and by referring to a table of sines, we find that cosine 0·7262 = angle 43° 26´, therefore angle DAC = 43½°, about. Also
| 142 + 42 − 112 | = | 196 + 16 − 121 | = | 91 | = | 0·8125, |
| 2 × (14 × 3) | 112 | 112 |
and by referring to a table of sines and cosines, we find cosine 0·8125 = angle 35⅔°.
We thus find that to move the connecting pin 1 inch to the front of the stroke, in the loom with 11-inch arm, the 4-inch crank will move through 35⅔°, and in the loom with the 12-inch arm the 3-inch crank will move through 43½° for the same movement of the slay. Assuming the force exerted by the latter to be 1, the force of the former will be as 35⅔ squared : 43½ squared
It may be as well here to give a short explanation of the system of obtaining angles by sines and cosines.
As the crank moves forward it is obvious that the line DQ will become shorter, and as the angle becomes larger the line DQ will increase in length. In trigonometry, the ratio between the length of the line DQ and the radius AD is called the sine of the angle, and if the radius is 1, the length of DQ will be the value of the sine. In an angle of 30° the sine is exactly ½ the radius, and the relation between the radius and the sine for every angle is known, and arranged in “tables of sines.” The length of AQ will also vary with the angle, and the length of this line is called the “cosine” of the angle QAD. The cosine of an angle of 30° is therefore the same as the sine of an angle of 60°. When the sine is known it is easy to obtain the cosine as follows:—
Cosine = √1 − sin2. Thus for an angle of 30°, cosine = √1 − 0·52, or cos2 = 1 − 0·52, therefore cos2 = 1 − 0·25, or cos2 = 0·75, ∴ cos = √0·75 = 0·866. By reversing, the sine may be obtained from the cosine.
The value given to the sines and cosines must not be taken for the actual length of the lines; they are simply the ratio to the radius. Thus in an angle of 30°, if the radius is 1 inch the length of the sine will be ½ inch and the cosine 0·866 inch. If the radius is 2 inches, the actual length of the sine will be 1 inch and of the cosine 1·732 inches.