EXAMPLE 1.-To find the logarithmic sine of 1° 3′ 45′′ (= A). SOLUTION.-Opening the table at page 43 (headed 1°), we look for 4' in the minute column, since 3′ 45′′ is nearer to 4' than to 3'. Horizontally opposite 4, and in the column headed S T, the sine correction 6.68555 (= S) is found. We now look in the minute column for the number of minutes (3) in the given angle; horizontally opposite it in the left-hand column is the number 3,780, number of seconds in 1° 3'; adding 45", we obtain 3,825 (= A") for the total number of seconds in the given angle. EXAMPLE 2.-To find the logarithmic tangent of 2° 36′ 17′′. SOLUTION.-On page 44, the correction for the tangent, opposite Number of seconds opposite 36' in the left-hand 36', is 6.68587 (= T). 3. log tan 2° 36′ 17" = 2.65793. Ans. To Find the Logarithmic Cotangent of an Angle Between 0° and 3°. Rule.-Find C, A", and log A" exactly as in the last article, C being taken from the correction column next to the cotangent column. Subtract log A" from C. The result will be the required logarithmic cotangent. EXAMPLE.-To find the logarithmic cotangent of 1° 52′ 37′′. SOLUTION.-On page 43, the correction under C, and horizontally opposite 53', is 5.31427; A" = 6,720 +37 = 6,757. 4. To Find the Logarithmic Tangent, Cosine, or Cotangent of an Angle Between 87° and 90°.-These functions also are to be taken from the first three pages of the table of logarithmic functions. The simplest way to proceed is to subtract the angle from 90° and look for the corresponding complementary function as explained in Arts. 2 and 3. Thus, log cos 88° 55′ 38′′ is obtained by looking for log sin (90° 88° 55′ 38′′) = log sin 1° 4′ 22′′. EXAMPLES FOR PRACTICE 1. Find the logarithmic sine of 1° 6′ 19′′. 2. Find the logarithmic sine of 0° 2′ 41′′. 3. Find the logarithmic tangent of 2° 56′ 57′′. Ans. 2.28532 Ans. 4.89240 Ans. 2.71196 Ans. 1.58049 Ans. 2.54157 5. To Find the Angle Corresponding to a Given Logarithmic Function, When the Function Lies Between Two of the Functions in the First Three Pages of the Table.-I. Sine and Tangent.-As explained in Art. 1, log sin AS+ log A"; therefore, From these formulas is derived the following Rule.-Find in the table the logarithm nearest to the given one. Take the correction horizontally opposite this logarithm, and subtract it from the given logarithm. The result will be the logarithm of the total number of seconds (A") in the given angle. Find the number corresponding to this logarithm, and reduce it to degrees, minutes, and seconds. It is here assumed that the given function lies between two functions in the column marked log sin or log tan, as the case may be, at the top. If the names of the functions are at the bottom, the sine should be treated as in Plane Trigonometry, Part 1; the tangent should be treated as if it were a cotangent, according to the directions to be given presently, and when the angle corresponding to that cotangent is found, it should be subtracted from 90°. = II. Cotangent.-Since log cot A Clog A" (Art. 3), we have log A" From this formula is derived the following Rule.-Find in the table the logarithmic function nearest the given cotangent. Take from the C column the correction horizontally opposite the logarithm just found, and from it subtract the given logarithmic cotangent. The result will be the logarithm of the total number of seconds in the angle. If Here, as before, it is assumed that the given cotangent lies between two of those marked log cot at the top. it lies between two logarithms in the column marked log cot at the bottom, it should be treated as if it were a tangent, and having found the angle corresponding to this tangent, it should be subtracted from 90° to obtain the required angle. III. Cosine. Rule. If the given cosine lies between two of those in the column headed log cos, apply the general rule given in Plane Trigonometry, Part 1. If it lies between two of the logarithms in the column marked log cos at the bottom, treat it as if it were a sine, find the angle corresponding to that sine as above, and subtract the result from 90°. EXAMPLE 1.-To find the angle whose logarithmic tangent is 2.32803. SOLUTION.-The logarithmic tangent nearest to 2.32803 is 2.32711, found in the column headed log tan on page 43. The T correction horizontally opposite 2.32711 is 6.68564. EXAMPLE 2.-To find the angle whose logarithmic cotangent is 2.49567. SOLUTION.-The nearest logarithmic cotangent found in the table is 2.49488. The number opposite this logarithm in the C column is 5.31442. C: = 5.31442 = 2.49567 2.81875; A" = 659′′ = 0° 10′ 59′′. Ans. NOTE.-Angles are here given to the nearest whole second. EXAMPLE 3.-To find the angle whose logarithmic cosine is 2.63723. SOLUTION.-The nearest logarithm, 2.63678, is found on page 44, in the column headed log sin. The given function is, therefore, to be treated as if it were a logarithmic sine, and the angle A, corresponding to this sine is to be subtracted from 90° to obtain the required angle A. The correction horizontally opposite 2.63678, in the S column, is 6.68544. 6. Use of the Column of Seconds for Obtaining the Angle Corresponding to a Given Function.-In order to avoid confusing the student by too many rules, the reduction of A" to degrees, minutes, and seconds was, in the preceding articles, effected by the ordinary rules of arithmetic, without any reference to the table. The following is a more expeditious method: Let the given function lie between the functions of two consecutive angles, A, and A, +1'. Then, the degrees and minutes in the required angle are those in A,, and may be at once written down. The number in the column of seconds on the left, horizontally opposite the number of minutes in A,, gives the total number of seconds in A,. Denoting that number by A," and the number of odd seconds in the required angle by s, we have s= A" - A EXAMPLE.-To find the angle whose logarithmic tangent is 2.30217. SOLUTION.-The given function lies between 2.29629 and 2.30263. The angle corresponding to the first of these two functions is 1° 8' (= A1); A," = 4,080". The subtraction A" - A," can usually be effected mentally. EXAMPLES FOR PRACTICE Apply the method just described to the Examples for Practice given after Art. 4. + GENERAL TRIGONOMETRIC FORMULAS ANGLES AND THEIR TRIGONOMETRIC FUNCTIONS -360° C20 180° B -90° 7. Angle of Any Magnitude. In trigonometry, an angle is considered as being generated by a straight line turning about one of its ends, which is the vertex of the angle. In this motion, any point in the turning line describes a circular arc, whose number of degrees is the measure of D the angle. The turning line is called the generating line. The position that this line occupies before it begins to turn, and from which arcs are meas ured, is called the initial FIG. 1 line, or the initial position of the generating line; and the position it occupies after turning through a certain angle |