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(8.) Besides the six trigonometrical lines now defined there are three others, sometimes, although but seldom, employed; these are the versed sine or sagitta, the coversed sine, and the suversed sine. The versed sine of an arc BC (see fig. to art. 5) is the line BS between the commencement of the arc and the sine; it is always equal to the radius minus, the cosine, and, therefore, is always positive. The coversed sine is the versed sine of the complement, so that the coversed sine of BC is Ds (see fig. to art. 6); also the suversed sine is the versed sine of the supplement. As the versed sine of any arc must be positive, it follows that the coversed sine and suversed sine must always be positive.

(9.) The following is the way in which the trigonometrical lines, connected with any arc or angle w, are expressed in computation;

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From knowing the numerical value of any one of these lines, those of all the others may be obtained; thus, let the sine be given, then since the radius sine and cosine always form a right-angled triangle, of which the hypotenuse is the radius = 1, (see the fig. in art. 5), we have

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Again, since the triangle formed by the radius, sine, and cosine, is always similar to that formed by the secant, tangent, and radius, and to that formed by the cosecant, radius, and cotangent, as the student will at once see by sketching these lines for any arc, it follows, from the proportionality of

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and, from these expressions, we at once see that the signs of the several lines, as well as their numerical values, are deducible from those of the sine and cosine.

Now the numerical expression for sin. w, for all values of w, from w=0 to w = 90°, (between which limits every possible value is comprised) are actually computed by methods to be hereafter explained, and thence the values of the other trigonometrical lines are deduced. These values are then arranged as in table III, at the end, and form a table of natural sines, cosines, &c. By help of such a table we may readily find the values of the same lines, computed to any other radius R; for as observed at (5) we shall merely have to multiply the tabular value by R. Writing; therefore, for distinction sake, the words sin., cos., &c. in capitals, when the value of the radius is other than unity, the foregoing equations are the same as

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SIN.W

COS.W

,

R

R

and thus by substituting in any trigonometrical formula

&c. for sin. w, cos. w, &c. the formula will become generalized so as to hold good for any value of the radius whatever.

(10.) It is obvious that when any trigonometrical formula is thus generalized every term in it will be the same abstract number as in the original formula; whatever powers or roots of the lines enter the formula they will always be divided by the same powers or roots of the radius R. The denominators will all be removed by multiplying each term by the highest power of R, which enters, and the result will ne

cessarily be a homogeneous expression; that is, every term will have the same dimensions, or will involve as factors the same number of lines. Hence, in order to generalize any trigonometrical formula, or to render it independent of any particular value of R, it will be necessary merely to introduce into the several terms such powers of R as will render them all of the same dimension. For example, the following formula, viz.

sin. (A+B)=sin. A cos. B + sin. B cos. A ;

in which the term on the left is of one dimension, and the terms on the right are each of two dimensions, will become homogeneous by introducing the factor R into the left hand number, so that when this is the value of the radius instead of unity, the formala will be

R sin. (A+B)=sin. A cos. B + sin. B cos. A ;

each term being the product of two lines.

In like manner the formula

cos. 4 A =8 cos.1 A - 8 cos.2 A + 1,

becomes when the radius is R instead of unity

R3 cos. 4 A= 8 cos.4 A -8 R2 cos.2 A + R1;

the powers of R being introduced so as to render each term of four di

mensions.

From the preceding definitions and remarks the following simple properties are immediately deducible, viz.

1. The sine of an arc is equal to half the chord of twice that arc. 2. The chord of 60° being equal to the radius (Geom. p. 119), therefore, the sine of 30°, or the cosine of 60°, is equal to half the radius.

3. Hence, from the expression for the secant at the top of the preceding page, the secant of 60° is equal to the diameter of the circle.

4. The tangent of 45° is equal to the cotangent, and, therefore, to the radius, (see fig. to art. 7.)

(11). We shall terminate this introductory chapter with a table exhibiting the correlative values of the trigonometrical lines, situated in different quadrants; it is readily constructed from the values of the sine and cosine, by help of the relations in (9), bearing in mind that an arc and its supplement have the same sine.

arc.

Table of the Correlative Values of the Trigonometrical Lines.

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This last line is the same as the first; and any line will, obviously, remain unaltered if we add to the corresponding arc a whole circumference or any number of circumferences. If we take w negatively, we may extend the table as follows:

ω

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90° -w+cos. w + sin. w + cot. w+tan. w+cosec. w+

cosec. w

sec. w

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and by continuing this series of arcs the same values of the trigonometrical lines would obviously recur as before.

It is obvious that the cosine of a negative arc, whether less or greater than a quadrant, is the same as the cosine of the same arc, taken positively; but the sine of a negative arc, although the same in magnitude as that of an equal positive arc, has an opposite sign: hence, by the equations at (9), the sine, tangent, cotangent, and cosecant, will have opposite signs to those of the same arc taken positively; but the cosine and secant will have the same signs.

11

CHAPTER II.

FORMULAS AND RULES FOR THE SOLUTION OF PLANE TRIANGLES.

(12.) We shall now proceed to investigate rules for the solution of all the cases of plane triangles.

Right-angled triangles.

As right-angled triangles are those whose several parts are the most easily determined we shall consider them first.

Let ABC be any right-angled plane triangle, and with AB as a radius describe the arc Ba. If AB were unity BC would be the tangent, and AC the secant of the angle A; as it is, however, these lines are equal to AB times the trigonometrical tangent and secant A (5), that is,

BC AB tan. A, AC AB sec. A.

Also, by taking the hypotenuse for the radius, we

have

BC AC sin. A, AB AC cos. A.

These four equations, together with the geometrical property

AC2AB+ BC2,

enable us to solve every case of right-angled triangles.

(13.) In applying these formulas, it must be remembered that the trigonometrical lines which they involve are according to the scale of radius =1; they are computed and registered in the tables of natural sines and tangents. The tables of logarithmic sines and tangents ar

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