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 = .7660
  which is as might have been expected. Since 40 degrees is the
  complement of 50 degrees, the cos 40 degrees should be the same in
  value as the sin 50 degrees.

Example 4.—Find the value of cos 87 degrees.

Solution—Columns 6 and 2 reading up, cos 87 degrees = .0523

Example 5.—Tangent and cotangent values. Proceed as with sines
  using columns 1 and 3, reading down, for tangent values between 0-45
  degrees inclusive, columns 6 and 4, reading up, for values between
  45-90 degrees.

  For cotangent values between 0-45 degrees use columns 1 and 4 reading
    down, and columns 6 and 3 reading up for cotangent values between
    45-90 degrees inclusive.

TABLE OF NATURAL SINES, TANGENTS, COSINES, AND COTANGENTS

Degrees Sine Tangent Cotangent Cosine
0 0 0 ∞ 1 90
1 .0175 .0175 57.2900 .9998 89
2 .0349 .0349 28.6363 .9994 88
3 .0523 .0524 19.0811 .9986 87
4 .0698 .0699 14.300 .9976 86
5 .0872 .0875 11.4301 .9962 85
6 1045 .1051 9.5144 .9945 84
7 1219 .1228 8.1443 .9925 83
8 1392 .1405 7.1154 .9903 82
9 1564 .1584 6.3138 .9877 81
10 .1736 .1763 5.6713 .9848 80
11 .1908 .1944 5.1446 .9816 79
12 .2079 .2126 4.7046 .9781 78
13 .2250 .2309 4.3315 .9744 77
14 .2419 .2493 4.0108 .9703 76
15 .2588 .2679 3.7321 .9659 75
16 .2756 .2867 3.4874 .9613 74
17 2924 .3057 3.2709 .9563 73
18 3090 .3249 3.0777 .9511 72
19 .3256 .3443 2.9042 .9455 71
20 .3420 .3640 2.7475 .9397 70
21 .3584 .3839 2.6051 .9336 69
22 .3746 .4040 2.4751 .9272 68
23 .3907 .4245 2.3559 .9205 67
24 .4067 .4452 2.2460 .9135 66
25 .4226 .4663 2.1445 .9063 65
26 .4384 .4877 2.0503 .8988 64
27 .4540 .5095 1.9626 .8910 63
28 .4695 .5317 1.8807 .8829 62
29 .4848 .5543 1.8040 .8746 61
30 .5000 .5774 1.7321 .8660 60
31 .5150 .6009 1.6643 .8572 59
32 .5299 .6249 1.6003 .8480 58
33 .5446 .6494 1.5399 .8387 57
34 .5592 .6745 1.4826 .8290 56
35 .5736 .7002 1.4281 .8192 55
36 .5878 .7265 1.3764 .8090 54
37 .6018 .7536 1.3270 .7986 53
38 .6157 .7813 1.2799 .7880 52
39 .6293 .8098 1.2349 .7771 51
40 .6428 .8391 1.1918 .7660 50
41 .6561 .8693 1.1504 .7547 49
42 .6691 .9004 1.1106 .7431 48
43 .6820 .9325 1.0724 .7314 47
44 .6947 .9657 1.0355 .7193 46
45 .7071 1.0000 1.0000 .7071 45
Cosine Cotangent Tangent Sine Degrees

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 from
    0-90 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 = 1.1918
                tan 51 degrees = 1.2349
  difference for an interval of 1 degree = .0431
  20 min. = 20/60 = 1/3 of 1 degree; ⅓ of .0431 = .0144
  tan 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. = .8988
        cos 27 deg. = .8910
  difference for interval of 1 deg. = .0078
  30 min. = ½ of 1 deg.; ½ of .0078 = .0039
  cos. 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 .5

Solution—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. = .0123
  123/218 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.