the calculus, for the present it will suffice to graph a few examples of this type. Let the expression be

A table of values of t and y must first be derived. There are three ways of proceeding: (1) Assign successive values to t irrespective of the period of the sine (see Table V and Fig. 112). (2) Select for the values of t those values that give aliquot parts of the period 2π of the sine (see Table VI and Fig. 113). (3) Draw the sinusoid y sin t carefully to scale by the method of §55; then draw upon the same coördinate axes, using the same units of measure

=

adopted for the sinusoid, the exponential curve y = e-/5; finally multiply together, on the slide rule, corresponding ordinates taken from the two curves, and locate the points thus determined.

The first method involves very much more work than the second for two principal reasons: First, tables of the logarithms of the trigonometric functions with the radian and the decimal divisions of the radian as argument are not available; for this reason 57.3° must be multiplied by the value of t in each case so that an ordinary trigonometric table may be used; second, each of the values written

in column (3) of the table must be separately determined, while if the periodic character of the sine be taken advantage of, the numerical values would be the same in each quadrant.

The second method, because of the use of aliquot divisions of the period of the sine, such as π/6 oг π/12 ог π/18 or π/20, etc., possesses the advantage that the values used in column (3) need be found for one quadrant only and the values required in column (2) are quite as readily found on the slide rule as in the first method.

The third method is perhaps more desirable than either of the others if more than two figures accuracy is not required. The curve can readily be drawn with the scale units the same in both dimensions, as is sometimes highly desirable in scientific applications.

In Figs. 112 and 113 a larger unit has been used on the vertical scale than on the horizontal scale. In Fig. 113 the horizontal unit is incommensurable with the vertical unit. To draw the curve to a true scale in both dimensions it is preferable to lay off the

coördinates on plain drawing paper and not on ordinary squared paper. Rectangular coördinate paper is not adapted to the proper construction and discussion of the sinusoid, or of curves, like the present one, that are derived therefrom.

y

Curves whose equations are of the form y

=

=

e-/5 sin t or 3e-1/5 sin t, etc., are readily constructed, since the constants 1/2, 3, etc., merely multiply the ordinates of (1) by 1/2, 3, etc., as the case may be. Likewise the curve y e-bx sin cx is readily drawn since sin cx can be made from sin x by multiplying all abscissas of sin x by 1/c.

=

CHAPTER IX

TRIGONOMETRIC EQUATIONS AND THE SOLUTION OF

TRIANGLES

A. FURTHER TRIGONOMETRIC IDENTITIES

159. Proof that p = a cos + b sin 0 is a Circle. I. Geometrical Explanation. We know (§64) that p1 = a cos 0 is the polar equation of a circle of diameter a, the diameter coinciding in direction with the polar axis OX; for example, the circle OA,

Fig. 114. Likewise, p2 = b sin 0 is a circle whose diameter is of length b and makes an angle of +90° with the polar axis OX, as the circle OB, Fig. 114. Also, pc cos (0 - 01) is a circle whose diameter c has the direction angle 01. See equation (4), §68. We shall show that if the radii vectores corresponding to any value of in the equations p1 = a cos 0 and p2 = b sin be added together to Cir- make a new radius vector a

FIG. 114.-Combination of the cles pa cos and p = b sin 0 into Single Circle p = a cos 0 + b sin 0.

X

a circle (the circle OC, Fig. 114) other words we shall show that:

p, then, for all values of 0, the extremity of p lies on of diameter Va2+b2. In

p = a cos 0 + b sin 0

is the equation of a circle.

(1)