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[3] 103 -9x+r=1, and 4 q' * 27 .

3\*
Assume sin 30=

the three values of x are
2
x= +2(}9)}. sin 0,

+2(}9).sin (60° – 0),

F2 (9)*.sin (60° + 0).

(C. 824-45; L. 427; Hind, 286_90.) (52.) Solution of the equation 005 +1=0.

2

2

2 COS

n

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n 1 COS

n

** – 1=(0–1){(*

2—cos ža+ņ– 1.sin:-)
(* -1.sin:-)..
{C

#+

-1.sin ". 2) (2–com " !--V-1.

V-1.sin”?-)}; =(1-1) (23–2003+*+1)(-+*+1)...

(12–2003";?<.+1); if n is odd : =(0–1)(3-2006.**+1) (2–200 ***+1)...

(**–2003" 1); if n is even. *+1=(+1)(x2–2006 *.*+2)(4–9.com ***+1)...

(2–2006 " 7.x+1); if n is odd: = (x2–2004. -.*+1)(+3–2004 ***+1)... (3-9 con ". .**.*+1); if niseven. (G.4. ch. xi

.)

2

7. + n

2 n

(53.) Solution of the equation in — 2 cos 6.00"+1=0.

e 2021 2 cos 0.00+1= (202 — 2 cos 8C +1)

n

2 +0

2(n-1) 7 + 0 (w? — 2 cos X + 1) ... (x2 – 2 cos

x + 1).

n

n

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37 +0

(2n-1) + 0 (202 — 2 cos x + 1) ... (202 — 2 cos

x + 1).

n

n

(Hind, 291, 2; G. A. 71.)

8 =

2

S

PROPERTIES OF A QUADRILATERAL INSCRIBED IN A CIRCLE. (54.) Let A, B, C, D, be the angles ;

a, b, c, d, the sides AB, BC, CD, DA, respectively ;
a; B, the diagonals AC, BD;

their angle of intersection ;
{ (a +b+c+d): then

a+ ď? 12 cos A=

2 (ad + bc)

2 sin A= (8—a)(s—b)(8–c)(s—d).

ad + bc

- a)(s-d) sin A

ad + bc

(8-5)(8 -c) cosA ad + bc

3

a)(s d) tanA= (8-5)(8-0)

i ad + bc

ab + cd a=(ac+bd)

B=(ac+bd) ab + cd

ad + bc The area =

(8—a)(s—b)(8—c)(8-d))";
žaß.sin .
2

(s—a)(s—b)(8—c)(s—d).

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=

sin =

ac+bd'

U

The radius of the circumscribed circle

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

1 / 1 를 (s - a)(s b)(sc)(c- d)

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

PROPERTIES OF POLYGONS.

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

, ,, &c. Qy, the sides A, A2, A, A3, &c. A,A,, respectively: 2 area=a, {ag.sin A, +az.sin (Ag + 4) + &c.

+@m.sin (A, + A, + &c. + An)} +a, {az.sin A, + &c. + an.sin (A, + &e. + An)}

+ &c. +0,-,a, .sin A, a (Lhuilier, Polygonom. viii.) a, = dg.cos aq,Q+ az.cos a gaz + &c. + an. cos a ,. a,' =a +az + &c. +0,2 – 2 {a, dz.cos agyag +a, ag.cos 22,Q4 + &c.}.

(Hamilton, Analytical Geometry, 41.) (56.) Let a=a2=&c.=Qne

R,, the radius of the circumscribed circle,
R,, that of the circumscribed circle: then

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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. i.) (3.) Relation of indeterminate equations to Geometry; and definitions. (H. A. G. 438; Biot, 29-40; L. A. G. . 2.)

ANALYTICAL GEOMETRY OF TWO DIMENSIONS.

(4.) The straight line.

The equation to a straight line is

y=aa + b.

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

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if rectangular,

a= tan lg. The equation to a straight line may be put under the form

y

1. 6

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The equation in terms of p, the perpendicular from the origin is

X.cos pyx +y.cos py=p. (H. A. G. 49–51; G. G. A. Ch. ii; Biot, 41—7; L. A. G. 8.3.) (5.) The equation to a straight line passing through the point (X1,4.) is

y-y=a (x - 2). If the line passes through two points, (x,y), (x,y), the

Y-Y equation is

Y - Y1

(x– x4). wa - N1

are

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The co-ordinates of the intersection of two lines
y=a, + b,

y=0,8 +b, ;
b, b,

a,be-a,b,

y = a

a 1 - 2 If a third line, y=a3x + bg, passes through the point of intersection, then

(a,b, a,b,) (a,bz az 61) +(a,bz azbe)=0. Let the two given lines be represented by li, lz; then

an az tanlıla=sin x,y

1 + (a + a,) cos x,y + a, a, If the two lines are perpendicular to each other,

1 + (az + a,) cos x,y +a, ag=0. If the co-ordinates are rectangular, then

&, -a, tan lille

1 + azaz

a-ag sin lle=

(1 + ,*)(1 + a,;)]'

1+ a, a,
cos bylą=

(1 + a)(1+ a.)
If in this case the lines are perpendicular to each other,

1+a, ag=0.
The distance between two points, (x,y), (X2992),

=(x, – æ,)* +(4.–4.)*). Let p be the perpendicular from a given point, (x1,4.), on the line y=aw+b, then

Y. -ax-b p=+

(1 + a*) : If (1941), (2;42), (X3993), are the angular points of a triangle,

=} {(«,92 x,y)-(*193 x3y2)+(x,y--,y)}. (H. A. G.52–9; H.C. S. Ch. ii.; Biot, 48–53; L. A. G. . 4.)

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