180. In order to illustrate the use of the tabular logarithmic functions we give the following extract from the table of logarithmic sines, cosines,... in Chambers' Mathematical Tables. 56 9-6706576 2382 670 2380 59 2378 57 9.6708958 9.9461362 670 9.9460692 671 9-9460021 672 9.9459349 4321O Sine 62 Deg. 181. We have quoted here the logarithmic sines, cosecants, secants, and cosines of the angles differing by 1' between 27° 0' and 27° 4′, and also between 27° 56′ and 27° 60′. The same extract gives the logarithmic functions of the complements of these angles, namely those between 62° O' and 62° 4′, and those between 62° 56′ and 62° 60'. The column of minutes for 27° is given on the left and increases downwards, the column for 62° is on the right and increases upwards. The names of the functions printed at the top refer to the angle 27°, the names printed at the foot refer to the angle 62°. Thus L cos 27° 3'9.9496876, Lcosec 27° 58′=10·3288662, The first difference column gives the differences in the logrithms of the sines and cosecants, the second difference column gives the differences in the logarithms of the cosines and secants, each difference corresponding to a difference of l' in the angle. Example 2. Given L sec 27° 39′=10·0526648, diff. for 10"=110, find A when L sec A=10·0527253. 1. Find sin 38° 3′ 35′′, having given that 2. Find tan 38° 24′ 37′5′′, having given that tan 38° 25'='7930640, tan 38° 24'7925902. 3. Find cosec 55° 21′ 28′′, having given that cosec 55° 22′ = 1.2153535, cosec 55° 21'=12155978. 4. Find the angle whose secant is 2-1809460, given sec 62° 43′ 2.1815435, sec 62° 42′ = 2.1803139. 5. Find the angle whose cosine is 8600931, given cos 30° 41'8600007, cos 30° 40''8601491. 6. Find the angle whose cotangent is 8766003, given cot 48° 46'8764620, cot 48° 45′ =8769765. 7. Find L sin 44° 17′ 33′′, given L sin 44° 18' 9'8441137, L sin 44° 17′=9·8439842. 8. Find L cot 36° 26′ 16′′, given L cot 36° 27'10·1315840, L cot 36° 26'=10·1318483. 9. Find L cos 55° 30′ 24′′, given L cos 55° 31'=97529442, L cos 55° 30′ 9.7531280. 10. Find the angle whose tabular logarithmic sine is 9.8440018, using the data of example 7. 11. Find the angle whose tabular logarithmic cosine is 9-7530075, using the data of example 9. 12. Given L tan 24° 50′ =9·6653662, diff. for l'=3313, find L tan 24° 50' 52.5". 13. Given L cosec 40° 5′-10·1911808, diff. for 1'=1502, find L cosec 40° 4' 17.5′′. 182. Considerable practice in the use of logarithmic Tables will be required before the quickness and accuracy necessary in all practical calculations can be attained. Experience shews that mistakes frequently arise from incorrect quotation from the Tables, and from clumsy arrangement. The student is reminded that care in taking out the logarithms from the Tables is of the first importance, and that in the course of the work he should learn to leave out all needless steps, making his solutions as concise as possible consistent with accuracy. Example 1. Divide 6.6425693 by 3873007. Example 2. The hypotenuse of a right-angled triangle is 3·141024 and one side is 2.593167; find the other side. Let c be the hypotenuse, a the given side, and ≈ the side required; then c=3*141024 a=2*593167 c+a=5734191 c-a 547858 1 [In this exercise the logarithms are to be taken from the Tables.] 1. Multiply 300-2618 by 0078915194. 2. Find the product of 235 6783 and 357·8438. 3. Find the continued product of 153-2419, 2.8632503, and 07583646. 4. Divide 1.0304051 by 27.093524. 5. Divide 357.8364 by 00318973. 6. Find a from the equation 0178345x=21.85632. 7. Find the value of 3.78956 x 0536872÷0072916. H. K. E. T. |