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Trigonometry Identities
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TRIGONOMETRIC RATIOS & IDENTITIES
1. INTRODUCTION
The word 'Trigonometry' is derived from two Greek words
(1) Trigonon and(2) MetronThe word trigonon means a triangle and the word metron means a measurement. Hence trigonometry means thescience of measuring triangles.
2. ANGLE
Consider a ray OAuuur
. If this ray rotates about its end point O and takes the position OB , then the angle AOB has
been generated.
Vertex O
= angle
Initial side A
B
Termin
al side
An angle is considered as the figure obtained by rotating a given ray about its end - point.
The initial position OA is called the initial side and the final position OB is called terminal side of the angle. The endpoint O about which the ray rotates is called the vertex of the angle.
3. SENSE OF AN ANGLE
The sense of an angle is said to be positive or negative according as the initial side rotates in anticlockwise orclockwise direction to get to the terminal side.
O = +ve
Anticlockwise directionA
B O = ve
Clockwise directionA
B
4. RIGHT ANGLE
When two lines intersect at a point in such a way that two adjacent angles made by them are equal, then eachangle is called a right angle.
9090
OX' X
A
5. A CONSTANT NUMBER pipipipipiThe ratio of the circumference to the diameter of a circle is always equal to a constant and this constant is denotedby the Greek letter pi
TRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIES
TRIGONOMETRIC RATIOS & IDENTITIES
i.e. Circumference of a circle
Diameter of the circle= pi
(constant)
The constant pi is an irrational number and its approximate value is taken as 227 . The more accurate value to six
decimals places is taken as 355113 .
6. SYSTEMS OF MEASUREMENT OF AN ANGLESThere are three systems for measuring angles
6.1 Sexagesimal or English system6.2 Centesimal or French system6.3 Circular system
6.1 Sexagesimal system : The principal unit in this system is degree (). One right angle is divided into 90 equalpart and each part is called one degree (1) . One degree is divided into 60 equal parts and each part is calledone minute. Minute is denoted by (1'). One minute is equally divided into 60 equal parts and each part is calledone second (1").In Mathematical form :
One right angle = 90 (Read as 90 degrees )1 = 60' (Read as 60 minutes )1' = 60" (Read as 60 seconds )
Ex.1 40 30' is equal to
(1) o41
2
(2) 81 (3) o81
2
(4) None of these
Sol. We know that , 30' = o1
2
; 40 + o1
2
=
o812
6.2 Centesimal system : The principal unit in system is grade and is denoted by (g). One right angle is dividedinto 100 equal parts, called grades, and each grade is subdivided into 100 minutes, and each minute into 100seconds.
In Mathematical form :
One right angles = 100g (Read as 100 grades)1g = 100' (Read as 100 seconds)1' = 100" (Read as 100 seconds)
Ex.2 25' is equal to -Sol. 100' is equal to 1g
so is equal to g g1 125
100 4
=
Relation between Sexagesimal and Centesimal systems :One right angle = 90 (degree system) ..... (1)
TRIGONOMETRIC RATIOS & IDENTITIES
One right angle = 100g (grade system) ..... (2)by (1) and (2)
90 = 100g or , D G90 100
=
then we can say, ; 1 = g10
9
, 1g = o9
10
Ex.3 80g is equal to
Sol. We know that 1g = o9
10
then, 80g = o9 80
10
80g = 72
6.3 Circular system : In circular system the unit of measurement is radian. One radian, written as 1C, is the measureof an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.Consider a circle of radius r having centre of O. Let A be a point on the circle. Now cut off an arc AB whoselength is equal to the radius r of the circle.
B O rIc
rr
C
A
In the adjacent figure OA = OC = arc AC = r = radius of circle, then measurement of AOC is one radianand denoted by 1c. Thus AOC = 1c .
6.3.1 Some Important conversion
pi Radian = 180 One radian = o180
pi 6pi
Radian = 30
4pi
Radian = 45 3pi
Radian = 60 2pi
Radian = 90
23pi
Radian = 120 34pi
Radian = 13556pi
Radian = 150
76pi
Radian = 210 54pi
Radian = 22553pi
Radian = 300
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.4 240 is equal to
[1] C4
3
p [2] C3
4
p [3] '4
3p [4]
'34p
Sol. We know that180 = pi
240 = C
x240180
p
=
C43
p
Ans. [1]
Ex.5 The difference between two acute angle of a right angle triangle is 9p . Then the angles in degree are -
[1] 50, 30 [2] 25, 45 [3] 20, 40 [4] 35, 55Sol. In triangle ABC let C = 90
So A B =
9p = 20 .......... (i)
and sum of all the angles in ABC
A + B + C = 180
C = 90
A + B = 90 ......... (ii)Solving (i) & (ii) A = 55, B = = 35 Ans. [4]
6.3.2 Relation between systems of measurement of angles
D G 2C90 100
= =
pi
Ex.6 The length of an arc of a circle of radius 5 cm subtending a central angle measuring 15 is -
[1] 312p
cm [2] 712p
cm [3] 512p [4] None of these
Sol. Let s be the length of the arc subtending an angle at the centre of a circle of radius r.
then, = s
r
Here, r = 5 cm, and = 15 = C
15x180
p
= C
12
p
= s
r = 12
p =
s
5
= 512
pcm
TRIGONOMETRIC RATIOS & IDENTITIES
7. TRIGONOMETRICAL RATIOS OR FUNCTIONSLet a line OA makes angle with a fixed line OX and AM is perpendicular from A on OX. Then in right-angledtriangle AMO, trigonometrical ratios (functions) with respect to are defined as follows :
sin = perpendicular(P)hypotenuse(H)
cos = base(B)
hypotenuse(H)
tan = perpendicular (P)
Base (B)
cosec = HP . sec =
HB , cot =
BP
Note :
(i) Since t-ratios are ratio between two sides of a right angled triangle with respect to an angle, so they arereal numbers.(ii) may be acute angle or obtuse angle or right angle.
8. RELATIONS BETWEEN TRIGONOMETRICAL RATIOS
(i) 1 1 1cosec , sec ,cotsin cos tan
= = = (ii)
sintancos
=
(iii) coscotsin
= (iv) sin
2 + cos2 = 1
(v) 1 + tan2 = sec2 (vi) 1 + cot2 = cosec2Ex.7 If cosec A + cot A = 11/2, then tan A is equal to
[1] 21/12 [2] 15/16 [3] 44/117 [4] 117/43Sol. Cosec A + cot A = 11/2
1cosecA cot A+ =
211 ....... (1)
cosec A cot A = 211 ..... (2)
(1) (2) = 2 cot A = 11 22 11- = 11722
= tan A = 44
117 Ans [3]
O B
M
H P
Y
X
A
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.8cos
1 tanq
q- + sin
1 cotq
q- is equal to
[1] sin cos [2] sin + cos [3] tan + cot [4] tan cot
Sol.cos
1 tanq
q- + sin
1 cotq
q-
=
cos sinsin cos1- 1cos sin
q qq qq q
+
-
=
2 2cos sincos sin cos sin
q qq q q q
-
- -
=
2 2cos sincos sin
q qq q
-
-
= cos + sin Ans [2]Ex.9 tan2 sec2 (cot2 cos2 ) equals
[1] 0 [2] 1 [3] 1 [4] 2Sol. tan2 sec2 (cot2 cos2 )
= sec2 (tan2 cot2 tan2 cos2 )
= sec2 2
22
sin1 coscos
q qq
- = sec2 (1sin2)
= sec2. cos2 = 1 Ans. [3]
9. SIGN OF TRIGONOMETRIC RATIOS(i) All ratios sin, cos, tan cot, sec and cosec are positive
in Ist quadrant.(ii) sin( or cosec) positive in IInd quadrant, rest are negative.(iii) tan( or cot) positive in IIIrd quadrant, rest are negative.(iv) cos( or sec) positive in IVth quadrant, rest are negative.
Ex.10 The value of sin and tan if cos = 1213
-
and lies in the third quadrant is -
[1] 513- and 5
12 [2] 5
12 and 5
13- [3] 1213- and
513
- [4] none of these
Sol. We have cos2 + sin2 = 1
sin = 21 cos q -
IInd Quadrant Ist Quadrant
IIIrd Quadrant IVth Quadrant
x-axis
y-axis
TRIGONOMETRIC RATIOS & IDENTITIES
In the third quadrant sin is negaitve, therefore
sin = 21 cos q- sin = 2121
13
- = 5
13
then, tan = sincos
qq tan =
5 13x
13 12-
-
=
512 Ans.[1]
Ex.11 If 2p
< < pi, then 1 sin1 sin
-
+ +
1 sin1 sin
+
-
is equal to
[1] 2 cosec [2] 2 cosec [3] 2 sec [4] 2 sec
Sol. Exp. = 2(1 sin ) (1 sin )
1 sinq q
q
- + +
- =
2cosq
-
= 2 sec Ans.[4]
10. DOMAIN AND RANGE OF A TRIGONOMETRICAL FUNCTIONIf f : X Y is a function, defined on the set X, then the domain of the function f, written as Domain is theset of all independent variables x, for which the image f(x) is well defined element of Y, called the co-domainof f.Range of f : X Y is the set of all images f(x) which belongs to Y , i.e.Range f = {f(x) Y:x X} Y The domain and range of trigonometrical functions are tabulated as follows
Trigo. function Domain Rangesin x R, the set of all the real number 1 sin x 1
cosx R 1 cos x 1
tan x R ( )2n 1 ,n2pi
+
I R
cosecx R { }n ,npi I R { x : 1 < x < 1 }
sec x R ( )2n 1 ,n2pi
+
I R { x : 1 < x < 1 }
cot x R { }n ,npi I R
TRIGONOMETRIC RATIOS & IDENTITIES
11. VARIATION OF VALUES OF TRIGONOMETRICAL RATIOS IN DIFFERENT QUADRANTS-
12. RELATION BETWEEN TRIGONOMETRICAL RATIOS AND IDENTITIES-
(1)
=cos
sintan (2)
=sincos
cot
(3) sin A cosec A = tan A cot A = cos A sec A = 1(4) sin2 + cos2 = 1 or sin2 = 1 cos2 or cos2 = 1 sin2(5) 1 + tan2 = sec2 or sec2 tan2 = 1 or sec2 1 = tan2.(6) 1 + cot2 = cosec2 or cosec2 cot2 = 1 or cosec2 1 = cot2(7) Since sin2A + cos2A = 1, hence each of sin A and cos A is numerically less than or equal to unity i.e.,
|sin A| 1 and |cos A| 1or 1 sin A 1 and 1 cos A 1
Note : The modulus of real number x is defined as |x| = x if x 0 and |x| = x if x < 0.(8) Since sec A and cosec A are respectively reciprocals of cos A and sin A, therefore the values of sec Aand cosec A are always numerically greater than or equal to unity i.e.,
sec A 1 or sec A 1
and cosec A 1 or cosec A 1
In other words, we never have
1 < cosec A < 1 and 1 < sec A < 1.
While tanA and cotA may take any real value
Sine decreases from 1 to 0
cosine decreases from 0 to 1
tangent increases from to 0
cotangent decreases from 0 to
secant increases from to 1
cosecant increases from 1 to
Sine increases from 0 to 1
cosine decreases from 1 to 0
tangent increases from 0 to
cotangent decreases from to 0
secant increases from 1 to
cosecant decreases from to 1
' Sine decreases from 0 to 1
cosine increases from 1 to 0
tangent increases from 0 to
cotangent decreases from to 0
secant decreases from 1 to
cosecant increases from to 1
IV Since increases from 1 to 0
cosine increases from 0 to 1
tangent increases from to 0
cotangent decreases from 0 to
secant decreases from to 1
cosecant decreases from 1 to
'
TRIGONOMETRIC RATIOS & IDENTITIES
13. TRIGONOMETRICAL RATIOS IN TERMS OF EACH OF THE OTHER
sin cos tan cot sec cosec
sin sin 2cos1 +
2tan1
tan
+ 2cot1
1
sec
1sec2eccos
1
cos 2sin1 cos + 2tan11
+
2cot1
cot
sec1
eccos
1eccos 2
tan
2sin1sin
cos
cos1 2 tan cot1
1sec2 1eccos
12
cot 21 sin
sin
2cos
1 cos
1
tan cot 21
sec 1 2cosec 1
sec 21
1 sin 1
cos21 tan+
21 cotcot+
sec 2
cosec
cosec 1
cosec 1
sin 21
1 cos
21 tantan+
21 cot+ 2
sec
sec 1 cosec
14. TRIGONOMETRICAL RATIOS OF STANDARD ANGLES
TRIGONOMETRIC RATIOS & IDENTITIES
15. TRIGONOMETRICAL RATIOS OF ALLIED ANGLES
Ex.12cos(90 )sec( ) tan(180 )
sin(360 )sec(180 )cot(90 )q q q
q q q+ - -
+ + - eqauls
[1] 2 [2] 1 [3] 1 [4] 0Sol. Given expression
( sin )(sec )( tan )(sin )( sec ) tan
q q qq q q
- -
-
= 1 Ans. [3]
Ex.13 The value of cot5 cot10 ......... cot 85 is
[1] 12 [2] 1 [3] 12 [4] 0
Sol. cot 5 cot 10 ......... cot 85= cot 5 cot 10 ........ cot(90 10) cot (90 5)= cot 5 cot 10 ...... tan 10 tan 5
= (tan 5 cot 5) (tan 10 cot 10) .......= (1) (1) (1) ........... = 1
Ex.14 Sin 10 + sin 20 + sin30 +...... + sin 360 is eqaul to[1] 1 [2] 0 [3] 1 [4] name of these
Sol. Q sin 190 = sin(180 + 10) = sin 10sin 200 = sin 20
sin 210 = sin 30
..............................
sin 360 = sin 180 = 0
\ Exp. = 0
TRIGONOMETRIC RATIOS & IDENTITIES
16. GRAPH OF DIFFERENT TRIGONOMETRICAL RATIOS
TRIGONOMETRIC RATIOS & IDENTITIES
17. SUM AND DIFFERENCE FORMULAE
(i) sin(A + B) = sinA cosB + cosA sin B (ii) sin(A B) = sinA cosB cosA sin B(iii) cos(A + B) = cosA cosB sinA sinB (iv) cos(A B) = cosA cosB + sinA sinB
(v) tan(A + B) =BtanAtan1BtanAtan
+ tan(A B) =
BtanAtan1BtanAtan
+
(vi) tan+
=
+pitan1tan1
4 tan +
=
pitan1tan1
4
(vii) cot(A + B) =BcotAcot1BcotAcot
+
(viii) cot(A B) = AcotBcot1BcotAcot
+
(ix) sin(A + B) sin(A B) = sin2 A sin2B = cos2B cos2A(x) cos(A + B) cos(A B) = cos2A sin2B = cos2B sin2A
(xi) sin2 = 2sin cos = )tan1(tan2
2 +
(cosA sin A)2 = 1 sin 2A
(xii) cos2 = cos2 sin2 = )tan1()tan1(
2
2
+
= 1 2 sin2 = 2 cos2 1
(xiii) tan2 = 2tan1
tan2
sin
)cos1( = tan 2
+
sin)cos1(
= cot 2
)cos1()cos1(
+
= tan2 2
)cos1()cos1(
+
= cot2 2
(xiv) A 1 cosAsin 2 2
= , cos
A 1 cosA2 2
+=
(xv) tan A 1 cos A2 1 coA
= +
(xvi) sin 3A = 3 sin A 4 sin3A or sin3 A = 14 (3 sinA sin 3A)
(xvii) cos 3A = 4cos3A 3 cosA or cos3A = 14 ( cos 3A + 3cosA )
(xix) tan 3A = 3
23 tanA tan A
1 3tan A
( A npi + pi/6 )
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.15 tan 20 + tan 40 + 3 tan 20 tan 40 is equal to
[1] 32
[2] 34
[3] 3 [4] 1
Sol. 3 = tan 60 = tan (40 + 20)
=
tan40 tan201 tan40 tan20
+
- +
\ 3 3 tan 40 tan 20 = tan 40 + tan 20
Hence tan 40 + tan 20 + 3 tan 40 tan 20 = 3 Ans.[3]
Ex.16 If tan A = 12 and tan B =
13 , then the value of A + B i.e. tan
1 12 + tan
1
13 is
[1] 6p [2] p [3] zero [4] 4
p
Sol. tan (A + B) = tan A tanB1 tanA tanB+
-
=
1 12 3
1 11 .2 3
+
-
=
5 / 65 / 6 = 1
\ A + B = 45 = 4p
Ans.[4]
Ex.17 If sin A = 35 , 0 < A < 2
p and cos B =
1213
-
, pi < B < 32p
then sin (A B) eqauls
[1] 1665- [2] 1665 [3]
6516 [4]
6516
-
Sol. We have : sin A = 35 , where 0 < A < 2
p
\ cos A = 21 sin A -
cos A = + 21 sin A - = 9125
- =
45 [Qcos is positive in first quadrant]
TRIGONOMETRIC RATIOS & IDENTITIES
It is given that : cos B = 1213
-
and pi < B < 32p
\ sin B = 21 cos B -
sin B = 21 cos B - [Q sin is negative in the third quadrant]
sin B = 2121
13 -
- = 5
13-
Now, sin (AB) = sin A cos B cos A sin B = 3 12 4 5x x5 13 5 13- -
-
=
1665
- Ans.[1]
Ex.18sin2
1 cosq
q+ equals
[1] cot [2] tan [3] sin [4] cosec
Sol.sin2
1 cosq
q+ = 22sin cos
2cosq q
q = tan Ans. [2]
Ex.19 If sin A = 12 , then 4 cos
3 A 3 cos A is equal to (0 < A < 90)
[1] 1 [2] 0 [3] 32
[4] 12
Sol. 4 cos3 A 3 cos A = cos 3A = cos 90 = 0 [Q sin A = 12 A = 30] Ans.[2]
Ex.20 If and be between 0 and 2p
and if cos ( + ) = 1213 and sin( ) = 35 , then sin 2 is equal
[1] 1615 [2] 0 [3] 5665 [4]
6465
Sol. sin (2) = sin ( + + )= sin ( + ) cos ( ) + cos ( + ) sin ( )
=
513 .
45 +
1213 .
35 =
5665 Ans.[3]
TRIGONOMETRIC RATIOS & IDENTITIES
18. FORMULAE FOR TRANSFORMATION OF SUM OR DIFFERENCE INTO PRODUCT
(i) sinC + sinD = 2sin
+
2)DC(
cos2
)DC(
(ii) sinC sinD = 2cos
+
2)DC(
sin2
)DC(
(iii) cosC + cosD = 2cos
+
2)DC(
cos2
)DC(
(iv) cosC cosD = 2sin
+
2)CD(
sin2
)DC(
(v) tanA tanB = BcosAcosBsinAcosBcosAsin
BcosBsin
AcosAsin
= BcosAcos
)BAsin( =
pi
pi+pi mB,
2nA
(vi) cotA cotB = BsinAsin)ABsin(
pi+pipi
2mB,nA
(vii) cosA sinA = 2 sin
pi A4 = 2 cos
pi A4 tanA + cotA = )AcosA(sin
1
(viii) 1 + tanA tanB = BcosAcos)BAcos(
1 tanA tanB = BcosAcos)BAcos( +
(ix) cotA tanA = 2cot2A tanA + cotA = 2cosec2A
(x) sin 2A
+ cos 2A
= Asin1+ sin 2A
cos 2A
= Asin1
Ex.21 cos 52 + cos 68 + cos 172 =
[1] 0 [2] 1 [3] 2 [4] none of theseSol. cos 52 + cos 68 + cos 172
= cos 52 + cos 68 + cos (180 8)
= 2 cos 52 68
2+
cos 68 52
2-
cos 8
= 2 cos 60 cos 8 cos 8
= cos 8 cos 8 = 0 Ans.[1]Ex.22 If sin 2 + sin 2 = 1/2, cos 2 + cos 2 = 3/2 then cos2 ( ) is equal to
[1] 3/8 [2] 5/8 [3] 3/4 [4] 5/4
TRIGONOMETRIC RATIOS & IDENTITIES
Sol. Using cosine formula
2 sin ( + ) cos ( ) = 1/2 ...... (1)2 cos ( + ) cos ( ) = 3/2 ...... (2)Squaring (1) and (2) and then adding
4 cos2 ( ) = 1 94 4+ = 52
cos2 ( ) = 58 Ans.[2]
Ex.23 Sin 47 + sin 61 sin 11 sin25 is equal to
[1] sin36 [2] cos 36 [3] sin 7 [4] cos 7Sol. Given value
= (sin 47 + sin 61) (sin 11 + sin 25)= 2 sin 54 cos 7 2 sin 18 cos 7
= 2 cos 7 (sin 54 sin 18)= 2 cos 7 2 cos 36 sin 18
= 2 cos 7 2sin36cos36
cos18 x cos 36
= cos 7 2sin36cos36
cos18
= cos 7 sin72cos18 = cos 7 [Qsin 72 = cos 18] Ans.[2]
19. FORMULAE FOR TRANSFORMATION OF PRODUCT INTO SUM OR DIFFERENCE(i) 2sinA cosB = sin(A + B) + sin(A B)(ii) 2cosA sinB = sin(A + B) sin(A B)(iii) 2cosA cosB = cos(A + B) + cos(A B)(iv) 2sinA sinB = cos(A B) cos(A + B)
Ex.24 2 cos 13p
cos 913
p + cos
313
p + cos
513
p equals
[1] 0 [2] 1 [3] 2 [4] 4
TRIGONOMETRIC RATIOS & IDENTITIES
Sol. LHS = 2 cos 13p
cos 913
p + cos
313
p+ cos
513
p
= cos 913 13
p p + + cos 913 13
p p - + cos
313
p + cos
513
p
= cos 1013
p + cos
813
p + cos
313
p+ cos
513
p
= cos 313
pp
- + cos513
pp
- + cos 313
p+ cos
513
p
= cos 313
p cos
513
p + cos
313
p + cos
513
p
= 0 = RHS [Q cos(pi ) = cos] Ans.[1]
Ex.25 2 sin 512
p sin 12p equals to
[1] 12 [2] 12 [3]
14 [4]
16
Sol. 2 sin A sin B = cos (A B) cos (A + B)
\ 2 sin 512
p sin 12p = cos
512 12
p p - cos
512 12
p p +
= cos 3p
cos 2p
=
12 0 =
12 Ans. [2]
20. TRIGONOMETRICAL RATIOS FOR SOME IMPORTANT ANGLES
(i) sin 17 2
=
4 2 62 2
(ii) cos 17 2
=
4 2 62 2
+ +
(iii) tan 17 2
= ( )( )3 2 2 1 (iv) sin15 = 22)13(
= cos75
(v) cos15 = 22)13( +
= sin75 (vi) tan15 = 2 3 = cot75
(vii) cot15 = 2 + 3 = tan75 (viii) sin22
21
= 2221
TRIGONOMETRIC RATIOS & IDENTITIES
(ix) cos22
21
= 2221
+ (x) tan22
21
= 2 1
(xi) cot22
21
= 2 + 1 (xii) sin18 = 41 ( 5 1) = cos72
(xiii) cos18 = 41
5210 + = sin72 (xiv) sin36 = 41
2210 = cos54
(xv) cos36 = 41 ( 5 + 1) = sin54
21. FORMULAE FOR SUM OF THREE ANGLES(i) sin (A + B + C) = sinA cos B cosC + cosA sin B cos C + cos A cos B sin C sin A sin B sin C
= cos A cos B cos C ( tanA + tan B + tanC tan A tan B tan C )(ii) cos (A + B + C) = cosA cosB cosC sinA sinB cosC sinA cos B sin C cos A sin B sin C
= cos A cos B cos C (1 tan A tan B tan B tan C tan C tanA )
(iii) tan (A + B + C) = tanA tanB tanC tanA tanBtanC1 tanA tanB tanBtanC tanCtan A+ +
(iv) 4sin(60 A) sinA sin(60 + A) = sin3A 4cos(60 A) cosA cos(60 + A) = cos3A tan(60 A) tanA tan(60 + A) = tan3A
22. CONDITIONAL IDENTITIES(1) If A + B + C = 180 , then
(i) sin 2A + sin 2B + sin2C = 4 sin A sin B sin C(ii) sin 2A + sin 2B sin 2C = 4 cosA cos B sin C(iii) sin (B + C A) + sin (C + A B) + sin (A + B C) = 4 sin A sin B sin C(iv) cos 2A + cos 2B + cos 2C = 14 cos A cos B cos C(v) cos 2A + cos 2 B cos 2C = 1 4 sinA sin B cos C
(2) If A + B + C = 180, then
(i) sin A + sin B + sin C = 4cos A2 cos B2 cos
C2
(ii) sin A + sin B sin C = 4 sin A2 sin B2 cos
C2
(iii) cosA + cos B + cosC = 1 + 4 sin A2 sin B2 sin
C2
(iv) cosA + cosB cos C = 1 + 4 cos A2 cosB2 sin
C2
(v) cos A cosB cosC 2sinBsinC sinCsinA sinA sinB+ + =
TRIGONOMETRIC RATIOS & IDENTITIES
(3) If A + B + C = pi , then(i) sin2A + sin2B sin2C = 2 sin A sin B cos C(ii) cos2A + cos2B + cos2C = 12 cos A cos B cos C(iii) sin2A + sin2B + sin2C = 2 + 2 cosA cos B cosC(iv) cos2A + cos2B cos2C = 12 sin A sin B cos C
(4) If A + B + C = pi , then
(i) sin2 A2
+ sin2B2 + sin
2 C2 =1 2sin
A2 sin
B2 sin
C2
(ii) 2 2 2A B C A B Ccos cos cos 2 2sin sin sin2 2 2 2 2 2+ + = +
(iii) 2 2 2A B C A B Csin sin sin 1 2cos cos sin2 2 2 2 2 2+ =
(iv) 2 2 2A B C A B Ccos cos cos 2cos cos sin2 2 2 2 2 2+ =
(5) If x + y + z = 2pi
, then
(i) sin2 x + sin2y + sin2z = 12 sin x sin y sin z(ii) cos2x + cos2y + cos2z = 2 + 2 sin x sin y sin z(iii) sin2x + sin2y + sin 2z = 4 cos x cosy cos z
(6) If A + B + C = pi , then(i) tanA + tan B + tan c = tan A tan B tan C(ii) cotB cot C + cot C cot A + cot A cot B = 1
(iii) B C C A A Btan tan tan tan tan tan 12 2 2 2 2 2+ + =
(iv) A B C A B Ccot cot cot cot cot cot2 2 2 2 2 2+ + =
(7) (a) For any angles A , B, C we have(i) sin (A + B + C) = sin A cos B cos C + cos A sin B cos C
+ cos A cos B sin C sin A sin B sin C(ii) cos (A+B+C) = cos A cos B cosC cos A sin B sin C
sin A cos B sin C sin A sin B cosC
(iii) tan (A + B + C) = tanA tanB tanC tanA tanB tanC1 tanA tanB tanB tanC tanCtan A+ +
(b) If A , B, C are the angles of a triangle, thensin(A + B + C) = sin pi = 0 andcos (A + B + C) = cos pi = 1then (a) (i) givessinA sin B sin C = sin A cos B cos C + cosA sin B cosC + cos A cos B sin Cand (a) (ii) gives1 + cos A cos B cos C = cos A sin B sin C + sin A cos B sin C + sin A sin B cos C
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.26 If A + B + C = pi, then sin 2A + sin 2B + sin 2C equals[1] 4 sin A sin B cos C [2] 4 sin A sin B sin C[3] 4 cos A sin B sin C [4] none of these
Sol. sin 2A + sin 2B + sin 2C
= 2 sin 2A 2B
2 + cos
2A 2B2
- + sin 2C
= 2 sin (pi c) cos (A B) + sin C sin 2C [Q A + B + C = pi, A + B = pi csin (A + B) = sin (pi C) = sin C]
= 2 sin C [cos (A B) + 2 sin C cos C]= 2 sin C [cos (A B) + cos C]= 2 sin C [cos (A B) cos (A + B)] [Q cos (A B) cos (A + B) = 2 sin A
sin B, by C & D formula]= 2 sin C [2 sin A sin B]= 4 sin A sin B sin C Ans.[2]
Ex.27 If A + B + C = pi, then cos2 A + cos2 B + cos2 C equal to[1] 2 cos A cos B cos C [2] 2 Sin A sin B sin C[3] 12cosA cos B cos C [4] 1 + 2 sin A sin B sin C
Sol. LHS = cos2 A + cos2 B + cos2 C= cos2 A +(1 sin2 B) + cos2 C = (cos2 A sin2 B) + cos2 C + 1= cos (A + B) cos (A B) + cos2 C + 1 [Q cos2A sin2 B = cos (A + B) cos(A B)]= cos (pi C) cos (A B) + cos2 C + 1 = cos C cos (A B) + cos2 C + 1= cos C [cos(A B) cos C] + 1 = cos C [(cos (AB) cos {pi(A + B)}] + 1= cos C [cos (A B) + cos (A + B)] + 1= cos C [(cos A cos B + sin A sin B) + (cos A cos B sin A sin B)] +1= cos C (2 cos A cos B) +1 = 1 2 cosA cosB cos C Ans. [3]
Ex.28 If A + B + C = pi, then tanA + tanB + tanC equals[1] cotA tanB tanC [2] tanA. cotB. tanC[3] tanA. tanB. tanC [4] None of these
Sol. A + B + C = piA + B = p C
tan (A + B) = tan (pi C)
tan A tanB
1 tanA tanB+
-
= tan C
tan A + tan B = tanC + tanC tanB tanC
tanA + tanB + tanC = tanA. tanB. tanC Ans.[3]
TRIGONOMETRIC RATIOS & IDENTITIES
23. GENERAL SOLUTIONS OF TRIGONOMETRICAL EQUATIONS*(i) If sin = sin then = npi + (1)n , n Z*(ii) If cos = cos then = 2npi , n Z*(iii) If tan = tan then = npi + , n Z(iv) If sin2 = sin2(v) If cos2 = tan2 then = npi , n Z(vi) If tan2 = tan2
(vii) If sin sin
cos cos
= =
then = 2npi + , n Z
24. MOTHOD OF COMPONENDO AND DIVIDENDO
If p aq b
= , then by componendo an dividendo we can write
p q a b q p b aor
q q a b q p b a
= =
+ + + +
or p q a b q p b a
orp q a b q p b a
+ + + += =
Note :- Reference of the above formulae will be given in the solutions of problems.
25. SOME IMPORTANT RESULTS
(i) 2 2 2 2a b asinx bcos x a b + + +(ii) sin2x + cosec2
x 2
(iii) cos2x + sec2 x 2(iv) tan2x + cot2 x 2
(v) 1 sin tan sec tan1 sin 4 2+ pi
= + = +
(vi) 1 sin tan sec tan1 sin 4 2 pi
= = +
(vii) 1 cos cot cosec cot1 cos 2+
= = +
(viii) 1 cos tan cosec cot1 cos 2
= = +
(ix) cos . cos 2 . cos 22 ............ cos 2n1 = n
n
sin22 sin
; ( )n pi
(x) cosA + cos (A +B) + cos (A + 2B) + ........ + cos { A + ( n 1) B } = sin nB / 2 Bcos A (n 1)sinB / 2 2
+
TRIGONOMETRIC RATIOS & IDENTITIES
26. THE GREATEST AND LEAST VALUES OF THE EXPRESSION [ a sin + b cos ]Let a = r cos ......... (1)and b = r sin ......... (2)
squaring and adding (1) and (2)then a2 + b2 = r2
or , r = 2 2a b+ a sin + b cos = r(sin cos + cos sin )= r sin ( + )But 1 sin 1so 1 sin ( + ) 1then r r sin ( + ) rhence,
2 2 2 2a b asin bcos a b + + +then the greatest and least values of
a sin + b cos are respectively 2 2 2 2a b and a b+ +
Least value of a sinx + b cos x + c is 2 2c a b + and greatest value is 2 2c a b+ +Ex.29 The maximum value of 3 sin + 4 cos is -
[1] 2 [2] 3 [3] 4 [4] 5Sol. 25 < 3 sin + 4 cos < 25 [By the standard resultes]
or 5 < 3sin + 4 cos < 5so the maximum value is 5. Ans. [4]
Ex.30 If a < 3 cos x + 5 sin (xpi/6) < b for all x, then (a, b) equals[1] ( 19, 19)- [2] (17, 17) [3] ( 21, 21)- [4] None
Sol. 12 cos x + 352
sin x = A cos x + B sin x
\ 2 2(A B )+ = 1 (76)2 = (19) Ans. [1] 27. MISCELLANEOUS POINTS(i) Some useful identities :
(a) tan (A + B + C) = tanA tanA tanB tanC1 tanA tanB
(b) tan = cot 2 cot 2(c) tan3 = tan . tan ( 60 ) . tan ( 60 + ) (d) tan (A+B) tanA tanB = tanA. tanB.tan(A+B)
(e) sin sin ( 60 ) sin (60 + ) = 1 sin34 (f) cos cos ( 60 ) cos (60 + ) = 1
cos34
(ii) Some useful series :
(a) sin + sin ( + ) + sin ( + 2) ......... + to n terms =
n 1 nsin sin
2 2; 2n
sin2
+
pi
(b) cos + cos ( + ) + cos ( + 2) + ........ + to n terms =
n 1 ncos sin
2 2; 2n
sin2
+
pi
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.31 The value of cos 11p
+ cos 311
p + cos
511
p + cos
711
p + cos
911
p is
[1] 0 [3] 1 [3] 12 [4] none of these
Sol. cos 11p
+ cos 311
p + cos
511
p + cos
711
p + cos
911
p
= cos 11p
+ cos 3
11 11p p + + cos
3 2.211 11
p p + + cos 3.2
11 11p p + + cos
4.211 11p p +
Use cos + cos ( + ) + cos ( + 2) + .... + cos { + (n 1) } =n
sin2
sin2
b
b cos 2 (n 1)
2a b+ -
Here = 11p
, = 211p
and n = 5 then
cos 11p
+ cos 311
p + cos
511
p + cos
711
p + cos
911
p =
5 5sin x
2 112
sin2.11
p
p =
2 2cos 4
11 112
p p +
=
5sin
11sin
11
p
p cos 511p
=
1 10sin
2 11sin
11
p
p =
sin1 112 sin
11
pp
p
-
=
12 Ans.[3]
Ex.32 The value of cos 9p
cos 29p
cos 39p
cos 49p
is
[1] 18 [2] 1
16 [3] 1
64 [4] none of these
Sol. cos 9p
cos 29p
cos 39p
cos 49p
=
2 4cos cos cos
9 9 9p p p x
3cos
9p
=
3
3sin(2 . / 9)
xcos / 32 .sin / 9
p pp
=
sin8 / 98sin / 9
pp x
12 =
18 x
12 =
116 Ans. [2]
TRIGONOMETRIC RATIOS & IDENTITIES
SOLVED EXAMPLES
Ex.1 2(sin6 + cos6 ) 3 ( sin4 + cos4 ) + 1 is equal to(1) 0 (2) 1 (3) 2 (4) None of these
Sol. (1) 2 [ (sin2 + cos2 )3 3 sin2 cos2 ( sin2 + cos2 ) ] 3 [ (sin2 + cos2 )2 ] 2sin2 cos2 +1 2 [ 1 3 sin2 cos2 ] 3 [ 1 2 sin2 cos2 ] + 1 26 sin2 cos2 3 + 6 sin2 cos2 + 1 = 0
Ex.2 If 3 sin + 4 cos = 5 then the value of 4 sin 3 cos is
(1) 0 (2) 1 (3) 2 (4) 3
Sol. (1) Let 4 sin 3 cos = a
Thus we want to eliminant from both 3 sinq + 4 cos = 5 and 4 sin q 3 cos = a, i.e. squaring and
adding these equations. We get
(3 sin q + 4 cos q)2 + (4 sin 3 cos )2 = 25 + a2
9 sin2 + 16 cos2 + 24 sin cos + 16 sin2 + 9 cos2 24 cos sin = 25 + a2
9 + 16 = 25 + a2
or a2 = 0
a = 0
4 sin 3 cos = 0
Ex.3 If x = r sin cos , y = r sin sin and z = r cos . Then the value of x2 + y2 + z2 is equal to(1) 2r2 (2) r2 (3) 0 (4) none of these .
Sol. (2) Herex
2 + y2 + z2 = r2 sin2 cos2 + r2 sin2 . sin2 + r2 cos2
= r2 sin2 ( cos2 + sin2 ) + r2 cos2
= r2 sin2 + r2 cos2
= r2 ( sin2 + cos2)
= r2
x2 + y2 + z2 = r2
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.4 If A + B = 45, then ( 1 + tanA) ( 1 + tanB ) is equal to(1) 2 (2) 1 (3) 0 (4) 4
Sol. (1) tan (A + B) = tan A tanB1 tanA tanB+
1= tan A tanB
1 tanA tanB+
[ as A + B = 45 , tan (A + B ) = 1 ]
tan A + tan B + tan A tan B = 1
or 1 + tan A + tan B + tan A tan B = 1 + 1 [ adding 1 to both side] (1 + tanA) + tanB (1 + tan A ) = 2 ( 1 + tan A) ( 1 + tanB ) = 2
Ex.5 If cos 2x + 2 cos x = 1 then sin2x ( 2cos2x) is equal to(1) 2 (2) 1 (3) 3 (4) 4 .
Sol. (2) Here, cos 2x + 2 cos x = 1 2cos2x1 + 2 cos x 1 = 0
cos2x + cosx 1 = 0
or cos x = 1 5
2 +
, neglecting 1 5
2
as 1 cos x 1 and 1 5
2
< 1
cos2 x =
25 1 6 2 5 3 52 4
= =
sin2x (2 cos2x) = 3 5 3 51 2
2 2
=
5 1 5 1 12 2
+
=
Ex.6 3 cosec20 sec 20 is equal to
(1) 0 (2) 1 (3) 2 (4) 4
Sol. (4) 3 1 3 cos20 sin20sin20 cos20 sin20 .cos20
=
3 14 cos20 sin202 22sin20 cos20
=
( )sin60 .cos20 cos60 .sin204 .sin40
=
sin(60 20 ) sin404 4 . 4sin40 sin40
= = =
TRIGONOMETRIC RATIOS & IDENTITIES
Ex.7 sin78 sin66 sin42 + sin 6 Is equal to
(1) 1/2 (2) 1/2 (3) 2 (4) 2Sol. (2) The expression
= (sin78 sin42) (sin 66 sin6)= 2cos (60) sin (18) 2 cos 36 . sin 30= sin18 cos36
=
5 1 5 1 14 4 2
+
=
Ex.8 Find set of all possible values of in [ pi , pi] such that 1 sin1 sin
+ is equal to ( sec tan ).
(1) ,2 2pi pi
(2) ,2 2pi pi
(3) ,2 2pi pi
(4) None of these
Sol. (3) Clearly / 2 pi .
as sec tan = 1 sin
cos
...(i)
and ( )2
2
1 sin1 sin 1 sin 1 sin1 sin cos coscos
= = =
+
From (i) and (ii) two expressions are equal only if cos > 0 , i.e. pi/2 < < pi / 2
1 sin1 sin
+ and sec tan are equal only
when ,2 2pi pi
Ex.9 sin 1672 + cos
1672 is equal to
(1) 1 4 2 22 + (2) 1 4 2 22
(3) 1 4 2 22 + (4) 1 4 2 22
+
Sol. (1) sin 167 2 + cos1672 =
11 sin135 12
+ = + (using cosA + sinA = 1 sin2A+ )
1 4 2 22
= + ....(i)
Ex.10 If 1 + sin 4pi
+
+ 2 cos 4pi
then the maximum value of is .
(1) 1 (2) 2 (3) 3 (4) 4
TRIGONOMETRIC RATIOS & IDENTITIES
Sol. (4) We have 1 + sin 4pi
+
+ 2 cos 4pi
= 1 + 12 (cos + sin ) + 2 ( cos + sin )
= 1 + 1 22
+
(cos + sin )
= 1 + 1 2 . 2 cos
42pi
+
the maximum value of 11 2 . 2 42
+ + =
Ex.11 The value of sin 20 sin 40 sin 60 sin 80 is
(1) 3/8 (2) 1/8 (3) 3/16 (4) none of theseSol. (3) sin 20 sin 40 sin 60 sin 80
= ( ) ( )3 sin20 sin 60 20 sin 60 202
+
= ( )2 23 sin20 sin 60 sin 202 = 23 3
sin20 sin 202 4
= 33 (3sin20 4sin 20 )
8
=
3sin60
8
=
3 3 3.
8 2 16=
Ex.12 The value of 1 cos 8p +
31 cos8p +
51 cos8p +
71 cos8p + is
[1] 12 [2] cos 8p [3] 18 [4]
1 22 2+
Sol. 1 cos 8p +
31 cos8p +
31 cos8pp
+ - 1 cos
8pp
+ -
= 1 cos 8p +
31 cos8p +
31 cos8p
- 1 cos
8p
-
= 21 cos
8p
- 2 31 cos
8p
-
=
1 2 1 cos4 4
p - -
32 1 cos4p
- -
TRIGONOMETRIC RATIOS & IDENTITIES
=
1 1 cos4 4
p -
31 cos4p
-
=
14
112
- 112
+ = 14
112
- =
18 Ans.[3]
Ex.12 If ABCD is a cyclic quadrilateral such that 12 tan A 5 = 0 and 5 cos B + 3 = 0 then the quadratic equationwhose roots are cos C tan D is
Sol. In a quadrilaterla no angle is greater then 180
Here tan A = 5
12 so, 0 < A < 2p
and 2p
< C < pi (Q A + C = 180)
\ tan (piC) = 512 , i.e. tan C = 5
12-
\ cos C = 1213
-
Also cos B = 35
- , so, 2p
< B < pi and 0 < D < 2p (QB + D = 180)
\ cos (pi C) = 35- , i.e., cos D = 35 \ tan D =
43
\ the required equation is x2 12 413 3
- + x +
1213
- .
43 = 0
39 x2 16 x 48 = 0 Ans.[1]
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