Related to Series

2 2 2 2
❖ sin 18° + sin 36° + sin 54° + sin 72° =? [BUTEX’12-13]

(a) − 2

(b) ± 2

c 0

d 2
 Related to Series
π
❑ If θ = , then determine the value of sin 3θ + sin 4θ + sin 5θ +. . . . . . + sin 15θ.
2 2 2 2
36

[BUET’13-14]
 Related to Series
2 2 2 2
❑ sin 3° + sin 9° + sin 15° + … … … … … … + sin 177° =?
 Related to Series
π
❑ If θ = , then determine the value of cot θ . cot 3θ . cot 5θ … … cot 19θ. [BUET’11-12]
20
 Related to Compound/Multiple/Sub-Multiple Angle
Formulae: 2 tan θ
(i) sin 2θ = 2 sin θ. cos θ =
1+tan2 θ
(i) sin A + B + sin A − B = 2 sin A. cos B
2 tan θ
(ii) sin A + B − sin A − B = 2 cos A. sin B (ii) tan 2θ = 1−tan2 θ

(iii) cos A + B + cos A − B = 2 cos A. cos B (iii) 2

cos 2θ = cos θ − sin θ 2

2
(iv) cos A − B − cos A + B = 2 sin A. sin B = 2 cos θ − 1
2
= 1 − 2 sin θ
1−tan2 θ
C+D C−D =
(i) sin C + sin D = 2. sin
2
. cos
2 1+tan2 θ
2
C+D C−D 1 + cos 2θ = 2 cos θ
(ii) sin C − sin D = 2. cos . sin
2 2 2
C+D C−D
1 − cos 2θ = 2 sin θ
(iii) cos C + cos D = 2. cos . cos
2 2 3
C+D D−C
(i) sin 3θ = 3 sin θ − 4 sin θ
(iv) cos C − cos D = 2. sin . sin 3
2 2 (ii) cos 3θ = 4 cos θ − 3 cos θ
3 tan θ−tan3 θ
(iii) tan 3θ =
1−3 tan2 θ
 cs Method

1 n
sin 2 A
❑ Prove that, cos A . cos 2A . cos 2 A . cos 2 A … … cos 2
2 3 n−1
A= .
2n sin A
 cs Method

2π 4π 8π 14π
❑ Show that, 16 cos cos cos cos =1 [BUET’00-01]
15 15 15 15
 cs Method
π π π π
❖ Which of the following is the value of cos 2 ⋅ cos 3 … … … cos 10 sin 10 ?
2 2 2 2

[2019 Main, 10 Jan II]

1
(a)
1024

1
(b)
2

1
(c)
512

1
(d)
256
 cs Method

π 5π 7π
❑ sin sin sin =?
18 18 18
 𝟐
𝟏 + 𝐜𝐨𝐬 𝐀 𝟏 + 𝐜𝐨𝐬 𝛑 − 𝐀 = 𝐬𝐢𝐧 𝐀

π 3π 5π 7π
❖ The value of 1 + cos 1+ cos 1+ cos 1+ cos is- [1984, 3M]
8 8 8 8

1
(a)
2

π
(b) cos
18

1
(c)
8

1+ 2
(d)
2 2
 𝟏 + 𝐭𝐚𝐧 𝐱 𝟏 + 𝐭𝐚𝐧 𝟒𝟓° − 𝐱 =𝟐

❑ (1 + tan 1°) 1 + tan 2° 1 + tan 3° … … 1 + tan 45° =?

 Related to Compound/Multiple/Sub-Multiple Angle

❖ What is the value of 3 sin x − cos x 4

+ 6 sin x + cos x 2 6 6
+ 4(sin x + cos x)?

[1995, 2M]

(a) 11

(b) 12

(c) 13

(d) 14
 Related to Compound/Multiple/Sub-Multiple Angle

1 3
❑ − =? [BUTEX’16-17]
sin 10° cos 10°

(a) 4

1
(b)
4

(c) 0

(d) 3
 Related to Compound/Multiple/Sub-Multiple Angle

❖ If tan α − tan β = p, cot β − cot α = q, α − β = θ, then what is the value of cot θ?

[KUET’12-13, BUTEX’15-16]

1 1
(a) −
p q

1 1
(b) −
q p

1 1
(c) +
p q

p
(d) 1 −
q
 Related to Compound/Multiple/Sub-Multiple Angle

❖ If cos A + B sin(C + D) = cos(A − B) sin(C − D), then what is the value of tan D?

[KUET’15-16]

(a) tan A tan B tan C

(b) cot A cot B cot C

(c) sin A sin B sin C

(d) cos A cos B cos C

(e) sec A sec B sec C

 Related to Compound/Multiple/Sub-Multiple Angle

❑ The angle  is divided into  and  in such a way that tan = k tan, prove that,
k−1
sin( α − β) = sin θ . [KUET’03-04]
k+1
 Related to Compound/Multiple/Sub-Multiple Angle

❑ Prove that, tan 20° tan 40° tan 80° = 3 [BUET’04-05, BUTEX’11-12]
 Related to Compound/Multiple/Sub-Multiple Angle

θ 1−p ϕ cos θ−p

❑ If tan = tan then show that, cos ϕ = [BUET’14-15]
2 1+p 2 1−p cos θ
 Related to Compound/Multiple/Sub-Multiple Angle
θ
a cos ϕ−b tan
❖ If cos θ = then what is the value of 2
ϕ ? [KUET’13-14]
a−b cos ϕ tan
2

a+b sin ϕ
(a)
b−a sin ϕ

a+b cos ϕ
(b)
a−b sin ϕ

a+b
(c)
a−b

a+b
(d)
b

a+b 2
(e)
a−b
 Related to Compound/Multiple/Sub-Multiple Angle

❖ If cos x + cos y = a and sin x + sin y = b then,

(i) cos x − y =? [CUET’14-15(similar)] (ii) sin x − y =? (iii) tan x − y =?
x−y x−y x−y
(iv) cos =? (v) sin =? [BUET’16-17] (vi) tan =?
2 2 2
 Related to Compound/Multiple/Sub-Multiple Angle

❖ If cos x + cos y = a and sin x + sin y = b then,

x+y x+y x+y
(i) cos =? (ii) sin =? (iii) tan =?
2 2 2
(iv) cos x + y =? [BUET’19-20, CUET’14-15, KUET’10-11, 11-12]
(v) sin x + y =? (vi) tan x + y =?
 Related to Compound/Multiple/Sub-Multiple Angle

❖ If sinx + siny = 1 and cosx + cosy = 0 then prove that, x + y = π.

[BUET’02-03, RUET’18-19]
 Related to Compound/Multiple/Sub-Multiple Angle

❖ If sin A + cos A = sin B + cos B, then which of the following is correct? [BUTEX’14-15]


(a) A − B =
4


(b) A + B =
6


(c) A + B =
2


(d) A − B =
2
 Related to Compound/Multiple/Sub-Multiple Angle

❑ Prove that, 4(sin 25° + cos 5°) = 3 3 sin 55°

3 3
[CUET’13-14]
 Related to Compound/Multiple/Sub-Multiple Angle

3 3 o 3 o
❖ sin x + sin ( 120 + x) + sin ( 240 + x) =? [RUET’11-12]

(a) −3 sin 3x

1
(b) − sin 3x
4

3
(c) sin 3x
4

3
(d) − sin 3x
4

1
(e) − sin 3x
3
 …… ……

π
❑ Prove that: 2 sin o ′
= 2 sin 11 15 = 2 − 2 + 2. [CUET’05-06, BUTEX’07-08]
16
 …… ……


(i) 2 cos n = 2 + 2 + 2 + ⋯ n − 1 no. of 2s
2


(ii) 2 sin = 2 − 2 + 2 + ⋯ n − 1 no. of 2s
2n


❑ 2 sin =?
32


❑ 2 sin =?
64


❑ tan =?
64
 Quiz–01
π
❑ sec =?
64
1
(a)
2+ 2+ 2+ 2

2
(b)
2+ 2+ 2+ 2

2
(c)
2+ 2+ 2

2
(d)
2− 2+ 2

(e) None
 …… ……


(iii) 2 cos n = 2 + 2 + 2 + ⋯ n − 1 no. of 2s + 3
3⋅2


(iv) 2 sin = 2 − 2 + 2 + ⋯ n − 1 no. of 2s + 3
3⋅2n


❑ 2 cos =?
96

❑ 2 sin =?
48
 Related to Trigonometric Expressions
π
❖ If A + B + C = 2n + 1 for n ∈ ℤ then value of tan A tan B + tan B tan C +
2

tan C tan A is- [CUET’15-16, IUT’19-20]

3 1
(a) (b) 1 (c) (d) 3
2 3
 Related to Trigonometric Expressions

❑ ABC is an obtuse triangle. Prove that, cot A cot B + cot B cot C + cot C cot A = 1.

[BUET’05-06, RUET’12-13]
 Related to Trigonometric Expressions

❑ If A + B + C = π then show that, cos A + cos B + cos C + 2 cos A . cos B . cos C = 1

2 2 2
 Related to Trigonometric Expressions
π
❖ In PQR triangle ∠R = , if the roots of the equation ax + bx + c = 0 a ≠ 0 are
2
2

p Q
tan and tan then, [1999, 2M]
2 2

(a) a + b = c

(b) b + c = a

(c) a + c = b

(d) b = c
 Related to Properties of Triangle

a b c
❑ Sine Rule: = = = 2R; [R Radius of the Circumcircle]
sinA sin B sin c

2 2
b +c −a 2 2 2
c +a −b2 2 2
a +b −c 2
❑ Cosine Rule: cos A = ; cos B = ; cos C =
2bc 2ca 2ab

1 1 1
❑ ∆= ab sin C = bc sin A = ca sin B = s s − a s − b s − c where half of the
2 2 2

a+b+c abc
perimeter, s = ; ∆= sr [r= radius of the inner circle]; ∆=
2 4R
 Related to Properties of Triangle
 a = b cos C + c cos B Tangent Rule:
 b = c cos A + a cos C  tan
A−B
=
a−b
cot
C
2 a+b 2
 c = a cos B + b cos A
B−C b−c A
 tan = cot
2 b+c 2
C−A c−a B
 tan = cot
2 c+a 2

A s−b s−c B s−c s−a C s−a s−b

 sin =  sin =  sin =
2 bc 2 ca 2 ab

A s s−a B s s−b C s s−c

 cos =  cos =  cos =
2 bc 2 ca 2 ab

A s−b s−c B s−c s−a C s−a s−b

 tan =  tan =  tan =
2 s s−a 2 s s−b 2 s s−c
 Related to Properties of Triangle

❑ The sides of a triangle are of length 3,5,7. Determine largest angle, area, inner
circle radius, circumcircle radius and the nature of the triangle.
 Related to Properties of Triangle

❑ Say, in-center of triangle ABC is I and radius of the inner circle is r. Again, let D,E,F
are the foot of the perpendicular line drawn to BC, CA and AB. If r1 , r2 , r3 are the
radius of the inner circle of AFIE, BDIF, CEID quadrilaterals respectively. Prove that,
r1 r2 r3 r1 r2 r3
+ + = . [2000, 3M]
r−r1 r−r2 r−r3 r−r1 r−r2 r−r3
 Related to Properties of Triangle

❖ Sum of the two sides of a triangle is x and product is y. If x − c = y, where c is

2 2

the third side of that triangle, then determine the ratio of radius of inner circle and
circumcircle. [2014 Adv]

3y
(a)
2x x+c

3y
(b)
2c x+c

3y
(c)
4x x+c

3y
(d)
4c x+c
 Related to Properties of Triangle

❖ If the sides of a triangle are 2x + 3, x + 3x + 3 and x + 2x, then the greatest

2 2

angle is: [BUET’11-12, IUT’14-15]

(a) 90 o

(b) 120 o

(c) 60o

(d) 180 o
 Related to Properties of Triangle
b+c c+a a+b
❑ If in ∆ABC, = = then determine the value of A, B, C.
11 12 13
 Related to Properties of Triangle
❑ If the length of the sides of a triangle are three consecutive numbers and maximum
angle is twice the minimum angle, determine the length of the sides of that
triangle. [1991, 4M]
 Related to Properties of Triangle

❑ The length of two corresponding sides of a quadrilateral of area 4 3 square unit

inscribed in a circle are 2 and 5 unit and angle between those sides is 60°.
Determine the length of the other two sides.
 Quiz–02

❖ In the triangle ABC, AB = 5 cm, BC = 6 cm and area of triangle is 11.25 cm .

2

What is the value of angle ABC? [IUT’19-20]

4
(a) sin−1
5

3
(b) sin −1
5

−1 3
(c) sin
4

−1 2
(d) sin
5
 Related to Properties of Triangle

❖ What is perimeter of ∆ABC? [IUT’17-18]

(a) 48

(b) 48 + 12 2

(c) 60 + 6 3

(d) 48 + 12 3
 Related to Properties of Triangle

❑ Length of the sides of a triangle are 9, 40 and 41. What is the radius of the
circumcircle of the triangle? [BUET’13-14]

(a) 24.5

(b) 30.0

(c) 20.5

(d) 25.0
 Related to Properties of Triangle

❑ If in ∆ABC, A ∶ B ∶ C = 1 ∶ 2 ∶ 3, then a ∶ b ∶ c =?
 Related to Properties of Triangle

❑ If a = 2b and A = 3B, then determine the angles of the triangle.

[IUT’18-19, BUET’03-04]
 Related to Properties of Triangle

❖ If the adjacent angles of a side of length 3 + 1 cm of a triangle are 30° and 45° .

What is the area of the triangle? [CUET’11-12]

1
(a)
2 2

(b) 2

1
(c) 3+1
2

(d) None of these

 Related to Properties of Triangle

❑ If in ABC cos A = sin B − cos C then show that, the triangle is a right triangle.

[KUET’13-14, BUET’04-05, 05-06]

 Related to Properties of Triangle

❑ If the angles associated with the base of a triangle are of 22.5° and 112.5° and
height is h, then determine the length of the base. [RUET’14-15]
 Related to Properties of Triangle

❑ If the median passing through the point A in ΔABC is perpendicular to AB then

prove that, tan A + 2 tan B = 0
C

A B
 Related to Properties of Triangle

A A
❑ In triangle ABC prove that, a sin + B = (b + c) sin [RUET’07-08]
2 2
 Related to Properties of Triangle

❖ If the area of the triangle PQR is ∆ where the opposite sides of P, Q, R are
7 5 2 sin P−sin 2P
a, b, c respectively a = 2, b = ,c = . Then the value of is-
2 2 2 sin P+sin 2P

[JEE Main 2012]

3
(a)
4∆

45
(b)
4∆

3 2
(c)
4∆

45 5
(d)
4∆
 Related to Properties of Triangle

❑ If length of two corresponding sides of a triangle are 3 + 1, 3 − 1 and angle

between them is 60° then solve the triangle. [BUET’02-03]
 Related to Properties of Triangle

❑ In a triangle ABC, C = 60° and A = 75°. D is a point on AC such that the area of
∆ABD is 3 times the area of ∆BCD. Determine the value of ∠ABD.
 Related to Properties of Triangle

❖ For triangle ABC a + b + c = 2c (a + b ), the value of cosC will be-

4 4 4 2 2 2

[CUET’14-15]

1
(a)
2

1
(b) ±
2

3
(c) ±
2

(d) None of these

 Related to Properties of Triangle

1 2
2 A 1 2B 1 2C s
❑ For a triangle ABC show that, cos + cos + cos =
a 2 b 2 c 2 abc

[RUET’03-04, BUET’00-01]
 Related to Progression

sin A sin(A−B)
❑ In ∆ABC, if = then show that, a , b , c are in arithmetic progression.
2 2 2
sin C sin(B−C)
 Related to Progression

❑ If in ABC the angles A, B, C are in arithmetic progression and the sides a, b, c are in
geometric progression, then prove that, a , b , c are in arithmetic progression.
2 2 2
 Miscellaneous

❑ Two circle lies on two sides of the diagonal BD (as shown in the figure) of ABCD
rectangular field with an area of 12 × 5 square unit. Ali stands at OA and Boni at
OB . What is the distance between them? [BUET’15-16 (similar)]

B C
OB
OA

A D