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By considering the arc AM, and its supplement AM', and recollecting what has been said, we readily see that,
sin (an arc)=sin (its supplement)
cos (an arc)=-cos (its supplement)
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 x=sin (2C+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 90°.
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, as has already been shown, that the negative cosines are separated from the positive cosines by the diameter DE; all the arcs whose extremities fall on the right side of DE, having a positive cosine, while those whose extremities fall on the left 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:
sin 0°=0 sin 90°-R sin 180°=0 sin 270°-R sin 360°=0 sin 450°=R. sin 540°-0 sin 630°-R
sin 720°=0 sin 810°-R
cos 90° 0
COS 0° R
And generally, k designating any whole number we shall
sin 2k. 90° 0,
cos (2k+1). 90°=0,
cos 4k. 90°=R,
sin (4k+1). 90°=R, sin (4k-1). 90° ——R, 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.
in other words, the sine of a third part of the right angle is equal to the half of 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 A+cos A=R2. 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 reciprocally, by means of the
formulas cos A=√(R2-sin A), and sin A=±√(R2-cos2A). The sign of these formulas is +, or —, because the same sine MP answers to the two arcs AM, AM', whose cosines CP, CP', are equal and have contrary signs; and 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°=1R, we may deduce from it cos 30°, or sin 60°= √(R2—R2)=√ R2=†Ř√3.
XVII. The sine and cosine of an arc A being given, it is required to find the tangent, secant, cotangent, and cosecant of the
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
R sin A
cos A R cos A
which are the four formulas required. It may also be observed, that the two last 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 their 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 developed already, it is unnecessary to say more of the law which regulates the tangents and secants.
By means of these formulas, several results, which have already been obtained concerning the trigonometrical lines, may be confirmed. If, for example, we make A=90°, we shall have sin A=R, cos A=0; and consequently tang 90°=
R2 an expression which designates an infinite quantity; for,
the quotient of radius divided by a very small quantity, is very great, and increases as the divisor diminishes; hence, the quotient of the radius divided by zero is greater than any finite quantity,
The tangent being equal to R.- S and cotangent to R. Cos
it follows that tangent and cotangent will both be positive when the sine and cosine have like algebraic signs, and both negative, when the sine and cosine have contrary algebraic signs. Hence, the tangent and cotangent have the same sign in the diagonal quadrants: that is, positive in the 1st and 3d, and negative in the 2d and 4th; results agreeing with those of Art. XII.
It is also apparent, from the above formulas, that the secant has always the same algebraic sign as the cosine, and the cosecant the same as the sine. Hence, the secant is positive on the right of the vertical diameter DE, and negative on the left of it; the cosecant is positive above the diameter BA, and negative below it: that is, the secant is positive in the 1st and 4th quadrants, and negative in the 2d and 3d: the cosecant is positive in the 1st and 2d, and negative in the 3d and 4th.
XVIII. The formulas of the preceding Article, combined with each other and with the equation sin 2A+cos 2A=R2, furnish some others worthy of attention.
First we have R2 + tang2 AR2 +
R2 (sin2 A+ cos2 A). cos 2A
R2 sin 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 right-angled triangle CDS, we have also R2+cot2 Acosec2 A. Lastly, by taking the product of the two formulas tang A= Resin A
and cot A=
R cos A
formula which gives. cot A=
we have tang Ax cot A=R2, a
Hence cot A cot B: tang B: tang A; that is, the colangents of two arcs are reciprocally proportional to their tangents. The formula cot Ax tang A=R might be deduced immediately, by 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, it is required to find the sine and cosine of the sum or difference of these arcs.
Let the radius AC=R, the árc AB-a, the arc BD=b, and consequently ABD=¿ + b.
the points B and D, let fall the
C FK KEP
The similar triangles BCE, ICK, give the proportions,
CB: CI: BE: IK, or R: cos b : sin a: IK="
CB: CI: CE: CK, or R: cos b:: cos a: CK=·
sin a cos b.
cos a cos b.
The triangles DIL, CBE, having their sides perpendicular, each to each, are similar, and give the proportions, CB: DI :: CE : DL, or R : sin b :: cos a: DL=
cos a sin b.. R
CB: DI: BE: IL, or R : sin b:: sin a:
But we have
IK+DL=DF=sin (a+b), and CK-IL=CF÷cos (a+b).
The values of sin (a-b) and of cos (a-b) might be easily deduced from these two formulas; but they may be found directly by the same figure. For, produce the sine DI till it meets the circumference at M; then we have BM=BD=b, and MI=ID=sin b. Through the point M, draw MP perpendicular, and MN parallel to, AC: since MIDI, we have MN =IL, and IN-DL. 'But we have IK-IN-MP-sin (a-b), and CK+MN=CP=cos (a-b); hence