x = ± 2 (}} q)1. sin 0,

= ±2 (39)1. sin (60° — 0),

= =2(g)*. sin (60° + 0).

(C. 824—45; L. 427; Hind, 286—90.)

(52.) Solution of the equation a"±1=0.

n

(a2-2 cos"=1.a+1);
π.x+1); if n is odd:

n

n-1

x2-2 cos T. +1); if n is even. (G. A. ch. xi.)

n

v2n +2 cos 0.x2 + 1 = (x2 - 2 cos x + 1)

n

(54.)

Let A, B, C, D, be the angles;

a, b, c, d, the sides AB, BC, CD, DA, respectively;

a, ẞ, the diagonals AC, BD;

their angle of intersection;
8 = (a+b+c+d): then
a2 + d2-b2- c2

=

2 (ad + bc)

2

sin 4 =

(s — a) (s — b)(s — c) (s — d) |* ;

The radius of the circumscribed circle

(ab+cd)(ac+bd)(ad+bc)

(s — a) (s—b) (s—c) (c–d)

(Hind, Trig. 166-74; Leg. Geom. Note v.)

PROPERTIES OF POLYGONS.

(55.) Let A1, A2, &c. A,, be the exterior angles,

a1, a, &c. a,, the

2 area=a1 {a,.sin 4

n

+ a2 { a,.sin A ̧ + &c. + a„. sin (A3 + &c. + A„)}

α

+ &c. +an-1an.sin A‚· (Lhuilier, Polygonom. VIII.)

a1 =α.cos a1,α1⁄2 + а ̧⋅ cos α ̧‚α3 + &c. + an⋅ cos a1‚a‚·

(56.) Let a1 = = &c. =αn'

R,, the radius of the circumscribed circle,

R2, that of the circumscribed circle: then

ANALYTICAL GEOMETRY.

(1.) METHOD of representing algebraical quantities geometrically. (H. A. G. Introd. 6-10; G. G. A. 1—6; Biot, Ch. i.) (2.) Analytical solution of determinate geometrical problems.

(H. A. G. 14—30; G. G. A. 7; Biot, Ch. ii.)

(3.) Relation of indeterminate equations to Geometry; and definitions. (H. A. G. 43-8; Biot, 29-40; L. A. G. §. 2.)

ANALYTICAL GEOMETRY OF TWO DIMENSIONS.

(4.) The straight line. The equation to a

The equation to a straight line is

y=ax+b.

Let the line be represented by l, then if the co-ordinates

The equation to a straight line may be put under the form

The equation in terms of p, the perpendicular from the origin is x.cos p,x+y.cos p,y=p.

(H. A. G. 49-51; G. G. A. Ch. ii; Biot, 41-7; L. A. G. §. 3.) (5.) The equation to a straight line passing through the point (x1,y1) is y-y1 = a(x-x1).

If the line passes through two points, (x,y), (x,y), the

(x-x1).

are

The co-ordinates of the intersection of two lines

If a third line, y=a3x+b,, passes through the point of intersection, then

(a,b, — a,b) — (a,b, — aşხ,) + (a,b, — ვხ.) = 0. Let the two given lines be represented by l1, l; then

tan l12 = sin x,y

1 + (α1 + α2) cos x,y + α1a1⁄2

If the two lines are perpendicular to each other,
1+ (a1 + a2) cos x,y + a12 = 0.

If the co-ordinates are rectangular, then

If in this case the lines are perpendicular to each other,

The distance between two points, (X1,Y1), (X2,Y2),

Let p be the perpendicular from a given point, (1,1), on the line y = ax + b, then

If (x ̧‚Y1), (¤ ̧‚Y1⁄2), (~,,y), are the angular points of a triangle,
Area =} {(x,y2— ∞2Y1) − (∞1Y 3 − ∞ ̧¥1) + (@2 Y3 −xzY2)}·

1

(X 2

(H. A. G. 52–9; H. C. S. Ch. ii.; Biot, 48—53; L. A. G. §. 4.)