To these rules may be added the following tables, which contain the solutions of all the cases of plane trigonometry before given; together with such additional formulæ for the tangents, as are better adapted, in certain instances, to the producing of accurate results than those derived from the sines and cosines (1).

N. B.— € L is used to denote the co-log, or the complement of the common tabular logarithm of the number answering to the letter or expression to which it is prefixed. And the sign expresses the difference of the two quantities between which it is placed, when it is not known which of them is the greater.

(1) The reason of this deficiency of the sines and cosines, in the cases alluded to, is, that if an arc near 90° be found in terms of its sine; or a very small arc, or one near 180°, be found in terms of its cosine, the variation of these lines is so small, that they will not change in the tables for many seconds.

Thus, if the log-sine, or cosine, of a required arc should come out 9.9999998, this number, in the tables, is the sine of an arc from 89° 56′ 19" to 89° 57′ 8′′, or the cosine of an arc from 2′ 52" to 3' 41"; and consequently it is impossible to say what arc or angle, between these limits, is to be taken, owing to the tables not being continued to more than seven places of decimals.

In these cases, therefore, it will be proper to employ the logtangents or cotangents, which are not liable to this defect, as the difference for 1" is 42 at an arc of 45°, and larger in every other part of the quadrant.

It may be remarked, however, that when this kind of sine or cosine enters into the calculation, as one of the data, it is rather favourable than otherwise to the accuracy of the result, or the value of the thing sought; as any small error in the given arc, or angle, will not affect the tabular value of its sine or cosine.

5

SOLUTIONS OF ALL THE CASES OF RIGHT-ANGLED PLANE TRIANGLES.

I. Given the hypothenuse and either of the oblique angles, to find either of the legs.

La= Lc + L sin A (or L cos B)-10; Lb = Lc + L sin B (or L COS A)—10.

II. Given the hypothenuse and either of the legs, to find either of the oblique angles.

τα

Or,

cos its adjt. <

Sin A (or cos B) = 7; sin B (or cos a) = +b

C

C

L Sin A (or L COS B) = € Lc + La; L sin B (or L

COS A) = € LC + Lb.

III. Given the hypothenuse and either of the legs, to find the other leg.

RULE I.

Find either of the oblique 4s by case II.; and then the required leg by case 1.

RULE II.

a = √ (c + b) x (c− b ) ; b = √ (c+a) × (c−a)

IV. Given either of the legs and either of the oblique angles, to find the other leg.

As rad given leg ::

RULE.

La = Lb + L tan A (or L Cot B)-10; Lb = La+

Area A=

b2 tan A or

V. Given either of the legs and either of the oblique angles, to find the hypothenuse.

RULE.

As sin 4 opp. given leg}: given leg :: rad : hyp

Or cos adj'. given leg

LCL sin A (or € L cos B) + La= € L sin B (or € L cos A) + Lb,

VI. Given the two legs, to find either of the ob

ra

Tan A (or cot B) =; tan B (or cot A)

L tan A (or L cot B) = € L b + La; L tan в (or L cot A) = € La + Lb.

r b

=

a

VII. Given the two legs, to find the hypothenuse.

RULE I.

Find either of the oblique 2 by case vi, and then find the hypothenuse by case v.

SOLUTIONS OF ALL THE CASES OF OBLIQUE-ANGLED PLANE TRIANGLES.

"

I. Given a side and two angles, to find either of the other two sides.

RULE.

Find the 3d or remaining 4, if necessary, by subtracting the sum of the two given 2s from 180°.

Then,

As sin opp. given side given side :: sin 4 opp. required side required side.

La € L sin B+L Sin A + Lb-10.

Either of the other sides may also be expressed in the same form, by taking the side and angles which are similarly situated with respect to the side whose value is sought. And the same may be observed in all the other cases, where only one side or an angle is exhibited by the formula.