Carpentry

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.

About This Book

APPENDIX II