and where the factors are formed therefore by a very easy and The exrapid arithmetical process, we shall find (Art. 880) which are in a form adapted to logarithmic computation. 2 k The log sin A, which is here sought for, is the tabulated logarithm, exceeding by 10 the proper logarithm of bc We have already shewn (Art. 881), that k is the area of the triangle, expressed in units which are the squares described upon the linear units in terms of which the sides are expressed, whether inches, feet, yards or miles. It should be kept in mind that the angle A determined by this formula is ambiguous; but inasmuch as only one angle of the triangle can be greater than 90°, and the greatest angle is opposite to the greatest side, this ambiguity is confined to that angle alone; and this angle will be greater or less than b2 + c2 - a2 90°, according as the expression for its cosine 2bc negative or positive: or in other words, according as b2+ c2 is less or greater than a3. is Again, if the three angles A, B and C be determined from the preceding formulæ, and if, assuming them to be acute, their sum or A+B+С= π, then their values are correctly determined: but if not, this equation will be satisfied, (assuming that the arithmetical and logarithmic processes are correctly performed), by taking that value of the equisinal (Art. 776) angle opposite the greatest side, which is greater than 90o. pressions for the sines of the angles, though ambiguous, are adapted to logarithmic computation. Let the three sides of the triangle be 107.9, 193.4 and 217.12 Example. yards: and let it be required to find the three angles of the triangle and its area. The angles A and B being found, the angle C is known: or it may be found as follows: Also k = 10423.06 square yards = 2.1535 acres, which is the area of the triangle. Modifications of the preceding formulæ might be proposed, which, in some cases, would enable us to obtain the required results, by shorter and more expeditious processes than those which are given above*: but it is not our object, in this Chapter, to give a complete treatise of practical Trigonometry, but merely to explain generally the method by which we adapt our formulæ, in one of their most useful applications, to arithmetical and logarithmic computation. * Thus, it will readily follow, from the investigation in Art. 880, that a symmetrical expression with respect to a, b, c, (and equal to the radius of the circle inscribed in the triangle) we get expressions which are not only more easily calculated than the expressions in the text, but are also free from ambiguity. See "Geometrical Problems and Analytical Formula, with their application to Geodetical Problems;" p. 21, a very ingenious and original work by the late Professor Wallace, of Edinburgh. CHAPTER XXXIV. ON THE USE AND APPLICATION OF SUBSIDIARY ANGLES. What is 887. A SUBSIDIARY angle is one whose sine, cosine, tanmeant by a gent, &c. does not exist in the primitive formula, but which subsidiary is introduced for the purpose of modifying its form, or of facilitating its computation, by means of logarithms or otherwise. angle. Used in the adaptation A subsidiary angle is generally necessary in the computation of formula of expressions consisting of two or more terms connected with to logathe signs and -, and which cannot otherwise be adapted to rithmic computa- logarithmic computation, as will more fully appear in many of the examples which follow. tion which are not otherwise adaptible. In the case of a + b. 888. Let the expression to be adapted to logarithmic computation be a + b, where a and b are positive quantities, whose separate numerical values are not easily found or added together, without the aid of logarithms. In the first place a+b=a (1 + 2 ): Example. Thus, as an example, let it be required to find the value of the expression Therefore log tan = 10+ (log bloga). |