BANSAL CLASSES

MATHEMATICS
TARGET IIT JEE 2007
XI (PQRS)

SOLUTIONS OF TRIANGLE
Trigonometry Phase-Ill

CONTENTS
KEY-CONCEPTS
EXERCISE-I
EXERCISE-II
EXERCISE-III
ANSWER KEY
 KEY CONCEPTS
I. SINE FORMULA : In any triangle ABC, = =
sinA sinB sinC

b 2 + c2 - a 2
n. COSINE FORMULA : (i) cosA = or a2 = b2 + c2 - 2bc. cos A
2bc
2
c' +• a~
2
b2 a + b - c2
(ii) cosB = (iii) cosC :
v
2ca ' 2ab
m. PROJECTION FORMULA: (i) a = b cosC + c cosB (ii) b = c cosA + a cosC
(iii) c = a cosB + b cos A
... B-C b - c A
IV. NAPIER'S ANALOGY - TANGENT RULE : (I)
v tan—— = cot—
' 2 b +c 2

.... C-A c-a B A-B a-b C

(11)
v tan——— = cot— (m)
v tan——— = cot—
' 2 c+a 2 ' 2 a +b 2

v. TRIGONOMETRIC FUNCTIONS O F HALF ANGLES :

. A j(s-b) (s-c) . B (s-c) ( s - a ) . c |(s-a) (s-b)

0) sin— = J 7 ; s m- sin =
2 V be ca i i

A s (s-a) B s (s-b) C s (s-c)

(ii) cos- — r — ; cos— cos =
be ' 2 ca i

A (s-b) (s-c) A a +b + c . . . ,
(iii) t a n
T =
V s(s-a) =
1(^1) Where S =
2 & A = arCa
° f tnangle
'

(iv) Area of triangle = Js(s-a) (s-b) (s-c) .

VI. M - N RULE : In any triangle,

(m+n) cot6 = m cota - n cotp
= ncotB-mcotC

VII. ~ ab sinC= ^ be sin A = ^ ca sin B = area of triangle ABC

2R
sinA sinB sinC

abc
Note that R =
4 A
; Where Ris the radius of circumcircle & A is area of triangle

Vin. Radius of the incircle 'r' is given by :

.. A , A+B+C A B C
(a) r = — where s = (b) r = (s-a)tan— = ( s - b ) t a n - = (s-c)tan—
Z Z ^
a sin I-
2 sin-?- „ ... . A . B . C
(c)r = ——- & soon (d) r = 4Rsin-j sin— sin—
COS I

IX. Radius of the Ex-circles r j , r 2 & r 3 are given by :

A A A A B C
(a) r = r = (b) r} = s tan— ; r2 = s t a n - ; r3 = s t a n -
i s-a ; 12
1 s- b ; 3 s-c

^Bansal Classes Trig.-t-III [2]

 acosl-cos5- „ A B C
(C) —- & so on (d) f j = 4 R sin— . cos— . cos—
COSy

AT> • B A C A JY
r,=4Rsin—
• C A
. cos — . cos—
B
r,2 = 4 R sin— . cos— . cos— 3
2 2 2
2 2 2

X. LENGTH O F ANGLE BISECTOR & MEDIANS :

If ma and (3a are the lengths of a median and an angle bisector from the angle A then,
1 I 2 2 2^0084
m. = - y2b + 2c - a and B = — -
2 b +c

Note that m2 + m2 + m2 = - (a2 + b2 + c2)

XL ORTHOCENTRE A N D P E D A L TRIANGLE :
The triangle KLM which is formed by joining the feet ofthe altitudes
is called the pedal triangle.
the distances of the orthocentre from the angular points of the
A ABC are 2 R cosA, 2 R cosB and 2 R cosC
the distances of P from sides are 2 R cosB cosC,
2 R cosC cosA and 2 R cosA cosB
the sides ofthe pedal triangle are a cosA (= R sin 2A),
b cosB (= R sin 2B) and c cosC (= R sin 2C) and its angles are
it — 2A, % — 2B and TC - 2C.
circumradii of the triangles PBC, PCA, PAB and ABC are equal.
xn EXCENTRAL TRIANGLE :
The triangle formed by joining
the three excentres I1? I2 and I3
of A ABC is called the excentral
or excentric triangle. Note that:
Incentre I of A ABC is the
orthocentre ofthe excentral A Ijl 2 I 3 .
A ABC is the pedal triangle of the
AIM
the sides ofthe excentral triangle are
A B C
4Rcos— , 4Rcos— and 4 R cos—
2 ' 2 2

7C B ,
and its angles are — - — and
2 2 2 2 2 2
A B C
IIj = 4 R s i n - j ; II 2 = 4 R s i n y ; II 3 = 4 R s i n y .

Xin. T H E DISTANCES B E T W E E N T H E S P E C I A L POINTS :

(a) The distance between circumcentre and orthocentre is = R. J 1 - 8 cos A cosB cosC

(b) The distance between circumcentre and incentre is = VR2 - 2Rr

(c) The distance between incentre and orthocentre is ^2r 2 - 4R2 cos A cosB cosC
[132]
faBansal Classes Trig.-0-III
XIV. Perimeter (P) and area (A) of a regular polygon of n sides inscribed in a circle of radius r are given by
71 1
• . 271
<>
P = 2nr sin— and A = — n r sin—
• ,

n 2 n
Perimeter and area of a regular polygon of n sides circumscribed about a given circle ofradius r is given by
n . tc
P = 2nrtan— and A = n r tan—
n n
XV. In many kinds oftrignometric calculation, as in the solution oftriangles, we often require the logarithms of
trignometrical ratios. To avoid the trouble and inconvenience ofprinting the proper sign to the logarithms
ofthe trignometric functions, the logarithms as tabulated are not the true logarithms, but the true logarithms
increased by 10. The symbol L is used to denote these "tabular logarithms". Thus:
L sin 15° 25' = 10 + log10 sin 15° 25'
and L tan 48° 23'= 10 + log 10 tan48° 23'
EXERCISE-I
Q. 1 With usual notation, if in a A ABC, ^ = ^ = — ; then prove that, = ^ = .
11 12 13 7 19 25

Q.2 For any triangle ABC , if B = 3 C, show that cosC = J ^ t l & sin^ = -.
y 4c 2 2c

•v3 Tt
Q.3 In a triangle ABC, BDisamedian. If /(BD) = —-/(AB) and Z DBC = ~ . Determine the ZABC.
»

Q.4 ABCD is a trapezium such that AB,DC are parallel & BC is perpendicular to them. If angle
+ q2) sin9
ADB = 9 , BC = p & CD = q, show that AB = .
pcosB + qsinG

Q.5 Let 1 < m < 3 . In a triangle ABC, if 2b = (m+1) a & cos A= | Prove that there

are two values to the third side, one of which is m times the other.
Q.6 If sides a, b, c of the triangle ABC are in A.P, then prove that

sin2 — cosec2A; sin2 — cosec2B ; sin2 — cosec2C areinH.P.

2 2 2
Q. 7 Find the angles of a triangle in which the altitude and a median drawnfromthe same vertex divide the
angle at that vertex into 3 equal parts.
A B C
Q.8 Inatriangle ABC, if tan—, tan— , tan— areinAP. Showthat cos A, cosB, cosC are in AP.

Q.9 Show that in any triangle ABC;

a 3 cosBcosC + b 3 cosC. cosA+c 3 cosAcosB = a b c ( l - 2 cos A cos B cos C).
3R
Q.10 A point'0'is situated on a circle ofradius Rand with centre O, another circle of radius — isdescribed.
Inside the crescent shaped area intercepted between these circles, a circle ofradius R/8 is placed. If the
same circle moves in contact with the original circle of radius R, thenfindthe length ofthe arc described
by its centre in moving from one extreme position to the other.
Q.ll ABC is a triangle. D is the middle point of BC. If AD is perpendicular to AC, then prove that

cos A. cos C :
3 ac
cos A + 2cosC sinB
Q. 12 Ifin a triangle ABC, = , prove that the triangle ABC is either isosceles cr right angled.
cosA + 2cosB sinC

faBansal Classes Trig.-0-III [4]

Q.13 In a A ABC, (i) —— = —— v(ii) 2 sinAcosB = sinC
cosA cosB
A A C
(iii) tan2 — + 2 tan — tan — - 1 = 0, prove that (i) => (ii) => (iii) (i).
Ai jL JL
Q.14 Sides a, b, c of the triangle ABC are in H.P., then prove that
cosec A (cosec A + cot A); cosec B (cosec B +cot B) & cosec C (cosec C + cot C) are in A.P.
A— C a+c
Q.15 In a triangle the angles A, B, C are in AP. Show that 2 cos
ac + c2

Q. 16 If pj, p2, p3 are the altitudes of a triangle from the vertices A, B, C & A denotes the area of the
, . l l l 2ab c
triangle, prove that — + = ——-—-— cos"2 —.
p, p2 p3 (a + b + c) A 2
Q.17 Let ABCD is a rhombus. The radii of circumcircle of AABD and AABC are Rj and Rj respectively then

show that the area of rhombus is TTz D 2 x2 .

(Kj + K 2 ;
Q.18 In a AABC, GA, GB, GC makes angles ot, P, y with each other where G is the centroid ot the
AABC then Show that, cot A + cot B + cot C + cot a + cot p + cot y = 0.
A +B
Q.19 If atanA+btanB = (a+b)tan—-— , prove that triangle ABC is isosceles.

Q.20 The two adjacent sides of a cyclic quadrilateral are 2 & 5 and the angle between them is 60°. If the area
of the quadrilateral is 4 fi, find the remaining two sides.
Q.21 The triangleABC (with side lengths a, b, c as usual) satisfies
log a2 = log b2 + log c2 - log (2bc cosA). What can you say about this triangle?
2
Q.22 Ifthe bisector of angle C oftriangle ABC meets AB in D & the circumcircle in E prove that, CE _ (a+b)
2
DE " c
Q.23 In a triangleABC, the median to the side BC is of length , 1 & it divides the angle A into
y n - 6V3
angles of 30° & 45°. Find the length ofthe side BC.
Q.24 Given the sides a, b, c of a triangle ABC in a G.P. (a, b, c * 1) . Then prove that;
x = rb2_c2) • ( c 2 . a * tanC + tanA . tanA + tanB
tanB + tanC
v =
X Y (C 4 Z (a b)
tanB - tanC ' > tan'C - tanA ' tanA-tanB
are also in G.P. Further, if a2 = logxe ; b j = l o ^ e & Cj=log z e are the sides of the triangle
A B C
Aj Bj Cj, then prove that : sin2-^- , sin2 - - , sin2-^- are in H.P.
Q.25 With reference to a given circle, Aj and B, are the areas of the inscribed and circumscribed regular
polygons of n sides, A^ and B2 are corresponding quantities for regular polygons of 2n sides. Prove that
(1) A2 is a geometric mean between Aj and B j
(2) B2 is a harmonic mean between A^ and B j.

EXERCISE-II
Q.l I +± +i = i Q.2 rj + r2 + r3 - r = 4R Q . 3 — ^ — r +^ — ^ — A -3
n r2 r3 r (s-b)(s-c) (s-c)(s-a) (s-a)(s-b) r
^ b-c c-a a-b „ „ r
i ~ r +. 2 ~ _ c abc A B C'
Q.4A Q.5c —~
r r

+ + =0 - 7 Q.6 cos— cos— cos— = A

r. r„ r„ a o r3 s 2 2 2

faBansal Classes T rig .-(f> - III [5]

Q.7 a cosB cosC +b cosC cosA+ c cos AcosB = —
R

Q.8 (ri+r2)tan| =(r3-r)cot|=c Q.9 a cot A+ b cotB + c cotC = 2 (R+r)

Q.10 4R sin A sinB sinC = a cos A + b cosB + c cosC Q.ll (r 1 -r)(r 2 -r)(r 3 -r) = 4 R ^
B-C , , , . C-A , , , . A-B
Q.12 (r + fj) t a n — + (r + r2) t a n - — + (r + r3) t a n — = 0
a„ 2 +. bv 2+ ,c„2
Q.13 _L _L _L J_
2 +
2 +
2 +
2
Q. 14 (r3+ fj) (r3+ r2) sin C = 2 r3 Jr2r3 + r ^ + r ^
r A r 2 r 3

1 1 1 1 Q.16 T, | I; | r3 _ 1
Q. 15 —

be
+ —

ca
+ —

ab
=

2Rr be ca ab r 2R
1 i 1 4R Q.18 bc - r2 r3 _ ca - r3 rt _ a b - rt r2 — r
Q.17
Vr V vr v Vr r
3/ r2 s2
1 1 1 1 4I I 1 1
Q. 19 - + — + — + — + —+—
r Tj r2 r3; r vr,i % r3;

Q.20 In acute angled triangle ABC, a semicircle with radius ra is constructed with its base onBC and tangent
to the other two sides. rb and rc are defined similarly. Ifr is the radius ofthe incircle of triangle ABC then
2 1 1 1
prove that, — = — + — + —
r ra r,b rc
Q.21 If I be the in-centre ofthe triangle ABC and x, y, z be the circum radii ofthe triangles IBC, ICA& IAB,
showthat 4 R 3 - R ( x 2 + y2 + z 2 )-xyz = 0.
Q.22 IfAq denotes the area of the triangle formed by joining the points of contact ofthe inscribed circle ofthe
triangle ABC and the sides of the triangle; Aj, Aj and A3 are the corresponding areas for the triangles
thus formed with the escribed circles of A ABC. Prove that A1 + A^ + A3 = 2A + Aq
where A is the area of the triangle ABC.
Q.23 Consider a A DEF, the pedal triangle ofthe A ABC such that A-F-B and B-D-C are collinear. If H is
the incentre of A DEF and Rj, R^ R3 are the circumradii of the quadrilaterals • AFHE; • BDHF
and • CEHD respectively, then prove that
I R p R + r where R is the circumradius and r is the inradius of A ABC.
A B C
Q.24 Prove that in a triangle, 8 r R(cos2— + cos2— + cos2—) = 2bc + 2ca + 2ab - a2 - b2 - c2.

Q.25 Prove that in a triangle — + — + — = 2R

b aJ vc by va c.

Q.26 The triangle ABC is a right angled triangle, right angle at A. The ratio of the radius of the circle
circumscribed to the radius of the circle escribed to the hypotenuse is, -Jl: (S+-Ii). Find the acute
angles B & C. Also find the ratio ofthe two sides ofthe triangle other than the hypotenuse.
Q.27 Let the points PJ, P2, , PN_1 divide the side BC of the triangle ABC into n parts. Let ij, i^ i3, in
be the radii of the inscribed circle ; el5 e2, e3, , e n bethe radii of the escribed circles corresponding
to the vertex A for the triangles A B P J , A P j P 2 , A P 2 P 3 , , A P N _ 1 C respectively, then show that
r + r,
(i) n i L = £ & ( i i ) t h = - , where R t , I^, , Rn are the radii ofthe circumcircle
R, R
of triangles ABP j, AP j P2, , AP n _jC &R is the circumradius, r is the inradius & r t is the exradius
as usual of A ABC.
faBansal Classes Trig.-0-III [6]
Q.28 In a plane of the given triangleABC with sides a, b, c the points A', B', C' are taken so that the
A A'BC, A AB'C and A ABC' are equilateral triangles with their circum radii Ra, Rb, Rc ; in-radii
ra, rb, rc & ex-radii r a ', r b ' & rc' respectively. Prove that ;

(i) n r a : IT R : IT r ' = 1: 8 :27 &(ii) r,r 2 r 3 = £ ( 3R * + 6 r * + 2 r *')] U t m A

648 V 3 2
Q.29 In a scalene triangle ABC the altitudes AD & CF are dropped from the vertices A& C to the sides BC
& AB. The area of A ABC is known to be equal to 18, the area of triangle BDF is equal to 2 and length
of segment DF is equal to 2V2 • Find the radius of the circle circumscribed.
Q.30 Consider a triangleABC with Aj, Bj, Cj, as the centres ofthe excirlces opposite to the verticles A B,
C respectively.
Ar.(AA1BC) + Ar.(AAB 1C)+Ar.(AABC1)_ 1
Showthat S(R?+R 2 +R 2 ) ~ 2R
Where R, Rj, R^, R3 are the circum radii of AABC, AAjBC, AABjC and AABCj respectively and S
is the semiperimeter of AABC.
EXERCISE-III
Q.l In a AABC, Z C = 60° & z A= 75°. If D is a point on AC such that the area of the A BAD
is S times the area of the A BCD, find the Z ABD. [REE'96,6]
Q.2 In a A ABC, a : b : c = 4:5:6. The ratio of the radius of the circumcircle to that ofthe incircle is .
[JEE '96,1]
Q.3 If in a A ABC, a = 6, b = 3 and cos(A- B) = 4/5 thenfindits area. [REE'97,6]
Q.4 If in a triangle PQR, sin P, sin Q, sin R are in A.P., then [JEE '98,2]
(A) the altitudes are in A.P. (B) the altitudes are in H.P.
(C) the medians are in G.P. (D) the medians are in A.P.
Q. 5 Two sides of a triangle are of lengths ^6 and 4 and the angle opposite to smaller side is 3 0°. How many
such triangles are possible ? Find the length of their third side and area. [REE '98,6]
Q. 6 Let ABC be a triangle having 'O' and T as its circumcentre and incentre respectively. If R and r are the
circumradius and the inradius respectively, then prove that, (IO)2 = R2 - 2 Rr.
Further show that the triangle BIO is a right triangle if and only ifb is the arithmetic mean of a and c.
[JEE'99,10 (out of200)]
Q.7 The radii r } , r2 , r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is
24 sq. cm and its perimeter is 24 cm,findthe lengths of its sides. [REE '99,6]
n
Q.8(a) In a triangleABC, Let Z C = —. If 'r* is the inradius and ' R' is the circumradius of the triangle, then
2 (r + R) is equal to :
(A) a + b (B) b + c (C) c + a (D) a + b + c
(b) In a triangleABC,
(A) a2 + b2 - c2 2 ac(B)sinc2~+(aA2 -- Bb2+ C) = (C) b2 - c2 - a2 (D) c2 - a2 - b2
[JEE '2000 (Screening) 1 + 1]
Q.9 LetABC be a triangle with incentre T and inradius 'r'.Let D,E,F be the feet ofthe perpendiculars
from I to the sides BC, CA & AB respectively. If rj, r2 & r3 are the radii of circles inscribed in the
quadrilaterals AFIE, BDIF & CEID respectively, prove that
— +— +— = 7 , J1 r2 \ , t . [JEE '2000,7]
( r - r,) ( r - r 2 ) (r - r3)
faBansal Classes Trig.-0-III [7]
Q. 10 If A is the area of a triangle with side lengths a, b, c, then show that : A < — ^(a + b + c)abc
Also show that equality occurs in the above inequality if and only if a = b = c. [JEE' 2001 ]
Q. 11 Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC
(R being the radius of the circumcircle)?
(A) a, sinA, sinB (B)a,b, c (C)a, sinB,R (D)a,sinA,R
[ JEE' 2002 (Scr), 3 ]
Q. 12 If I n is the area of n sided regular polygon inscribed in a circle of unit radius and On be the area of the
polygon circumscribing the given circle, prove that

On
1 + , 1- [JEE 2003, Mains, 4 out of 60]
v « J

Q. 13 The ratio ofthe sides of a triangle ABC is 1: ^3 :2. The ratio A: B : C is

(A) 3 : 5 : 2 (B) 1 : :2 (C) 3 : 2 : 1
(D) 1 :2 : 3
[JEE 2004 (Screening)]
Q. 14 In AABC, a, b, c are the lengths of its sides and A, B, C are the angles of triangle ABC. The correct
relation is
B-C^j f\ \ B-C
(A) (b-c)sin
2
=
J
a cos —
(B) (b-c)cos
V^J
= asm
V 2 ,

B + C> fA> ' B + C^

(C) (b + c) sin = a cos —
(D) (b-c)cos = 2a sin
2
J U ; v 2 y
[JEE 2005 (Screening)]

ANSWER KEY
EXERCISE-I
7tiR
Q.3 120° Q.7 TT/6, tc/3, n/2 Q.10 Q20. 3 cms & 2 cms Q 23. a = 2
12

EXERCISE-II
Q.26B=f Q.29 - units
c 2
EXERCISE-III
Q.l angleABD = 30° Q.2 Q.3 9 sq. unit Q.4 B

Q.5 2, (2V3-V2) , (2V3+V2) , (2V3-V2) & (2V3+V2) sq. units Q.7 6, 8, 10 cms
Q.8 (a) A, (b) B Q.ll D Q.13 D Q.14 D

o
faBansal Classes Trig.-^-III 15]