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(20) Prove that in any triangle ABC

4bc +

sin? (6

21/5c A Hence prove that if tan A= sin then a=(b-c) sec 8.

b

2

a®=(6–c)* {2+

}

-C

COS

(21) Prove that in any triangle ABC

21bc

A = (b+c) cos ß, where sin ß

b+c 2 Use this method, or that of Example (20), to find a by the aid of the Tables, when b=347, c=293, A=39° 42'. [See Art. 260.]

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(22) Prove that in any triangle A

b + c

A log a=log (b-c)+ L cos - L cos 0; where p=tan-1

tan 2

b-c 2

(23) If D, E, F are the middle points of BC, CA, AB, then

b sin A sin BAD=

[See Ex (11) (15) page 205.] Jāx + 2bc cos A + c2 (24) If x=111, y=11, z=113, d=2R, prove that

xyz +0 (22+ y + z)=4d3.

(25) The circle escribed to the side BC of the triangle ABC will be the inscribed circle of the triangle whose sides are

sb
a' s-a

sa

SC

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3- a

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(26) If a point Q be taker within ABC such that the angles BQC, CQ A, AQB are each 120°, prove that

412A +16 (62 +02 - a2) 04=;

{12132 +3 (a +62 + c2)}} (27) If sin A and cos A are both given, prove that in general

A ne different values and no more can be assigned to sin

.

(28) If from a point on a circular arc (of radius r) which subtends an angle a at the centre, perpendiculars are drawn to its bounding radii, then the distance between the feet of the perpendiculars is r sin a.

(29) If two circles, radii a and b, touch externally, the angle

between their

common tangents – 2 tan-14 (VÖ-VA).

(30) If the inscribed circle touches the sides in DEF, prove

A B that EF: FD:DE=COS

2 2 2

: COS

: COS

COS

sin

COS

COS

.

. COS

(31) If the bisectors of the angles A, B, C meet the opposite

2bc

a sin B sides in D, E, F then (i) AD= 54.

b+c

(ii) CD=

sin B+sin C (iii) The area of DEF

A B с 2 sin

sin 2abc

2 2 2 =S (6+c)(C+a) (a+b)=S. B C-A A-Bo

2

2 (32) If a, B, 8 are the angles of a quadrilateral inscribed in a circle, then cos (a+b). cos (8+y). cos (y+8). cos (8+a)=(1 – cosa a - cosa B)2.

(33) If , and , be two values of 8 found from the equation a cos 0+ b sin 0=c, then 1 0,+0,1

1
sin

0,
2
2

2

(34) If a=b cos x + c cos 0

b=
= c cos 0 +a cos

Х c=a cos 0 +b cos 0 prove that cos 20+ cos 20+cos 2x + 4 cos 0. cos $. cos x+1=0, hence (see F.x. (20), p. 143], prove that (0+0+x)=(2n+1) T.

(35) A circle is drawn touching the inscribed circle and the sides AB, AC (not produced) of the triangle ABC. Another circle is drawn touching the circle and the same side. Prove

1-sin

2

that the radius of the nth circle thus drawn is r|

and

1 +sin

that the sum of the radii of all the circles, when n=00, is

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(36) If AO, BO, CO pass through the centre of the circumscribing circle and meet the opposite sides on DEF, then

2 AD + + =

BE* CFER

(37) If h and k be the lengths of the diagonals of a quadrilateral and 0 the angle between them, prove that the area of the quadrilateral is hk sin 0.

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(38) If

sin &

sin Z

Y

and X + Y+2=180°, prove that

sin Y x=y cos Z+2 cos Y.

(39) Two regular polygons of n and 2n sides are described such that the circle inscribed in the first circumscribes the second ; also the radius of the circle inscribed in the second is to that of the circle circumscribing the first as 3+13 is to 4 12. Prove that n= = 6.

(40) Prove by the aid of a figure like that of Art. 282 that 1,02=R2+2Rri. (41) If y2 + 2x + xy=1, prove by Trigonometry that

4.xyz
1 – 22 * 1- y2 * 1 32 (1 – 22) (1 – y2) (1 – 22)
2 1 +

[See Ex. (33), (32), p. 194.]

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ANSWERS TO THE EXAMPLES.

I. Page 2

a.c

(1) 80.
(2) 10.

(3) 16. (4) 10911

1760a (5) 5 acres. (6)

(7) yds.

b (8) A shilling and a three-penny piece.

II. Pages 4, 5. (1) 77440.

(2) 6 ft. (3) . (4) 68, 47, 47. (5) 15.625, 155, 136, d (6) 917, 9 shillings.

80a 80a 80a 80a (7)

(8) 21 shillings. d k

с

(1) 10 ft. (5) 90 ft.

III. Pages 7, 8.
(2) 80 yds. (3) 20 ft. (4) 50 ft.
(6) 2014 nearly. (7) 5a feet.
V2

(12) 2/3 a leet.
2
(14) V84 ft. (15) 2 V90% -67 ft.

(8) 12a yards.

(10) ya a yards.

3

(13) 1:72.

IV. Page 13. (1) 9.899 ft. (2) 2489 yds. (3) 5836 in. (4) 1138 in. (5) 933•4 ft. (6) 2504 in. (7) 8.66 ft.

(8) 4.607 ft. (9) 48 in. (10) 85 ft. 11 in.

V. Pages 21, 22. (1) 34 yds. (2) 254 ft. (3) 1509 in. (4) 34 ft. (5) 771 ft. (6) 560. (7) 154 nearly. (8) 33600. (9) 327. (10) 7 ft. (11) 5537, 13.8 in.

(12) 339; ft. (13) 443 in. (14) 235 in.

(15) 203 in. (16) 274 in. (17) 1886 in.

VII. Page 28. (1) •632118 of a right angle. (3) •021827 of a right angle. (2) 1.0426991

(4) •03294894

(5) •006241 of a right angle. (9) •6900071 of a right angle. (6) 10-000812

(10) 1.19030045 (7) 3204052

(11) 10.061801 (8) "0102034

(12) •0226048 (13) 365 78' 91". (14) 1048 30* 21". (15) 18 20' 3". (16) 10* 20".

(17) 68 25. (18) 3028 12' 50". (19) 100% 10".

(20) 18 1' •1". (21) 6458 10'. (22) 26 30'. (23) 1' 10".

(24) 10".

VIII. Page 30. (1) •09175 of a right angle=98 17* 50". (2) •0675

= 68 75. (3) 1.07875

=1078 87' 50". (4) 180429012345679 =188 4' 29", etc.

=1468 77 77.9.

=548 44 44.4". (7) 1° 14' 15". (8) 7° 52' 30". (9) 153° 24' 29.34". (10)

21° 36' 8.1". (11) 16° 12' 37.26". (12) 31° 30'.

(5) 1.467 (6) •54

IX.

Page 35. 1. (1) 2 right angles or 180°. (2) of a right angle. 2

6 (3) right angles.

(4) right angles.

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пи (7)

2009

50 (4)

(2) Y.

(3) 1.

(5) ff. (6)

4. (1)

37 *

180*

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