double arc.

For the radius CA, per- s pendicular to the chord MN, bisects this chord, and likewise the are MAN; hence MP, the sine of the arc MA, is half the chord MN which subtends the arc MAN, the double of MA.

The chord which subtends the sixth part of the circumference is equal to the radius; hence sin 360°

12

or sin 30°=R, in other words, the sine of a third part

of the right angle is equal to the radius.

XVI. The square of the sine of an arc, together with the square of the cosine, is equal to the square of the radius; so that in general terms we have sin aA+cos 2A=R3.*

This property results immediately from the right-angled triangle CMP, in which MP2+CP2=CM2.

It follows that when the sine of an arc is given, its cosine may be found, and vice versa, by means of the formulas cos A=±√(R3—sin2 A), and sin A=±√(R2—cos2 A). The sign of these formulæ is ambiguous, because the same sine MP answers to the two arcs AM, AM', whose cosines CP, CP'are equal and have contrary signs; as the same cosine CP answers to the two arcs AM, AN, whose signs MP, PN are also equal, and have contrary signs.

Thus, for example, having found sin 30°=R, we may deduce from it cos 30°, or sin 60°= √(R2—¡R3) = √ ̧R2 · R√3.

XVII. The sine and cosine of the arc A being given, the

*By sin 2A is here meant the square of sin A; and, in like manner, by cos 2A is meant the square of cos A.

tangent, secant, cotangent, and cosecant of the same are, may be found by the following formulas:

For, the triangles CPM, CAT, CDS, being similar, we have the proportions:

CP: PM :: CA: AT; or cos A: sin A:: R: tang A=

CP : CM :: CA: CT; or cos A: R::R: sec A =

PM: CP :: CD: DS; or sin A: cos A :: R : cot A=

PM: CM :: CD: CS; or sin A: R::R: cosec A

RsinA

cos A

R

cos A ReosA

sin A

Ri

sin A

from which are derived the four formulas required. It may also be observed, that the last two formulas might be deduced from the first two, by simply putting 90°-A instead of A.

From these formulas, may be deduced the values, with the proper signs, of the tangents, secants, &c. belonging to any arc whose sine and cosine are known; and since the progressive law of the sines and cosines, according to the different arcs to which they relate, has been sufficiently developed in the preceding chapter, it is unnecessary to say more of the law which tangents, secants, &c. likewise follow.

By means of these formulas, several results, which have already been obtained concerning tangents, may be confirmed. If, for example, we make A=90°, we shall have sin A=R,

cos A=0; and consequently tang 90°=

R

" 0

an expression which designates an infinite quantity; for the quotient of radius divided by a very small quantity, is very great; hence the quotient of radius divided by zero is greater than any finite quantity. And since zero may be taken with the sign+, or with the sign, we have the ambiguous value tang 90°∞.

Again suppose A=180°-B; we have sin A=sin B, and RsinB RsinB cos A=cos B; hence tang (180°—B) :

tang B, which agrees with Art. 12.

=

-cosB

COS B

XVIII. The formulas of the preceding Article, combined with each other and with the equation sin'A+cos2 A=R2, furnish some others worthy our attention.

First we have R2 + tang2 A

R2 (sin3 A+cos3 A)

R4

=

R2 +

R2 sin2 A

cos2 A

cos2 A; hence R2 + tang2 A=sec2 A, a formula which might be immediately deduced from the rightangled triangle CAT. By these formulas, or by the rightangled triangle CDS, we have also R2+cot A=cosec2 A. Lastly, by taking the product of the two formulas tang A=

R sin A

and cot A

R cos A

sin A, we have tang Ax cot A=R2, a

cos A

R2

Hence cot A: cot B::

Likewise we have cot B = tangB

tang B: tang A; that is, the cotangents of two arcs are in the inverse ratio of their tangents.

This formula cot Axtang A=R2 might be deduced immediately from comparing the similar triangles CAT, CDS, which give AT: CA:: CD: DS, or tang A: R:: R: cot A.

XIX. The sines and cosines of two arcs a and b, being given, the sine and cosine of the sum or difference of these arcs may be found by the following formulas.