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Mathematics JEE MAINS-2014-15
(K-CET and other CET’s)
Anand S. Khot
Dpt of Mathematics
TRIGONOMETRY
SYNOPSIS
Radian: Radian is a constant angle subtended at the centre of the circle whose arc
length is equal to the radius of the circle.
One radian or 1c = 57017’45’’
One degree or 1o = 0.017453 radians
* Conversation of units:
1. Radian to degree
xc = (
πx
180
)radians
2. Degree to radian
xo = (
180x
π
) degrees
* Relation between the arc length, radius of the circle and angle formed by the
arc in radian at the centre of the circle is,
l = rθ
* Relation between Area of the sector(A), radius of the circle(r) and the angle
formed by the sector in radian at the centre of the circle(θ) is,
1
A = r2θ
2
* Circumference of the circle = 2πr
or
Circumference of a circle
Diametere of the circle(2r)
* Angle between two consecutive digits of a clock is = 300 or
π
6
* Angle moved by hour hand in one hour = 300
* Angle moved by hour hand in one minute =
10
2
* Angle moved by minute hand in one minute = 60
* Short cut method
The angle between the hour hand and the minute hand
when the time is x hours and y minutes is,
1
| [60x − 11y]| degrees
2
* Minute hand hour hand coincide after intervals of
𝟕𝟐𝟎
𝟏𝟏
minutes
* The gain of minute hand over the hour hand in one minute is equal to
60 -
𝟏𝟎
𝟐
=
𝟏𝟏𝟎
𝟐
# TYPES OF ANGLES:
1) Acute angle,
if
0 ≤ θ < 90
2) Obtuse angle,
if
90 < θ < 180
3) Reflex angle,
if 180 < θ < 360
4) right angle,
if
θ = 90
5) Straight angle
if
θ = 180
=π
* Co-terminal angles: Two angles having different measure but same
initial and terminal sides are said to be co-terminal angles.
Ex: -300 and 3300
* Angles of a triangle are in A.P then one of the angle is 300
* Angles of a triangle are in A.P then they are of the form,
(a – d), a, (a + d)
* Angles of a quadrilateral are in A.P then they are of the form,
(a – 3d), (a – d), (a + d) (a + 3d)
* The sum of interior angles of a polygon of n sides is,
(n – 2)1800 or
(n – 2) π
* In a regular polygon
1) All the interior angles are equal.
2) All the exterior angles are equal.
3) All the sides are equal.
4) Sum of all the exterior angles is 360.
5) Each exterior angle is =
360
number of exterior angles
6) Each interior angle = 180 – exterior angle.
# RELATION BETWEEN T-RATIOS
1. Sin2θ + cos2θ = 1
Sin2θ = 1 - cos2θ ; cos2θ = 1 - Sin2θ
2. 1 + tan2 θ = sec2θ
sec2θ - tan2 θ = 1 ;
3. 1 + cot2θ =cosec2θ
cosec2θ - cot2θ = 1
4. Sinθ =
5. cosθ =
6. tanθ =
1
cosecθ
1
secθ
1
cotθ
7. tanθ =
sinθ
cosθ
and cotθ =
8. sec θ + tan θ =
1
secθ − tanθ
9. cosec θ + cot θ =
10.
11.
1−cosθ
sinθ
sinθ
sinθ
,
1
cosecθ − cotθ
= tan(θ/2)
1 + cosθ
cosθ
= cot(θ/2)
θ ≠ nπ + π/2
,
θ ≠ nπ
Periods trigonometric functions:
Function
Sinnθ, Cosnθ,
Cosecnθ, Secnθ
Tannθ, cotnθ
Period
Condition
2π
π
If n is even
π
If n is odd
{
If n is odd
or
even
|sinθ|, |cosecθ|,
|cosθ|, |secθ|,
|tanθ|, |cotθ|
π
|sinθ| + |cosθ|,
|cosecθ| + |secθ|,
|tanθ| + |cotθ|
π
2
DOMAIN AND RANGE OF TRIGONOMETRIC FUNCTIONS
Range
Function
Domain
ℝ
Sinθ
[-1, 1]
ℝ
Cosθ
[-1, 1]
π
ℝ - {(2n+1) ; n ∈ ℤ}
2
Tanθ
ℝ- { nπ; n ∈ ℤ}
Cosecθ
(-∞, ∞) = ℝ
(-∞, 1] ∪[1, ∞)
π
ℝ - {(2n+1) ; n ∈ ℤ}
2
Secθ
ℝ - { nπ; n ∈ ℤ}
cotθ
i.e. – 1 ≤ Sinθ ≤ 1
⟹ |𝐒𝐢𝐧θ| ≤ 1
– 1 ≤ Cosθ ≤ 1
⟹ |cosθ| ≤ 1
Secθ ≥ 1 or Secθ ≤ - 1
Cosecθ ≥ 1 or Cosecθ ≤ - 1
(-∞, 1] ∪[1, ∞)
(-∞, ∞) = ℝ
Maximum and Minimum values of T-functions
T-function
Max value
Min value
Sinθ
1
-1
Cosθ
1
-1
Tanθ
R
R
Sinθ. Cosθ
1/2
-1/2
aSinθ + bCosθ
aSinθ + bCosθ + c
√a2 + b 2
C
+ √a2
+
b2
- √a2 + b 2
C - √a2 + b 2
T-FUNCTIONS OF STD ANGLES
Function/Angles
00
300
450
600
900
Sinθ
0
1/2
1/√2
√3/2
1
Cosθ
1
√3/2
1/√2
1/2
0
Tanθ
0
1/√3
1
√3
Function/Angles
180
360
540
720
Sinθ
√5 − 1
4
√10 − 2√5
√5 + 1
4
√10 + 2√5
√10 + 2√5
√5 + 1
4
√10 − 2√5
√5 − 1
4
Cosθ
4
4
Function/Angles
Sinθ
Cosθ
4
4
150
750
√3 − 1
√3 + 1
2√2
2√2
√3 + 1
√3 − 1
2√2
2√2
* ALLIED ANGLES:
Co-function of Sin θ is Cosθ and vice-versa
Co-function of tan θ is cot θ and vice-versa
1. For the angles (90 ± θ) and (270 ± θ)
T-function changes to its co-function. Then respective sign will come
according to ASTC rule
2. For the angles (180 ± θ) and (360 ± θ)
T-function remains in its original form. Then respective sign will come
according to ASTC rule
* COMPOUND ANGLES
1) Sin(A ± B )= SinA CoSB ± CosA SinB
2) Cos (A ± B )= CosA CoSB ± SinA SinB
3) tan(A ± B) =
tanA ± tanB
1 ∓ tanA tanB
4) Sin(A – B) Sin(A + B) = Sin2A –Sin2B
5) Cos(A + B) Cos(A – B) = Cos2A – Sin2B = Cos2B - Sin2A
Sum or Difference rule
1. 2SinA CosB = Sin(A +B) + Sin(A – B)
2. 2CosA SinB = Sin(A + B ) – Sin (A – B)
3. 2CosA CosB = Cos(A + B ) – Cos (A – B)
4. - 2SinA SinB = Cos(A + B ) – Cos (A – B)
Product rule
C+D
C−D
2
2
1. SinC + SinD = 2Sin(
) Cos(
C+D
2. SinC - SinD = 2Cos(
2
3. CosC + CosD = 2Cos(
)
C−D
) Sin(
)
2
C+D
C−D
2
2
) Cos(
C+D
4. CosC - CosD = -2Sin(
2
C−D
) Sin(
2
)
)
Multiple Angle Formulae
2tanA
1) Sin2A = 2SinA CosA =
2) Cos2A =
Cos2A
3) Tan2A =
–
Sin2A
1 + tan2 A
=1–
2tanA
1 − tan2 A
5) Cos3A = 4Cos3A – 3CosA
7) 1 – cosA = 2Sin2(A/2)
2Sin2A
=
2cos2A
–1=
𝟏 − 𝐭𝐚𝐧𝟐 𝐀
𝟏 + 𝐭𝐚𝐧𝟐 𝐀
4) Sin3A = 3SinA – 4Sin3A
6) Tan3A =
3 tanA − tan3 A
1 − 3 tan2 A
8) 1 + cosA = 2Cos2(A/2)
A
A
2
2
9) Sin9 = Cos81; Cos9 = Sin81
10) Sin + Cos = √1 + SinA
A
A
2
2
11) Sin - Cos = √1 − SinA
12) If A + B + C = 180
i) Sin2A + Sin2B + Sin2C = 4SinA SinB SinC
ii) Cos2A + Cos2b + Cos2C = -1 – 4CosA CosB CosC
A
B
C
2
2
2
iii) SinA + SinB + SinC = 4 Cos Cos Cos
A
B
2
2
iv) CosA + CosB + CosC = 1 + 4 Sin Sin
C
Sin 2
v) tanA + tanB + tanC = tanA tanB tanC
vi) CotA CotB + CotB CotC + CotC CotA = 1
A
B
C
A
B
C
2
2
2
2
2
2
vii) Cot + Cot + Cot = Cot Cot Cot
A
B
2
2
viii) tan tan
B
C
C
A
2
2
2
2
+ tan tan + tann tan
=1
i) Sinθ1 + Sinθ2 + Sinθ3 +………………+ Sinθn = n
Then Sinθ1 = Sinθ2 = Sinθ3 =…………………..= Sinθn = 1
ii) Cosθ1 + Cosθ2 + Cosθ3 +………………+ Cosθn = n
Then Cosθ1 = Cosθ2 = Cosθ3 =………………= Cosθn = 1
iii) Sinθ + Cosecθ = 2
Cosθ + Secθ
=2
Sinθ + Cosecθ = -2
Cosθ + Secθ
then Sinθ = 1
then Cosθ = 1
then Sinθ = -1
= -2 then Cosθ = -1
EXAMPLES:
1. The angle between the minute hand and the hour hand, in degrees,
at 3 : 15 P.M.
a) 70
b) 50
c) 90
d) 7.50
2. The angle between the minute hand and the hour hand, in degrees,
at 1 : 05 P.M.
a) 00
b) 2.50
c) 20
d) 50
3. The angle between the minute hand and the hour hand, in degrees,
at 2 : 10 P.M.
a) 100
b) 150
c) 50
d) 7.50
4. The angle between the minute hand and the hour hand, in degrees,
at 6 : 30 P.M.
a) 170
b) 150
c) 190
d) 12 0
5. The angle between the minute hand and the hour hand, in degrees,
at 11 : 55P.M.
a) 27.50
b) 250
c) 290
d) 17.50
6. The angles of a triangle are in A.P. and the radian measure of the smallest to
the degree measure of the mean is as π : 200. The greatest angle in degree is
a) 660
b) 650
c) 700
d) 680
7. The angle between hands of a clock when the time is 4:25 A.M is [COMED’K-09]
a) 12.50
b) 17.50
c) 300
d) 47.50
8. The ratio of the areas of the squares that can be inscribed in a circle and
semicircle of same radius is
a) 3:4
b) 5:2
c) 2:5
d) 4:3
9. The perimeter of a certain sector of a circle is equal to length of the arc of
the semicircle. Then the angle at the centre of the sector in radian is
a) π
b) π +2
c) π - 2
[CET-08]
d) π/2
10. Which of the following is possible?
a) Sinθ=
a2 + b2
a2 −
b2
4
b) secθ =
5
c) tanθ = 45
d) cosθ =
7
3
11. ∆ABC is a right angled at C, then tanA + tanB is equal to
a)
b2
b) a + b
ac
12. If tanθ =
a)
−4
5
but not
−𝟒
𝟑
4
5
c)
a2
d)
bc
c2
ab
, then sinθ is
b)
−4
5
or
4
5
c)
4
5
but not
−4
5
d) none of these.
13. If sinθ and cosθ are the roots of the equation ax2 – bx + c =0, then a, b and
c satisfy the relation
a) a2 - b2 + 2ac= 0
b) a2 + b2 + 2ac= 0
c) a2 + c2 + 2ab= 0
d) a2 - b2 - 2ac= 0
14. The maximum value of sinθ + cosθ is
a) 1
b)1/√2
c) √2
d) 2
15. The minimum value of sinθ - cosθ is
a) -√2
b)- 2√2
c) √2
d) -2
16. The maximum value of sinθ. cosθ is
a) 1
b)2
c) √3/2
d) 1/2
17. The minimum value of sinθ.cosθ is
a) -1
b) -1/2
c) 0
d) 1/√2
18. If SinA + CosA = 1then sin2A is equal to
a) 1
b) 2
c) 0
d) 1/2
19. If ABCD is a cyclic quadrilateral, then
a) Sin(A + B) = 1
b) Sin(B +D) =-1
c) Cos(A +C) = 1
d) Cos(A +C) = -1
20. If Sin𝛉𝟏 + Sin𝛉𝟐 + Sin𝛉𝟑 = 3 Then Cos𝛉𝟏 + Cos𝛉𝟐 + Cos𝛉𝟑 =
a) 3
b) 2
c) 0
d) 4
21. If tan(θ + x)tan(θ - x)= 1 for all x, then value of θ must be,
a) 450
b) 900
c) 00
d) 600
ANSWER KEY:
1. Ans – d
2. Ans – b
3. Ans – c
4. Ans -b
5. Ans – a
6. Ans – a
7. Ans – b
8. Ans – b
9. Ans – c
10. Ans – c
11. Ans – d
12. Ans – b
13. Ans – a
14. Ans – c
15. Ans –a
16. Ans –d
17. Ans –b
18. Ans – c
19. Ans –d
20. Ans – c
21. Ans – a