APPENDIX II
Directions. An examination of the table of natural functions will indicate in the column at the left, angles of degrees to and including 45 degrees, reading down. The column to the extreme right will be found to contain degrees from 45-90 inclusive, reading up.
This compact arrangement of table is made possible thru the fact that sines and cosines, tangents and cotangents are reciprocals one of the other. That is, as the sine (column 2, reading down) increases in value, the cosine of the complementary angle (columns 6 and 2, reading up) decreases.
Example 1.—Find the value of the sine of 40 degrees.Solution—Columns 1 and 2, reading down, sin 40 degrees = .6428.Example 2.—Find the value of sin 50 degrees.Solution—Columns 6 and 5, reading up, sin 50 degrees = .7660.Example 3.—Find the value of cos 40 degrees.Solution—Columns 1 and 5, reading down, cos 40 degrees = .7660which is as might have been expected. Since 40 degrees is thecomplement of 50 degrees, the cos 40 degrees should be the same invalue as the sin 50 degrees.Example 4.—Find the value of cos 87 degrees.Solution—Columns 6 and 2 reading up, cos 87 degrees = .0523Example 5.—Tangent and cotangent values. Proceed as with sinesusing columns 1 and 3, reading down, for tangent values between 0-45degrees inclusive, columns 6 and 4, reading up, for values between45-90 degrees.For cotangent values between 0-45 degrees use columns 1 and 4 readingdown, and columns 6 and 3 reading up for cotangent values between45-90 degrees inclusive.
TABLE OF NATURAL SINES, TANGENTS, COSINES, AND COTANGENTS
TO FIND THE VALUE OF AN ANGLE, THE VALUE OF A FUNCTION BEING KNOWN
Example 6.—sin = .5150, find the angle.Solution—Looking in columns 2 and 5 (sine values from0-90 degrees) Ans. 31 degrees (Columns 2 and 1).Example 7.—cot = 1.3764, find the angle.Solution—Looking in columns 3 and 4, Ans. = 36 degrees.
Interpolation.—Frequently one must find a functional value for fractional degrees, or degrees and minutes. Also, it becomes necessary to find the value of an angle with greater accuracy than even degrees, as given in the table herewith. This process of finding more accurate values is known as interpolation.
TO FIND THE VALUE OF A FUNCTION WHEN THE ANGLE IS IN FRACTIONAL DEGREES
Example 8.—Find the value of tan 50 degrees 20 min.Solution.—tan 50 degrees = 1.1918tan 51 degrees = 1.2349difference for an interval of 1 degree = .043120 min. = 20/60 = 1/3 of 1 degree; ⅓ of .0431 = .0144tan 50 degrees 20 min. = 1.1918 + .0144 = 1.2062.
The value of a fractional degree would be similarly treated for the sine, these functions increasing as the value of the angle increases. The cosine and cotangent, however, decrease in value as the angle increases. For this reason the fractional value of the cosine and cotangent must be subtracted from, instead of added to, the value of the function of the next lower number of degrees.
Example 9.Find the value of cos 26 deg. 30 min.Solution—cos 26 deg. = .8988cos 27 deg. = .8910difference for interval of 1 deg. = .007830 min. = ½ of 1 deg.; ½ of .0078 = .0039cos. 26 deg. 30 min. = .8988 - .0039 = .8949.
TO FIND THE VALUE OF AN ANGLE WHEN THE FUNCTIONAL VALUE CANNOT BE FOUND IN EXACT FORM IN THE TABLE
Example 10.—Find the angle whose tan is .5Solution—From the table, .4877 = tan 26 deg..5095 = tan 27 deg.difference for interval of 1 deg. = .0218.5000 = tan angle X..4877 = tan 26 deg.difference for interval between tan angle X and tan 26 deg. = .0123123/218 of 1 deg. or 60 min. = 34 min.Therefore, angle whose tangent = .5 = 26 deg. 34'.
Rule: (1) Search the body of the table for the functional values next above and next below that given. (2) Find the difference between these functional values. This difference is for an interval of 1 degree or 60 minutes. (3) Find the difference between the given functional value and that of the lower angle of the two used above. (4) Express this last difference as the numerator of a fraction whose denominator is the first difference found, or the difference for the interval of 1 degree. This gives the fractional part of 1 degree or 60 minutes which the second difference is. (5) Express this difference in minutes and add if the function be a sine or tangent, and substract if a cosine or cotangent to the number of degrees representing the angle whose function was the lower of the two functions found given in the table.