smaller. When OC has made a quarter turn and reached the position OA, the cosine is zero and the sine equals OA or the radius of the circle. After OC passes OA, the sine becomes smaller, and the cosine is to the left of A.

In the above definitions we have assumed that the radius of the circle is 1. For example, if the radius is 1 inch, then the line CB is a certain fraction of an inch, and this fraction is the sine of the angle COB. If the radius is not 1, then we must divide CB by the radius to get the sine of the angle, and for the same angle we shall get the same fraction as before. The clearest idea of the sine, however, can be obtained by assuming the length of the radius to be 1. Then

the actual length of the line CB is the sine, the actual length of EC is the cosine, and the length of line marked tangent is the tangent of the angle.

It is possible with any angle to draw a circle whose radius is one unit, with the center of the circle at the

point where the lines forming the angle meet. Then the sine, cosine and tangent lines can be drawn as in Fig. 17.

It can readily be seen, by referring to Fig. 17, that the farther the line OC turns in the direction of the arrow the longer the sine and tangent lines become, and the shorter the cosine line becomes. So it is clear that as the angle increases from 0° to 90° the sine and

Sine, cosine, and tangent of an angle greater than 90° and less than 180:

FIG. 18.

tangent become larger and the cosine becomes smaller. At 90° the cosine is 0, the sine is 1, or equal to the radius, and the tangent is so long that we can not measure it, and we call it infinity.

If the angle is greater than 90° and less than 180°, the cosine line, as Fig. 18 shows, is measured to the left from the vertical line AO. We call it a minus

cosine, because it is opposed in direction to the plus cosine of our first angle, which is between 0° and 90°. Between 180° and 270° the sine is minus, because it is measured downward from the horizontal line OB and is opposed in direction to the plus sine of the first angle. Between 270° and 360° the sine is minus and the cosine is plus. Table XIII gives the sines, cosines and tangents of some common angles.

Fig. 19 shows the sine curve for a single armature coil. You will observe that the dotted lines marked 30°, 60°, etc., are equal to the sines of these angles. The first dotted line in the curve for example represents the distance the coil is from the horizontal line when it has turned 30 degrees. It also represents the rate at which the wire is cutting magnetic lines of force. It also represents the electrical pressure, and that is the important thing. The pressure is proportional to the sine of the angle.

The wavy line to the right of the circle is therefore

the pressure curve. It corresponds exactly to the steam-pressure curve marked out by the steam-engine indicator. The height of this curve above the horizontal line for any angle from 0° to 180° represents the electrical pressure, or voltage, when the coil is at that particular angle. This height, as we have seen, is the sine of the angle.

At 90° the curve has its greatest height, and the coil is perpendicular to the O line. The pressure is greatest at this point. The pressure for any other angle is, therefore, the maximum pressure; that is, the 90° pressure multiplied by the sine of the angle. For example, the sine of 30° is one-half, and the electrical pressure at 30° is one-half of the pressure at 90°.

From 180° to 360° the curve is below the horizontal line, which means that the pressure is minus, or in the reverse direction from the first. The pressure is, therefore, alternating in direction, and we have an alternating current in the armature. If armature is connected to line by collector rings instead of commutator, there will be an alternating current in the line. The distance below the line, that is, the sine of the angle, measures the amount of this minus pressure for any angle.

Fig. 20 is the pressure curve for two armature coils wound at an angle of 90° apart on the same armature. The heavy wave lines are the curves for the two coils separately. The dotted wave line shows the line pressure produced by the two coils acting together. For any angle the height of the dotted line above the horizontal is the sum of the heights of the two heavy lines. At 45° the dotted line crosses the horizontal. At this point one heavy line is as far above the horizontal as the other is below it; that is to say, the pressure in one coil is plus and the pressure in the other coil is equal and in the opposite direction to the first. The second pressure is, therefore, minus. These two pres

sures acting together actually produce zero pressure in the line. The last statement explains the fact that one pressure is called plus and the other minus. If we add a plus quantity to an equal minus quantity the sum is zero. Fig. 20 is the pressure curve for a

AA

190°

180

210 360

FIG. 20.

two-phase current. When one pressure is at its greatest value the other is at zero, because when one coil is at 90° the other is at 0°. Fig. 21 is the pressure curve for three coils wound on the same armature and 120° apart. This is the curve for a three-phase current.

Idle Current and Power Factor.

In an alternating-current circuit the actual power lags a certain angle behind the apparent power. The curves, Figs. 19 and 20, represent voltage. Voltage