sin2 a

42. Having given a short abstract of the more useful formulæ relative to multiples and powers of arcs, we shall now proceed to show the method of constructing the tables of sines, tangents, &c.

When the radius of a circle is unity, the semicircumference is 3.1415926536 nearly. Now there are 180° or 10800' in a semicircle, consequently, if the former be divided by the latter, the result will

1

be 0.0002908882, the measure of an arc of one minute, which as the arc is so small, may be considered its sine.

Now, art. 35. 2, cos = (1 sin) consequently cos l' = 0.9999999577. If these values are substituted in formulæ, (32), and (33), art. 41 the sines and cosines may be obtained through the whole quadrant.

Thus let the arc a=1', and, therefore, sin a=0.0002908882. Let 5 x 3.1415926536 a=5°, then =0.08726646 the length of a, and

180

arc.

a5 therefore, a- + &c. 0.08715574— the natural sine 1.2.3 1.2.3.4.5' of 5°, the logarithm of which is 8.940296, the log. sine of the same This method is easy when the arc is small, as the series then converges very rapidly; but it is rather laborious when the arc is large, in which case recourse must be had to other methods depending upon the properties of multiple arcs, as may be seen in most of our treatises on trigonometry.

As the sines are computed, the cosines of the same arcs may be found from art. 41, formula (33), or from art. 36, formula (2,) the tangents and cotangents, from formula (7) and (8), and the secants and cosecants from (9) and (10).

SECTION IV.

Of the Application of Tables of Sines, Tangents, Secants, &c. to Plane Trigonometry.

CASE I.

43. In any plane triangle it is shewn in our usual treatises,* that the sides are proportional to the sines of their opposite angles, or The sine of any one angle,

Is to the sine of another angle;
As the side opposite to the first,

Is to the side opposite to the second.

These terms may be taken alternately, inversely, &c.

44. When one of the angles is a right angle, then the preceding rule may either be applied, or a modification of it derived from the properties which are peculiar to right-angled triangles.

In right-angled triangles, it is usual to call that side subtending the right angle the hypotenuse, and the other sides, which contain the right angle, the legs, or the one the base and the other the perpendicular.

Then if one of the sides of any triangle ABC, be assumed equal to the radius, the names of the other sides must be determined by art. 28, as follows:

:

* Leslie's and Legendre's Geometry, Ingram's, Playfair's, Simson's Euclid, &c.

The names of the sides being thus known when three of the parts of a triangle including a side are given, the rest may be found by the following rules:

I.-To find a side.

As the name of the given side,

Is to the name of the required side;

So is the given side,

To the required side.

II. To find an angle.

As the side made radius,

Is to the other given side,
So is radius,

To the name of this side.

Any side may be made radius to find a side, but one of the given sides must be made radius to find an angle.

In the solution of plane triangles, it must be recollected that all the angles in any triangle are together equal to two right angles, or 180°. Whence if two of the angles are given, the other may be found by subtracting their sum from 180°; when one angle is given the sum of the other two may be found by subtracting it from 180°; and if one be right or 90°, the sum of the other two is also 90°, and the one is the complement of the other.

CASE II.

45. In a plane triangle when the two sides and contained angle are given.

I. As the sum of the given sides

Is to their difference;

So is the tangent of half the sum of the opposite angles

To the tangent of half their difference.

Half the difference added to half the sum of those angles gives the greater, and subtracted from half the sum gives the less.

All the angles being now known, the third side may be found by the rules in Case I.

Or, after having found half the sum and half the difference of the angles, the remaining side may be found without determining the actual angles, in the following manner:*

II. As the sine of half the difference of the opposite angles
Is to the sine of half their sum,

So is the difference of the containing sides to the remaining
side; or,

III. As the cosine of half the difference of the opposite angles

Is to the cosine of half their sum;

So is the sum of the containing sides

To the remaining side.

* Playfair's Euclid, Plane Trigonometry, Prop. IX. and X.

C

These two methods may be used as a verification to each other, and will be found somewhat more easy in practice than the first method, as several of the quantities may be taken out from the trigonometrical tables at the same time.

Should the sides come out in logarithms from some previous operation, then Gauss' table for finding the logarithm of the sum and difference of two numbers from their logarithms, without first determining the natural numbers themselves, would be some advantage, though it was not thought sufficient to warrant an insertion of it among the tables.

The following method of resolving this problem is convenient, particularly when the logarithms of the sides are given.*

IV. From the logarithm of the greater of the two given sides, having its index increased by 10, subtract the logarithm of the less side, the remainder will be the logarithm tangent of an arc, from which, 45° being subtracted, there will be obtained a remainder. To the logarithm tangent of this remainder add the log. tangent of half the sum of the opposite angles, the sum, rejecting 10 in the index, will be the log. tangent of half their difference, from which the angles themselves may be found.

CASE III.

46. In any plane triangle, when the three sides are given, I. As the base

To the difference of the segments of the base made by a perpendicular upon it, or upon it produced from the opposite angle. It may perhaps be convenient to call the longest side the base, in order that the perpendicular may fall within the triangle.

When the three sides of a triangle are given, the difference of the segments of the base may thus be found. Then half the difference added to half the sum, that is, to half the base, will give the greater segment adjacent to the greater side; and half the difference taken from half the sum will give the less. From these the angles may be found by Rule II. § (44).

II. In a plane triangle, as the rectangle under any two sides, is to the rectangle under the excesses of the semiperimeter above those sides; so is the square of the radius to the square of the sine of half their contained angle, as shown in Leslie's Geometry. In practice, this rule, when logarithms are employed, may be stated as follows:

To the arithmetical complements of the logarithms of the two sides containing the required angle, add the logarithms of the differences between those sides and half the sum of the three sides, then half the sum of these four logarithms will be the log. sine of half the required angle.

III. To the arithmetical complements of the sides containing the required angle, add the logarithm of half the sum of the three sides, and the logarithm of the difference between this half sum and the side opposite the required angle; half the sum of these four logarithms will be the log. cosine of half the required angle.

IV. To the arithmetical complement of the logarithm of half the sum of the three sides, add the arithmetical complement of the difference between half the sum of the three sides and the side opposite the required angle, and the logarithms of the differences between

* See Simson's Euclid, Plane Trigonometry, Prop. IV.

that half sum and the sides containing the required angle; half the sum of those four logarithms will be the log. tangent of half the required angle.

It may be remarked, that these three last rules will, in general, be the most commodious in practice, though, in particular cases, each may have its peculiar advantage when great accuracy is required.

When the required angle does not exceed 90°, Rule II. may be used; when it does, Rule III. may be employed; and in either case Rule IV. will give correct solutions. These observations depend upon the variation of the trigonometrical lines in certain parts of the circle, as, for example, near 90°, the sines vary very slowly, so that the true value of an arc cannot be obtained by our ordinary tables, while the tangents always vary by such perceptible quantities as to leave no doubt of the real value of the required arc. These remarks may be easily verified by examining any of our tables extended to six or seven places of decimals.

Of the Construction of Triangles.

47. Previous to the numerical solution of any triangle, it is generally first constructed geometrically. This is accomplished by means of what are termed mathematical instruments, consisting of scales, compasses, &c. contained in a case, at various prices, to suit the convenience of purchasers. Printed descriptions of these, as well as of many others, are to be found in Jones' edition of Adams' Geometrical and Graphical Essays.

In the construction of plane triangles, the sides are taken from a scale of equal parts, and the angles are laid down by a scale of chords, or more conveniently by a protractor.

EXAMPLES.

CASE I.

48. 1. Given the angles and hypotenuse of a right-angled triangle, to find the base and perpendicular.

Let the hypotenuse AC of the right-angled triangle ABC be 288, and the angle A 39° 22′; it is required to find the sides AB and BC.

b

a

c B

Construction. In the indefinite straight line AB take any point A, and by a protractor, or scale of chords, make the angle A equal to 39° 22′; from any convenient scale of equal parts take AC equal to 288, and from C draw CB, perpendicular to AB; then ABC will be the triangle required. In order to simplify and preserve uniformity, the angles may, in general, be denoted by the capital letters A, B, C, and the opposite sides by the small letters a, b, c. The sides a and c being measured by the same scale from which b was taken, will be found to be 182.7 and 222.7.

1:0-634281: : 288:

sin B

0.634281 × 288

=182.673-a