Tables, therefore, do not go beyond 45°; or, rather, are so arranged that each number answers as a function of both an angle less than 45° and its complement greater than 45°. TABLE OF LOGARITHMIC SINES, COSINES, &c. 98. A TABLE of LOGARITHMIC SINES, COSINES, &c. contains the logarithms of the numbers expressing the natural sines, cosines, &c. 99. Since the sines and cosines are never greater than 1, and tangents likewise, when under 45°, their logarithms properly have negative characteristics. But to avoid the inconvenience of these, the characteristics are, by common consent, increased by 10. Thus the characteristic 9 is used in the place of —1, 8 in place of -2, &c. The radius, therefore, of the logarithmic sines, cosines, &c. is, as arbitrarily assumed, 101o, or 10,000,000,000. 100. In the accompanying table the degrees are given at the top and bottom of the page, and the minutes in the columns at the sides designated by M. The column headed D contains the increase or decrease for 1 second. This is obtained by taking one sixtieth of the difference between the logarithmic sine, cosine, &c. of an angle or arc, aud that next exceeding it by 1 minute. The result is placed against the lesser angle or arc. TO FIND THE LOGARITHMIC SINE, &C. OF ANY ANGLE OR ARC. 101. If the angle or arc is less than 45°, look for the degrees at the top of the table, and for the minutes on the left; then, opposite to the minutes, on the same horizontal line, and in the column headed Sine, will be found the logarithmic sine; in the column headed Cosine will be found the logarithmic cosine, &c. Thus, the logarithmic sine of 19° 23' is 9.520990, 66 66 66 66 cosine of 31° 47' " 9.929442, tangent of 43° 5' " 9.970922. 102. If the angle or arc is between 45° and 90°, look for the degrees at the bottom of the table, and for the minutes on the right; then, opposite to the minutes, and in the column designated at the bottom Sine, will be found the logarithmic sine; in the column designated at the bottom Cosine will be found the logarithmic cosine, &c. Thus, the logarithmic sine of 80° 11' is 9.993594, 66 66 66 cosine of 65° 59' " 9.609597, 103. If the angle or arc is between 90° and 180°, subtract it from 180°, and take the logarithmic sine, &c. of the remainder. Thus, the logarithmic sine of 112° is the logarithmic sine of 68°. 104. If the angle or arc is expressed in degrees, minutes, and seconds, find the logarithmic sine, &c. of the degrees and minutes as before; then multiply the number opposite, in the column headed D, by the seconds, and add the product to the number first found, for sines and tangents, but subtract it for cosines and cotangents. Thus, if the logarithmic sine of 30° 25′ 42′′ is required, The logarithmic sine of 30° 25' is 9.704395 It is customary to omit the decimal figures at the right, but to increase the last figure retained, by 1, when the figure at the left of those omitted is 5 or greater than 5. 105. The secants and cosecants are not included in the table, since they may be readily derived from the cosines and sines. By (5), sec A cos A 1, and log sec A+ log cos A = 0 ; but as log sec and log cos are each increased by 10 (Art. 99), the second member of the equation must be increased by 20, that is, logarithmic secant = 20 logarithmic cosine. In like manner, logarithmic cosecant = 20 logarithmic sine. Hence, to find the logarithmic secant, subtract the logarithmic cosine from 20; and to find the logarithmic cosecant, subtract the logarithmic sine from 20. Thus, The logarithmic secant of 65° 59' 66 is 10.390403 66 cosecant of 30° 25′ 42′′ 66 10.295454. TO FIND THE ANGLE OR ARC CORRESPONDING TO ANY LOGARITHMIC SINE, &C. 106. Look in the column designated by the same name with the given logarithm for the sine, &c. which is nearest to the given one, and if the name be at the head of the column, take the degrees at the top of the table, and the minutes on the left; but if the name be at the foot of the column, take the degrees at the bottom, and the minutes on the right. Thus, The angle or arc corresponding to the logarithmic sine 9.681443 is 28° 42'. The angle or arc corresponding to the logarithmic tan 9.984079 is 43° 57'. The angle or arc corresponding to the logarithmic cos 9.731603 is 57° 23'. 107. If the given logarithmic sine, &c. is not found exactly, or very nearly, then, to find the seconds, subtract from the given logarithm that next less in the table, to the remainder annex two ciphers, divide the result by the number in the column headed D, and the quotient will be the number of seconds to be added to the degrees and minutes of the lesser logarithm for sines and tangents, or to be subtracted for cosines and cotangents. Thus, to find the angle or arc corresponding to the logarithmic sine 9.938070. The log sine 9.938070 has for its cor. angle or arc, 60° 7' 25' The angle or are corresponding to the logarithmic tangent 9.497200 is 17° 26' 33". The angle or arc corresponding to the logarithmic cosine 9.792477 is 51° 40' 30". EXAMPLES. Ans. 9.681443. 1. Required the logarithmic sine of 28° 42'. Ans. 9.704657. 3. Required the logarithmic cotangent of 127° 2'. Ans. 9.877640. 4. Required the logarithmic sine of 81° 20'. Ans. 10.207523. 6. Required the logarithmic tangent of 74° 21′ 20′′. Ans. 10.552778. 7. Required the logarithmic cosecant of 102° 24′ 41′′. Ans. 10.010270. 8. Required the logarithmic tangent of 1° 59′ 51′′.8. Ans. 8.542587. 9. Required the angle of the logarithmic sine 9.999969. Ans. 89° 19'. 10. Required the arc of the logarithmic tangent 9.645270. Ans. 23° 50′ 17′′. 11. Required the angle of the logarithmic cosine 9.598075. Ans. 66° 39'. 12. Required the angle of the logarithmic cotangent 10.301470. Ans. 26° 32′ 31′′. 13. Required the arc of the logarithmic sine 9.893410. Ans. 51° 28' 40". 14. Required the angle of the logarithmic cosine 9.421157. Ans. 105° 17' 29". 15. Required the arc of the logarithmic tangent 9.692125. Ans. 26° 12′ 20′′. 16. Required the angle of the logarithmic cotangent 9.421901. Ans. 75° 12' 6" BOOK III. SOLUTION OF PLANE TRIANGLES. 108. THE SOLUTION OF TRIANGLES is the process by which, when the values of a sufficient number of their elements are given, the values of the remaining elements are computed. The elements of every triangle are the three sides and the three angles. Three of these elements must be given, one of which must be a side, in order to solve a plane triangle. The solution of plane triangles depends upon the following FUNDAMENTAL PROPOSITIONS. 109. In a right-angled triangle, the side opposite to an acute angle is equal to the product of the hypothenuse into the sine of the angle; and the side adjacent to an acute angle is equal to the product of the hypothenuse into the cosine of the angle. Let ABC be a triangle having a right angle at C; then, by (1), 110. In a right-angled triangle, the side opposite to an acute angle is equal to the product of the other side into the tangent of the angle; and the side adjacent to an acute angle is equal to the product of the other side into the cotangent of the angle. |