BY CONSTRUCTION. Make B C = 327, and B = 54° 17'; then raise the perpendicular C A, meeting BA in A; and A B C will be the triangle required: in which a B is found to measure 560, a c 454%, and 4 A, which is the complement of B, 351°. Extend the compasses from 45° to 54° (4 B) on the tangents, and that extent will reach from 327 (BC) to 454, on the line of numbers, for the side a c. And the extent from 544 (4A) to 90° on the adjacent to that angle to the hypothenuse. Or, As radius is to the cosecant of either of the acute angles, so is the leg opposite to that angle to the hypothenuse. But the rule for this case is as readily performed by the sines and cosines, which are always to be found in the logarithmic tables, where the secant is frequently omitted. sines, will reach from 327 (в c) to 560, on the line of numbers, for the hypothenuse a B. To find the other sides and angle. To these rules it has been judged necessary to add 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 in the former analogies. The reason of this deficiency of the sines and cosines, in the cases alluded to, is, that if an arc near 90° is to be found by means of its sine; or a very small arc, or one near 180°, by means 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 log-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. An error also, of an opposite kind, will arise in finding arcs near 90° by their tangents, or those near 0° by their cotangents, as the largeness of the logarithmic differences, in this part of the quadrant, will render the method of determining them by proportional parts, incorrect, when they are to be found to seconds, by the common tables, or to parts of seconds, by Taylor's tables. But in all other cases, the magnitude and great variation of the logarithmic differences render the use of the tangents preferable to that of the sines or cosines; besides which, the above error, in the use of the common tables, may always be avoided, by subtracting the log-tangent, or log-cotangent, from 20, and then finding the corresponding arc, or angle, in the first part of those tables, where it is usually given to seconds for the first two or three degrees of the quadrant. To this we may further add, that when a sine, cosine, &c of the kind here mentioned 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 much affect the tabular value of its sine or cosine. Note. L is used, in the following formulæ, 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. ~ SOLUTIONS OF ALL THE CASES OF RIGHT-ANGLED PLANE TRIANGLES. A a C I. Given the hypothenuse and either of the oblique angles, to find either of the legs. |