SECT. IV.

ON CIRCULAR FUNCTIONS.

DEF. IF the vertex of any angle be made the centre of a circle, whose radius is equal to a linear unit, the numerical value of this angle is represented by that number, which expresses the length of the circular arc intercepted between the two straight lines, containing the angle.

Thus, let BAC be an angle, CA equal a linear unit, and let CB be a circular arc, described from centre A, with radius CA, then if (0) be the

numerical value of the length of the arc CB, it will also, by the definition, be the numerical value of the angle BAC.

In general, when we call an angle (0), we mean by (0) its numerical value, and when we make use of the symbols, sin 0, cos 0, and tan 0, we intend them to represent the sine, cosine, and tangent of the angle (0), to radius unity.

If (x) be the arc, which the angle (0) subtends at the centre of a circle, whose radius = a; and if sin %, cos and tan %, be the sine, cosine, and tangent of the angle to this radius, we have, (by Trigonometry)

In the succeeding propositions, we shall take for granted the following well-known axiom, viz. that the length of a circular arc is less than that of its tangent, and greater than that of its chord or sine.

PROP. XXXV.

If (0) be an angle, the differential coefficient of its sine, considered as a function of (0), is equal to cos 0.

For if we suppose (h) to be the increment of (0),

we have, by Trigonometry,

2 cos 0.sin h=sin (0+h) — sin (0 – h)

Now, since by the axiom we have assumed (h) is greater than sin h, h-sin h, or the series

than 1. (For if it were so, the coefficient of the

f. sin o

cannot be greater

first term in the series would be negative, and, consequently, by diminishing (h), so that the first term becomes greater than the sum of all the rest, we might make the value of the whole series negative, which is absurd).

as will appear by substituting the value of sin h, which was found in the equation (A); but, by the assumed axiom, tan h is greater than (h); hence, tan h-h, or the series

sin cos e

1). h+Bh3 + &c.

must be positive; and, therefore,

f.sin e

cos e

less than 1, and, consequently, we have

cannot be

COR. 2. Hence, d. cos 0= sin 0. do.

For sin2+cos 0=1, and differentiating,

COR. 5. Hence, d. tan 0 sec2 0.do.

=

COR. 6. Hence, if (x) be a circular arc, whose radius is (a), and if (y) (x) (x') (s) (t) be its sine, cosine, versed sine, secant, and tangent respectively to radius (a), then we have the following equations :

For since, by Trigonometry, we have, supposing (k) the increment of (≈)

2 cos z. sin k=a {sin (x+k) – sin (≈ − k)}

Thus we have the differential coefficient of (y)

considered as a function of (x)

√ a2 - y2

=

; hence,

a

the differential coefficient of (z) considered as a

In the same manner the other formulæ may be proved.

COR. 7. Supposing, as before, sin 0, and cos 0, to represent the sine and cosine of the angle (0) to radius unity, while sin %, and cos %, represents them to radius (a), we have, (by Trigonometry),