39. Let us take an example by which to illustrate their application, and as upon a former occasion, one which shall at the same time exhibit the practical utility of Trigonometry. A roof is to have a height of 15 feet in the interior at the centre, and an inclination of 35°. Required the length of the inner line of the rafters. A right angled triangle will be formed in which the angle at the base will be 35°, and the side oppo 35° 7391 site 15 feet, and of which the hypothenuse is required. Formula (1) of the last article applied to this case gives* 1 sin 350::a: 15 Multiplying the extremes and dividing by the first mean the value of the other mean which is a, the hypothenuse required, will be obtained Or formula (2) by a simple transformation gives the same thing. Looking out the sine of 35° in the tables and performing the operations indicated in the last equation, the value of a will be known, which will be the length of the rafters required. The answer will be in feet. Sin 35° is found from the tables to be 57358. thus α= 15 57358 =26.1 feet If (to vary the problem) half the interior breadth of the roof had been given, say 20 feet, and the angle of inclination, instead of 35° as in the last example, had been 15°, then to find the length of the rafters, it would * Since in the demonstration of the formulas, the sides and angles of the triangle were supposed to have no particular values, it follows that any numbers, or any other letters compatible with the properties of a triangle, may be put in the place of those employed, and the formulas will still be true. This must be borne in mind through. out the work. be necessary to find first the angle opposite the given side 20 feet; which is done by subtracting the given angle 15° from 90°, since the two acute angles of a right angled triangle are complements of each other. The remainder is 75°. Applying the same formula as before, there results the proportion The same result might be obtained by using the given angle 15°, and employing formula (3) above, which contains the cosine of one of the acute angles. The proportion would stand thus the same as before. In fact cos 15° sin 75°. (See Art. 23.) 40. Had the height and half the breadth of the interior of the roof been given, the length of the rafters might have been obtained, by employing the property of the right angled triangle demonstrated at Theorem 26, of Plane Geometry, that the square on the hypothenuse is equivalent to the sum of the squares upon the other two sides. height of the roof be 12 feet, and the semi-breadth 16 feet, then Let the If the length of the rafters had been given equal to 20 feet, and the height of the roof equal to 12 feet, then the semi-breadth would have been expressed thus 41. Had the semi-breadth or base of the triangle and the inclination of the roof been given, and the height of the roof or perpendicular of the triangle been required, the hypothenuse not entering into the problem, neither of the above formulas, all of which contain the hypothenuse, would serve to find the side required in a direct manner. It might, however, be found indirectly by first finding the hypothenuse, using one of the above proportions, and then by means of the hypothenuse, using the same proportion, the required side might be obtained. It is, however, objectionable to find one of the required parts in terms of the part which has itself been calculated from the given parts; because in the use of the tables which give the trigonometrical lines of the different angles not with perfect accuracy, but truly for as many decimal places as the table employs, a small error arises from the decimals neglected beyond the last place, and this, though so small as to be unimportant, becomes magnified by repetition, as in the case where one part of a triangle itself not perfectly accurate, is employed to calculate another. It is therefore desirable to find each of the required parts, in terms of the given parts; and this may always be done in right angled triangles. We proceed, therefore, to demonstrate a formula for the direct solution of the last case supposed above. is a perpendicular to the radius at one extremity of the arc, and is terminated by the radius which passes through the other extremity, according to definition of Art. 19. It is also the tangent of the angle c. The equiangular and similar triangles CLT and CAB give the proportion or, C tan c= (3) Let the angle of the roof in the above problem be 20° and the semibreadth 25 feet, then whence, 1: tan 200 :: 25: c tan 200 × 25 1 Had the angle в been used instead of c, the resulting proportion would have been Both proportions may be expressed together in common language thus: Radius: the tangent of one of the acute angles of a right angled triangle the side adjacent that angle: the side opposite. This last rule applied to the problem at Art. 11, gives whence,* 1 tan 30°:: 200: c C=== tan 30°×200=57735 × 200 115°47000 c is the height of the tower. The same rule will evidently serve to determine either of the acute angles of a triangle when the two perpendicular sides are given. If the side c were given and the angle B, the side 6 might be found in the same manner, using the proportion (4) which contains the angle B. 42. We have now exhibited all the cases which can possibly occur in the solution of right angled triangles, with some specimens of their application. The right angle of the triangle is fixed; and any two of the five remaining parts being given, the other three may be found. Let the student select at pleasure any two of the five parts, the two selected to be considered as given, and he will find the case for solution with which he will then be presented, solvable by some one of the formulas above. The operations in the cases already exhibited, though of the most simple kind, nevertheless involve multiplications and divisions, which, from the number of places of figures, are somewhat tedious. In more complicated cases this evil would be much increased. * The tangent is found from Table XXIV. by dividing the sine by the cosine. (Art. 32.) Should the cotangent be required, divide the cosine by the sine. (Art. 34.) To find the secant divide 1 by the cosine. (Art. 33.) For the cosecant divide one by the sine. (Art. 35.) On this account it is customary to employ in trigonometrical calculations, that ingenious invention of Lord Napier's for facilitating numerical calculations, the table of logarithms;* before explaining the use of which we shall give some exposition of the THEORY OF LOGARITHMS. 43. The logarithm of any given number is the exponent of the power to which it is necessary to raise some particular number in order to produce the given number. Thus, let 10 be the number raised to the power; then 2 is the logarithm of 100, because 102 = 100 and 3 is the logarithm of 1000, because 10'1000. Every given number will have a corresponding logarithm or exponent of the power to which it is necessary to raise 10 in order to produce the given number. The number 10, which is the only number that does not change in the above equalities, is called a constant. Should the constant number which has been employed be changed for another, the logarithms of numbers would be different from those derived by the use of the first constant. Logarithms derived from different constants are said to belong to different systems of logarithms, and the constant number belonging to each system is called the base of that system. The system most in use has the number 10 for a base, and is called the common system. The relation which this number sustains to the decimal system of notation will readily suggest some reasons for its selection; it will be found, as we proceed, to have many advantages. 44. If b be the base of a system, n a number, and its logarithm, then by the definition b' = n If we put b in the place of n, this equation becomes b' = b Here is evidently equal to 1. Hence the logarithm of the base of every system is 1. Table XXVI. at the end. The tables most in repute are the French tables of Callet. N. B. The tables in this volume having been printed from the accurate stereotype plates of Bowditch's tables, by permission of the proprietor, are numbered as in Bowditch's edition, and as but a part of his tables are necessary to the present work, the Nos. of the table must not be expected to occur in regular order. |