or

c=153° 38′ 42′′.4, c'= 90° 5′41′′.0.

B'=137°22′42′′.5, C' 50° 18′55′′.2,

64. Given a= 99° 40'48", find B= 65° 33'10", (No ambiguity; why?)

A= 95° 38' 4"; c=100° 49'30".

b= 64°23′15′′, C= 97°26' 29",

find b= 55°25' 2", c= 81°27'261",

C=119° 22' 27",

C'=164° 41'55".

find A=138°15′45′′.4, B= 31°11′14′′.0, C= 35° 49′ 58′′.2.

B=108°30',

b=107°19'52",

B=118° 40',
b=120° 8'49",

B= 98°20',
b= 83° 5'36",

B=142° 12′42′′,
b=144°37'15",

B= 31°11'10",
b= 49° 59′56′′.4,
B= 54°32'24",

C=116°15';

c=115° 28'131".

C= 93°20';

c=100° 18'113".

C= 63° 40';

c= 64° 3'20".

C=105° 8'10";

c= 60° 4'54". C= 35° 50'; c= 60° 0'11".2.

C= 89° 5'46"; c= 97° 44'18".

b= 53° 49'25",

90. If a, b, c are each <show that the greater angle may

exceed

2

π

91. If a alone >, show that A must exceed 7.

92. If a and b are each >, and c<, prove that:

(1) The greatest angle A must be >;

(2) B may be >;

(3) C may or may not be <1⁄2Ã.

93. If cos a, cos b, cos c are all negative, prove that cos A, cos B, cos C are all necessarily negative.

94. In a spherical triangle, of the five products, cos a cos A, cos b cos B, cos c cos C, cos a cos b cos c,

- cos A cos B cos C, show that one is negative, the other four being positive.

CHAPTER XII.

THE IN-CIRCLES AND EX-CIRCLES. - AREAS.

215. The In-Circle (Inscribed Circle). To find the angular radius of the in-circle of a triangle.

B

Let ABC be the triangle; bisect the angles A and B by the arcs AO, BO; from O draw OD, OE, OF perpendicular to the sides. Then it may be shown that O is the in-centre, and that the perpendiculars OD, OE, OF are each equal to the required angular radius. Let 2s the sum of the sides of the triangle ABC. The right triangles OAE, OAF are equal.

=

Now

tan OF=tan OAF sin AF

or, denoting the radius OF by r, we have

(Art. 195) (2)