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40. If, in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent be calculated for every minute or second of the quadrant, and arranged in a table, such table is called a table of sines and tangents. If the radius of the circle be 1, the table is called a table of natural sines. A table of natural sines is, therefore, a table which shows the values of the sines, cosines, tangents, and cotangents, to seconds or minutes, of all the arcs of a quadrant. The corresponding values of the secants and cosecants are usually omitted, being readily found from those of the cosine and sine (37).

41. If the values of the sine, cosine, tangent, and secant be known for arcs or angles less than 90°, they are also known for arcs or angles which are greater. For, if an arc or angle is greater than 90, its supplement is less than 90', and the values of these lines are the same for an angle and its supplement (39).

42. We have not considered the sines, cosines, &c. of arcs greater that 180o; for, as the sum of the three-angles of a plane triangle is equal to 180, it follows, that no larger arc can enter into the calculations of the sides and angles of plane triangles.

THEOREM.

43. The sides of a plane triangle are proportional to the sines of their opposite angles. Let ABC (PI. I. Fig. 2) be a triangle; then,

CB : CA :: sin. A : sin. B. For, with A as a centre, and AT, equal to the less side BC, as a radius, describe the arc DI ; and with B as a centre, and BC as a radius, describe the arc CL. Now, ED is the sine of the angle A (30), and CF is the sine of the angle B, to the same radius AD or BC. But, by similar triangles, AD : DE :: AC : CF; AD, being equal to BC, we have,

BC : sin. A :: AC : sin. B, or

BC : AC :: sin. A : sin. B. By comparing the sides AB, AC, in a similar manner, we should obtain,

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THEOREM.
44. In any plane triangle, the sum of the two sides contain-
ing either angle, is to their difference, as the tangent of half
the sum of the other two angles, to the tangent of half their
difference.

Let ABC (Pl. I. Fig. 3) be a triangle ; then,
AC+AB : AB-AC :: tang. :(C+B) : tang. (C-B).

With A as a centre, and a radius AC, the less of the two given sides, let the semicircle IFCE be described, meeting AB in I, and AB produced, in E. Draw CI and AF. Since CAE is an outward angle of the triangle ACB, it is equal to the sum of the inward angles C and B (78)*.

But the angle CIE, being at the circumference, is half the angle CAE (126)*, that is, half the sum of the angles C and B, or, equal to į (C+B).

The angle AFC=ACB, is equal to ABC+BAF; therefore, BAF=ACB-ABC. But ICF= BAF=} (ACB-ABC), or 1 (C-B).

With I and C as centres, and the common radius IC, let the arcs CD and IG be described, and let the lines CE and IH be drawn perpendicular to CI. EC passes through the extremity E of the diameter IE, since the right angle ICE is an angle in a semicircle (128)*Now, CE is the tangent of CIE =} (C+B); and IH the tangent of ICB=} (C-B), to the common radius CI.

But since the lines CE and IH are parallel (57)*, the triangles BHI and BCE are similar, and give the proportion,

BE, or AB+AC : BI, or AB--AC :: CE : IH;
That is, AB+AC:AB-AC :: tang. (C+B): tang. i (C-B).

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THEOREM. 45. In any plane triangle, if a line be drawn from either angle perpendicular to the opposite sidr, dividing it into two segments, the whole side or sum of the segments, is to the sum of the other two sides, as the difference of those sides, to the difference of the segments.

* The references marked thus, ( )*, refer to the articles of the translations of Legendro's Geometry.

Let ABC (Pl. I. Fig. 4) be a triangle, and BD perpendicular to the base AC: then,

AC : AB+CB :: AB - CB : AD-DC ; For, ABP =AD2+BD2, and BCP=CD? +BD- (186)* ; by subtraction, AB2-BC2= ADP – CD ; but, the difference of the squares being equal to the rectangle under the sum and difference (184)*, we have,

ABP – BC=(AB+BC)(AB - BC),

and ADP –CD? =(AD+CD).(AD-CD); therefore, (AB + BC).(AB-BC)=(AD+CD).(AD-CD);

hence, AD+CD : AB+BC :: AB-BC : AD-CD.

THEOREM. 46. In any right-angled plane triangle, radius is to either leg, as the tangent of the adjacent angle, to the opposite side.

Let ABC (Pl. I. Fig. 5) be a plane triangle, right-angled at B; then,

Radius : AB :: tang. A : BC. For, with any radius, as AD, and the vertex of the angle A as a centre, let the arc DE be described, and the tangent DF drawn. Then, by similar triangles,

AD : AB :: DF : BC; That is, calling the radius R, R : AB :: tang. A : BC.

If the vertex of the angle C were used as a centre, and an arc described similar to DE, we should have,

R: CB :: tang. C: AB. 47. The relations between the sides and angles of plane triangles, demonstrated in the last four articles, are sufficient to solve all the cases of Plane Trigonometry. In every plane triangle, there are six parts, three sides, and three angles. Of these six parts, at least three must be given, and one of these a side, to enable us to determine the others. If the three angles are given, it is plain, that an indefinite number of similar triangles may be constructed, whose angles shall be respectively equal to the angles that are given. In such case, the sides being proportional to the sines of the opposite angles (43),] the ratio, or proportion of the sides, is known, although Assuming, with this restriction, any three parts of a triangle as given, one of these three cases will always be presented.

1. When there are two angles and a side given. 2. When there are two sides and an angle given. 3. When the three sides are given.

In the first case, add the given angles together, and subtract their sum from 180', the remainder is the third angle of the triangle (73)* The remaining parts are then found by Art. 43.

In the second case, the given angle must either be included by the given sides, or opposite one of them; if the latter, the remaining parts are found by Art. 43; if the former, by Art. 44.

In the third case, a perpendicular is let fall on the greatest side, the segments found by Art. 45, and then the hypothenuse is to radius, as the base of either of the right-angled triangles, to the sine of the corresponding vertical angle.

A table of sines, cosines, tangents, &c. is necessary to afford the sines, cosines, or tangents of such angles as enter into the proportions. To find the fourth terms of these proportions, the second and third terms must be multiplied together, and their product divided by the first. This process is tedious. To avoid it, we have recourse to logarithms. As the addition of logarithms corresponds to the multiplication of their numbers (4), and their subtraction to the division of their numbers, it is requisite only to add the logarithms of the second and third terms together; this gives the logarithm of their product, and from this sum, to subtract the logarithm of the first term, and the remainder is the logarithm of the quotient, or fourth term of the proportion.

But we are unable to avail ourselves of the logarithmie computation, unless we are possessed of a table, showing the logarithms of the sines, cosines, tangents, &c. calculated for a given radius, of all the arcs of a quadrant. Such a table iş annexed, and is called a table of logarithmic sines and tangents.

TABLE OF LOGARITHMIC SINES. 48. In this table are arranged the logarithms of the numerical values of the sines, cosines, tangents, and cotangents, of all the arcs or angles of the quadrant, divided to minutes, and calculated for a radius of 10000000000. The logarithm of this radius is 10 (9). In the first and last horizontal line of each page, are written the degrees whose logarithmic sines, &c. are expressed on the page. The vertical columns on the left and right, are columns of minutes.

49. To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle.

1. If the angle be less than 45°, look in the first horizontal line of the different pages, until the number of degrees be found; then descend along the column of minutes, on the left of the page, till you reach the number representing the minutes; then pass along the horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be : the number so indicated, is the logarithm sought. Thus, the sine, cosine, tangent, and cotangent of 19° 55', are found on page 37, opposite 55, and are, respectively, 9.532312, 9.973215, 9.559097, 10.440903.

2. If the angle be greater than 45', search along the bottom line of the different pages, till the number of degrees are found; then ascend along the column of minutes, on the righthand side of the page, till you reach the number expressing the minutes; then pass along the horizontal line into the columns designated tang., cotang., sine, cosine, which correspond to the degrees indicated at the bottom of the page; the number so pointed out, is the logarithm required.

50. It will be seen, that the column designated sine at the top of the page, is designated cosine at the bottom; the one designated tang., by cotang; and the one designated cotang., by tang.

The angle found by taking the degrees at the top of the page, and the minutes from the first vertical column on the left, is the complement of the angle, found by taking the corresponding degrees at the bottom of the page, and the minutes traced up in the right-hand column to the same horizontal line. This being apparent, the reason is manifest, why the columns designated sine, cosine, tang., and cotang., when the degrees are pointed out at the top of the page, and the minutes counted

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