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The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc and limited by the tangent drawn through the other extremity. Thus CT is the secant of the arc AM, or of the angle ACM.
The versed sine of an arc, is the part of the diameter intercepted between one extremity of the arc and the foot of the sine. Thus, AP is the versed sine of the arc AM, or the angle ACM.
These four lines MP, AT, CT, AP, are dependent upon the arc AM, and are always determined by it and the radius; they are thus designated :
MP-sin AM, or sin ACM,
AT=tang AM, or tang ACM,
AP-ver-sin AM, or ver-sin ACM.
VI. Having taken the arc AD equal to a quadrant, from the points M and D draw the lines MQ, DS, perpendicular to the radius CD, the one terminated by that radius, the other terminated by the radius CM produced; the lines MQ, DS, and CS, will, in like manner, be the sine, tangent, and secant of the arc MD, the complement of AM. For the sake of brevity, they are called the cosine, cotangent, and cosecant, of the arc AM, and are thus designated:
MQ=cos AM, or cos ACM,
DS=cot AM, or cot ACM,
In general, A being any arc or angle, we have
The triangle MQC is, by construction, equal to the triangle CPM; consequently CP MQ: hence in the right-angled triangle CMP, whose hypothenuse is equal to the radius, the two sides MP, CP are the sine and cosine of the arc AM: hence, the cosine of an arc is equal to that part of the radius intercepted between the centre and foot of the sine.
The triangles CAT, CDS, are similar to the equal triangles CPM, CQM; hence they are similar to each other. From these principles, we shall very soon deduce the different relations which exist between the lines now defined: before doing so, however, we must examine the changes which those lines undergo, when the arc to which they relate increases from zero to 180o.
The angle ACD is called the first quadrant; the angle DCB, the second quadrant; the angle BCE, the third quadrant ; and the angle ECA, the fourth quadrant.
VII. Suppose one extremity of the arc remains fixed in A, while the other extremity, marked M, runs successively throughout the whole extent of the semicircumference, from A to B in the direction ADB.
When the point M is at A, or when the arc AM is zero, the three points T, M, P, are confounded with the point A ; whence it appears that the sine and tangent of an arc
zero, are zero, and the cosine and secant of this same arc, are
each equal to the radius.
the circle, we have
Hence if R represents the radius of
sin 0-0, tang 0=0, cos 0=R, sec 0=R.
VIII. As the point M advances towards D, the sine increases, and so likewise does the tangent and the secant; but the cosine, the cotangent, and the cosecant, diminish.
When the point M is at the middle of AD, or when the arc AM is 45°, in which case it is equal to its complement MD, the sine MP is equal to the cosine MQ or CP; and the triangle CMP, having become isosceles, gives the proportion
MP: CM:: 1: √2,
sin 45° R: : 1: √2.
In this same case, the triangle CAT becomes isosceles and equal to the triangle CDS; whence the tangent of 45o and its cotangent, are each equal to the radius, and consequently we have
tang 45°-cot 45°=R.
IX. The arc AM continuing to increase, the sine increases till M arrives at D; at which point the sine is equal to the radius, and the cosine is zero. Hence we have
sin 90°-R, cos 90°=0;
and it may be observed, that these values are a consequence of the values already found for the sine and cosine of the arc zero; because the complement of 90° being zero, we have
sin 90° cos 0°-R, and
cos 90°-sin 0°-0.
As to the tangent, it increases very rapidly as the point M approaches D; and finally when this point reaches D, the tangent properly exists no longer, because the lines AT, CD, being parallel, cannot meet. This is expressed by saying that the tangent of 90° is infinite; and we write tang 90°—∞ The complement of 90° being zero, we have
tang 0=cot 90o and cot 0=tang 90o.
X. The point M continuing to advance from D towards B, the sines diminish and the cosines increase. Thus M'P' is the sine of the arc AM', and M'Q, or CP' its cosine. But the arc M'B is the supplement of AM', since AM'+M'B is equal to a semicircumference; besides, if M'M is drawn parallel to AB, the arcs AM, BM', which are included between parallels, will evidently be equal, and likewise the perpendiculars or sines MP, M'P'. Hence, the sine of an arc or of an angle is equal to the sine of the supplement of that arc or angle.
The arc or angle A has for its supplement 180°-A: hence generally, we have
sin A sin (180°-A.)
The same property might also be expressed by the equation sin (90°+B)=sin (90°-B),
B being the arc DM or its equal DM'.
XI. The same arcs AM, AM', which are supplements of each other, and which have equal sines, have also equal cosines CP, CP'; but it must be observed, that these cosines lie in different directions. The line CP which is the cosine of the arc AM, has the origin of its value at the centre C, and is estimated in the direction from C towards A; while CP', the cosine of AM' has also the origin of its value at C, but is estimated in a contrary direction, from C towards B.
Some notation must obviously be adopted to distinguish the one of such equal lines from the other; and that they may both be expressed analytically, and in the same general formula, it is necessary to consider all lines which are estimated in one direction as positive, and those which are estimated in the contrary direction as negative. If, therefore, the cosines which are estimated from C towards A be considered as positive, those estimated from C towards B, must be regarded as negative. Hence, generally, we shall have,
cos A= cos (180°-A)
that is, the cosine of an arc or angle is equal to the cosine of its supplement taken negatively.
The necessity of changing the algebraic sign to correspond
unless we suppose the cosine AM to become negative as soon as the arc AM becomes greater than a quadrant.
At the point B the cosine becomes equal to -R; that is,
For all arcs, such as ADBN', which terminate in the third quadrant, the cosine is estimated from C towards B, and is consequently negative. At E the cosine becomes zero, and for all arcs which terminate in the fourth quadrant the cosines are estimated from C towards A, and are consequently positive.
The sines of all the arcs which terminate in the first and second quadrants, are estimated above the diameter BA, while the sines of those arcs which terminate in the third and fourth quadrants are estimated below it. Hence, considering the former as positive, we must regard the latter as negative.
XII. Let us now see what sign is to be given to the tangent of an arc. The tangent of the arc AM falls above the line BA, and we have already regarded the lines estimated in the direction AT as positive: therefore the tangents of all arcs which terminate in the first quadrant will be positive. But the tangent of the arc AM', greater than 90°, is determined by the intersection of the two lines M'C and AT. These lines, however, do not meet in the direction AT; but they meet in the opposite direction AV. But since the tangents estimated in the direction AT are positive, those estimated in the direction AV must be negative: therefore, the tangents of all arcs which terminate in the second quadrant will be negative.
When the point M' reaches the point B the tangent AV will become equal to zero: that is,
When the point M' passes the point B, and comes into the position N', the tangent of the arc ADN' will be the line AT:
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 årcs 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.
Let 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 ares 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 ≈ the arc, we have in general,
sin -x)=—sin x
cos (x)cos x