log. tan log. R+ log. blog. a thus becomes known from the logs. of a and b, without calcu Taking CD for radius, DB will be the tangent of the angle DCB, and DA, the tangent of therefore, AB is the difference of those tangents. By the table of natural sines and cosines,* DCA, From the top of a mountain three miles high, the angle of depression of a line tangent to the earth's surface is taken, and found to be 20 13' 27"; it is required thence to determine the diameter of the earth, supposing it to be a perfect sphere. * A table of natural tangents which some collections of tables contain is often convenient. Let o be the centre of the earth, BA the mountain, AC the visual ray or line touching the earth's surface at c. Draw the tangent BD, and join OD, OC; then the angle of depression EAC being given, we have also the angle BAD, the complement of it, equal to 87° 46' 33". Also since the tangents BD, CD, are equal (Geom. p. 83), we have the angle comp. A1° 6' 43', and, there BOD DOC = Given the distances between three objects, A, B, C, and the angles subtended by these distances at a point D in the same plane with them; to determine the distance of D from each object. Let a circle be described about the triangle ADB, and join AE, EB, then will the angles ABE, BAE, be respectively equal to the given angles ADE, BDE (Geom. p. 44), thus all the angles of the triangle AEB are known, as also the side AB; we may find, therefore, the remaining sides AE, EB. Again, the sides of the triangle ABC being known, we may find the angle BAC; hence the angle CAE becomes known, so that in the triangle CAE we shall have the two sides AE, AC, and the included angle given, from which we may find the angle AEC in fig. 1, or the angle ACE in fig. 2, с E B A D opposite to A, B, C, we have and thence its supplement AED or ACD; this with the given side AE and angle ADE, in the first figure, or with the given side AC, and angle ADC in the second, will enable us to find AD, one of the required lines, and thence DC and DB the other two. Or the solution may be conducted more analytically as follows: Put x for the angle DAC, and x' for the angle DBC; also call the given angles ADC, BDC a and a' then a, b, c, representing as usual the sides This is one equation between the unknown quantities x, x'. Another is easily obtained; for since the four angles of the quadrilateral ABCD make up four right angles or 360°, we have x + x' + a + a' + ACD + BCD = 360°; the sum of the two latter angles may become known, since in the triangle ABC the angle c is determinable from the three given sides; therefore all the terms in the first member of this equation are known except x and x'. Call the sum of the known quantities B, and we shall thus have x' =ẞ-x, and, consequently by substitution, equation (2) becomes The first term of this second member may be easily calculated by logarithms, and this added to the natural cotangent of ẞ gives the nat. cot. of x, and thence r' is known from the equation x' = ß — x, and CD from either of the equations (1). This problem has a useful application in the survey of harbors. Let the angles be taken with a sextant, from a boat, at a point where a sounding is made, to three stations on the shore. After having drawn upon a map the triangle, of which these three stations are the vertices, the following simple and elegant construction will determine the point where the sounding was made. Upon the line joining two of the stations, on the map, make a segment, capable of containing the angle observed from the place of sounding, and subtended by this line (Plane Geom., Prob. 21); upon a line joining one of these two stations and the third, make another segment that will contain the angle observed to be subtended by this last line, and the intersection of the arcs of these two segments will determine the point on the map, corresponding to that at which the sounding was made. PROBLEM IV. B Given the angles of elevation of an object taken at three places on the same horizontal straight line, together with the distances between the stations; to find the height of the object and its distance from either station. Let AB be the object, and c, c', c'', the three stations, then the triangles BCA, BCA, BCA, will all be right angled at A; and, therefore, to radius BA, AC, AC', AC", will be the tangents of the angles at B, or the cotangents of the angles of elevation; hence, putting a, a', a'', for the angles of elevation, r for the height of the object, and a, b, for the distances cc', c'c'', we shall have in order to eliminate c'r, multiply the first by b, the second by a, and add, and we shall have x2 (b cot2 a+ a cot3 a'') = (a + b) x2 cot2 a' + ab (a+b) ..I = ab (a + b) b cota + a cot? a'' — (a + b) cot3 a' If the three stations are equidistant, then a = b, and the expression becomes The height A B being thus determined, the distances of the stations from the object are found by multiplying this height by the cotangents of the angles of elevation. EXAMPLE. 22 Given the hypothenuse a = 6512.4 yards, b= 6510.6, to find c Upon inspecting the tables that are calculated to seven places of decimals only, it will be seen that, when the angles become very small, the cosines differ very little from each other. The same remark applies, of course, to the sines of angles nearly 900. In cases, therefore, where great accuracy is required, we may commit an important error by calculating a small angie from its cosine, or a large one from its sine. We must consequently endeavor to avoid this, by transforming the expression employed. In the example before us, c is a small angle which has been calculated from its cosine; we must, therefore, if possible, calculate this angle by means of its sine, or. some other trigonometrical function. Now, by formula (8), Art. 72, we have generally Instead of 1° 20' 50", as obtained by the former process. Or c might first be calculated from a and b, and then c by means of its sine. No angle which is nearly 90° ought to be calculated from its tangent, for the tangents of large angles increase with so much rapidity, that the results, derived from the column of proportional parts found in the tables, cannot be depended on as accurate. |