112. Having given tan 8, to find tan no.

And equating the possible and impossible parts of this equation,

COR.

If n be a positive integer, the series (1) will termi

nate, and the last term is (1 tan 0)".

(√1 tan 0)" = (√− 1)2'" (tan 0)" = (− 1)" . (tan 0)" ;

n

therefore the last term is (-1) (tan 0)", and

(√1 tan0)"=√1. (√-1)-1. (tan 0)"=√1.(-1) (tan 0)" ;

n

therefore the last term is (-1) (tan 0)", and

[The sign of n (tan 0)-1 is the same as that of (tan 0)", because every odd term in the expansion (1) has the same sign as the term immediately following it; and when n is odd, n (tan 0)”-1 is an even term, and has therefore the same sign as (tan 0)".]

113. As the angle, (which is supposed to be less at first than a right angle), is continually diminished, each

O vanishes, each of them does become 1.

Let AB be the arc of a circle whose centre is C and radius AC.

Let ACB = 0; and let AT touch the circle at A.

From B draw BN perpendicular, and

BM parallel, to AC.

Then MN is a rectangle, and

AM = NB, AN = MB.

Now arc AB > the straight line joining A and B, and .. >NB; and arc AB <AM+ MB; (Legendre's Geometry; IV. 9.)

Ꮎ

tan

Now if be continually diminished till it at last vanishes, continually decreases, and at last vanishes, in which

114.

The area of a circle of radius r is πr2.

The area of a regular polygon of n sides inscribed in the circle is n times the area of a triangle two of whose

2π

sides are r, r and the included angle ;

Now as the number of the sides increases, the area of the polygon approaches nearer and nearer to that of the circle, and when n is infinite becomes identical with it,-in which

Now this is true whatever be the value that is given

to 0; and if o be written for 0,-in which case

and cos=1,—we have

tan

1,

By a similar substitution in the expression of Art. 112, (3),

Sin n0 = (cos 0)". {n tan (

(1.)

NOTE. To arrive at these expressions we have supposed

to become 1 when 0 is written for 0, which is the case

only when the angle is referred to the circular measure

arc

radius

Hence a, which is equal to ne, is also referred to the same measure in the series (1) and (2).