13. If we take an arc, ABET, L greater than 90°, its sine 'will be FH; OH will be its cosine; AQ its tangent, and OQ its secant. But FH is the sine of the arc GF GLI ODHA which is the supplement of AF, and OH is its cosine; hence, the sine of an arc is equal to the sine of its supplement; and the cosine of an arc is equal to the cosine of its supplement* Furthermore, AQ is the tangent of the arc AF, and OQ is its secant: GL is the tangent, and OL the secant of the supplemental arc GF. But since AQ is equal to GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tangent of its supplement; and the secant of an arc is equal to the secant of its supplement.* TABLE OF NATURAL SINES. 14. Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have · been calculated for every minute or second of the quad rant, and arranged in a table; such a table is called a table of sines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the values of the sines, cosines, tangents, and cotangents of all the ares of a quadrant, which is divided to minutes or seconds. If the sines, cosines, tangents, and secants are known for arcs less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the nunerical values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20°, we also know the sine of its supplement 160°; for the two are equal to each other. The Table of Natural Sines is not given, a it is much easier to make the computations by the Table which we are about to explain. ---* These relations are between the numerical values of the trigonometrical linus; the algebraic signs, which they have in the different quadrants, are not considored TABLE OF LOGARITHMIC SINES. 15. In this table are arranged the logarithms of the nunerical values of the sines, cosines, tangents, and cotangents of all the arcs of a quadrant, calculated to a radius of 10,000,000,000. The logarithm of this radius is 10 In the first and last horizontal lines of each page, are writ ten the degrees whose sines, cosines, &c., are expressed on the page. The vertical columns on the left and right, are columns of minutes. CASE I. To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle. 16. If the angle is less than 45°, look for the degrees in the first horizontal line of the different pages: when the degrees are found, descend along the column of minutes, on the left of the page, till you reach the number showing the minutes: then pass along a horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be: the number so indicated is the logarithm sought. Thus, on page 37, for 19° 55', we find, sine 19° 55'.... 9.532312 cot 19° 55'.... 10.440903 17. If the angle is greater than 45°, search for the de. grees along the bottom line of the different pages: when the number is found, ascend along the column of minutes on the right hand side of the page, till you reach the number express: ing the minutes: then pass along a horizontal line into the column designated tang, cot, sine, or cosine, as the case may be: the number so pointed out is the logarithin required. 18. The column designated sine, at the top of the page, is designated by cosine at the bottom; the one designated tang, by cotang, and the one designated cotang, by tang. The angle found by taking the degrees at the top of the page, and the minutes from the left hand vertical columu, is the complement of the angle found by taking the degrees at the bottom of the page, and the minutes from the right hand column on the same horizontal line with the first. Therefore, sine, at the top of the page, should correspond with cosine, at the bottom; cosine with sine, tang with cotang, and cotang with tang, as in the tables (Art. 12). If the angle is greater than 90°, we have only to sub tract it from 180°, and take the sine, cosine, tangent, or cotangent of the remainder. The column of the table next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner. Opening the table at any page, as 42, the sine of 24° is found to be 9.609313; that of 24° 01', 9.609597: their difference is 284; this being divided by 60, the number of seconds in a minute, gives 4.73, which is entered in the column D. Now, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 4.73 is the increase of the sine for 1". Similarly, if the arc were 24° 20', the increase of the sine for 1", would be 4.65. The same remarks are applicable in respect of the column D, after the column cosine, and of the column D, between the tangents and cotangents. The column D, between the columns tangents and cotangents, answers to both of these columns. Now, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before ; then, multiply the corresponding tabular difference by the seconds, and add the product to the number first found, for the sine of the given arc. Thus, if we wish the sine of 40° 26' 28". · Product, 69.16 to be added 69.10 Gives for the sine of 40° 26' 28". 9.812021. • The decimal figures at the right are generally omitted in the last result; but when they exceed five-tenths, the figure on the left of the decimal point is increased by 1; the logarithm obtained is then exact, to withịn less than one unit of the right hand place. . • The tangent of an arc, in which there are seconds, is found in a manner entirely similar. In regard to the cosine and cotangent, it must be remembered, that they increase while the arcs decrease, and decrease as the arcs are increased; consequently, the proportional numbers found for the seconds, must be subtracted, not added. EXAMPLES Ans. 9.884592. 3. Find the cotangent of 87° 57' 59". Ans. 8.550356. CASE II. To find the degrees, minutes, and seconds answering to any .: given logarithmic sine, cosine, tangent, or cotangent. 19. Search in the table, in the proper column, and if the number is found, the degrees will be shown either at the top or bottom of the page, and the minutes in the side column either at the left or right. But, if the number cannot be found in the table, take from the table the degrees and minutes answering to the nearest less logarithm, the logarithm itself, and also the corresponding tabular difference. Subtract the logarithm taken from the table from the given logarithm, annex two ciphers to the remainder, and then divide the remainder by the tabular difference: the quotient will be seconds, and is to be connected with the degrees and minutes before found : to be added for the sine and tangent, and subtracted for the cosine and cutangent. EXAMPLES. 1. Find the arc answering to the sine 9.880054 · Sine 49° 20', next less in the table 9.879963 Tabular difference, 1.81)91.00150". Hence, the arc 49° 20' 50" corresponds to the given sine 9.880054. 2. Find the arc whose cotangent is 10.008688 cot 44° 26', next less in the table 10.008591 · Tabular difference, 4.21)97.00(23". Hence, 44° 26' — 23" = 44° 25' 37" is the arc answering to the given cotangent 10.008688. 3. Find the arc answering to tangent 9.979110. Ans. 43° 37' 21". 4. Find the arc answering to cosine 9.944599. Ans. 28° 19' 45'. 20. We shall now demonstrate the principal theorems of Plane Trigonometry. THEOREM I. The sides of a plane triangle are proportional to the sings of their opposite angles. · 21. Let ABC be a triangle; then CB : CA :: sin A : sin B. For, with A as a centre, and AD equal to the less side BC, as a radius, describe the arc DI: and with B as a centre and the equal radius A¢- EI L F BC, describe the arc CL, and draw DE and CF perpen. D |