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hence, the tangents of all arcs which terminate in the third quadrant are positive.
At E the tangent becomes infinite: that is,
When the point has passed along into the fourth quadrant to N, the tangent of the arc ADN'N will be the line AV: hence, the tangents of all arcs which terminate in the fourth quadrant are negative.
The cotangents are estimated from the line ED. Those which lie on the side DS are regarded as positive, and those which lie on the side DS' as negative. Hence, the cotangents are positive in the first quadrant, negative in the second, positive in the third, and negative in the fourth. When the point M is at B the cotangent is infinite; when at E it is zero: hence,
cot 180°; cot 270°=0.
q stand for a quadrant; then the following table will show the signs of the trigonometrical lines in the different quadrants.
XIII. In trigonometry, the sines, cosines, &c. of arcs or angles greater than 180° do not require to be considered; the angles of triangles, rectilineal as well as spherical, and the sides of the latter, being always comprehended between 0 and 180°. But in various applications of trigonometry, there is frequently occasion to reason about arcs greater than the semicircumference, and even about arcs containing several circumferences. It will therefore be necessary to find the expression of the sines and cosines of those arcs whatever be their magnitude.
We generally consider the arcs as positive which are estimated from A in the direction ADB, and then those arcs must be regarded as negative which are estimated in the contrary direction AEB.
We observe, in the first place, that two equal arcs AM, AN with contrary algebraic signs, have equal sines MP, PN, with contrary algebraic signs; while the cosine CP is the same for both.
The equal tangents AT, AV, as well as the equal cotangents DS, DS', have also contrary algebraic signs. Hence, calling r the arc, we have in general,
-x) ——sin x
tang (-x)=—tang x
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)
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 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 180°-0
sin 360° 0 sin 540°=0 sin 720° 0
COS 90°-=0 cos 270°=0 cos 450°=0 cos 630°=0 cos 810° 0 &c.
And generally, k designating any whole number we shall have
sin 2k. 90°=0,
sin (4k+1). 90°=R,
sin (4k-1). 90°-R, cos (4k+2). 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.
For the radius CA, perpendicular to the chord MN, bisects this chord, and likewise the arc MAN; hence MP, the sine of the arc MA, is half the chord MN which subtends the arc MAN, the double of MA.
cos (2k+1). 90°=0,
THEOREMS AND FORMULAS RELATING TO SINES, COSINES, TANGENTS, &c.
The chord which subtends the sixth part of the circumference is equal to the radius ; hence
sin or sin 30°=R,
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.
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—sin3A), 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 sines 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—¿R2)=√ 3R2=¿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
CP: PM:: CA: AT; or cos A: sin A:: R: tang A=
The triangles CPM, CAT, CDS, being similar, we have the proportions:
PM: CP:: CD: DS; or sin A: cos A :: R: cot A=
CP : CM :: CA: CT; or cos A: R::R: sec A
PM: CM:: CD: CS; or sin A: R::R: cosec A:
R sin A
R2 cos A R cos A sin A R2
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. ; and cotangent to R. ;
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.
The Algebraic signs of the secants and cosecants are readily determined. For, the secant is equal to radius square divided by the cosine, and since radius square is always positive, it follows that the algebraic sign of the secant will depend on that of the cosine: hence, it is positive in the 1st and 4th quadrants and negative in the 2nd and 3rd.
Since the cosecant is equal to radius square divided by the sine, it follows that its sign will depend on the algebraic sign of the sine hence, it will be positive in the 1st and 2nd quadrants and negative in the 3rd and 4th.
and cot A=
XVIII. The formulas of the preceding Article, combined with each other and with the equation sin 2A+cos 'A=R2, furnish some others worthy of attention.
R2 sin2 A First we have R2+ tang? A R2 + cos2 A R2 (sin2 A+ cos A). R1 -; hence R2+tang? A sec2 A, a cos A cos 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= R sin A cos A
R cos A
we have tang A× cot A==R2, a
formula which gives cot A
We likewise have cot B=. tang B'
R2 tang A
R2 cot A.