Important Formulas: Functions With Examples | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


                
  
Relations and Functions
? A relation R from a set A to a set B is a subset of the cartesian product A × B obtained
by describing a relationship between the first element x and the second element y of the
ordered pairs in A × B.
? Function : A function f from a set A to a set B is a specific type of relation for which every
element x of set A has one and only one image y in set B. We write f : A ? B, where
f (x) = y.
? A function f : X ? Y is one-one (or injective) if
f(x
1
) = f(x
2
) ? x
1
 = x
2
 ? x
1
, x
2
 ? X.
? A function f : X ? Y is onto (or surjective) if given any
y ? Y, ? x ? X such that f(x) = y.
? Many-One Function :
A function f : A ? B is called many- one, if two or more different elements of A have the
same f-image in B.
? Into function :
A function f : A ? B is into if there exist at least one element in B which is not the f -
image of any element in A.
? Many One-Onto function :
A function f : A ? R is said to be many one- onto if f is onto but not one-one.
? Many One-Into function :
A function is said to be many one-into if it is neither one-one nor onto.
? A function f : X ? Y is invertible if and only if f is one-one and onto.
Trigonometric Functions and Equations
? General Solution of the equation sin? = 0:
when sin? = 0
? = n? : n ? I i.e. n = 0, ± 1, ±2...........
General solution of the equation cos? = 0 :
when cos? = 0
? = (2n + 1)?/2, n ? I. i.e. n = 0, ±1, +2.......
General solution of the equation tan? = 0:
General solution of tan? = 0 is ? = n?; n ? I
? General solution of the equation
(a) sin? = sin? : ? = n? + (?1)
n
? ; n ? I
(b) sin? = k, where ?1 ? k ? 1.
? = n? + (?1)
n
?, where n ? I and ? = sin
?1
k
(c) cos? = cos? : ? = 2n? ± ?, n ? I
(d) cos? = k, where ?1 < k < 1.
? = 2n? ± ?, where n ? I and ? = cos
?1
 k
(e) tan? = tan? : ? = n? + ? ; n ? I
(f) tan? = k, ? = n? + ?, where n ? I and ? = tan
?1
k
(g) sin
2
? = sin
2
? : ? = n? ± ?; n ? I
(h) cos
2
? = cos
2
? : ? = n? ± ? ; n ? I
(i) tan
2
? = tan
2
? : ? = n? ± ? ; n ? I
  JEE  Mathematics Imporatant  Formulas
Page 2


                
  
Relations and Functions
? A relation R from a set A to a set B is a subset of the cartesian product A × B obtained
by describing a relationship between the first element x and the second element y of the
ordered pairs in A × B.
? Function : A function f from a set A to a set B is a specific type of relation for which every
element x of set A has one and only one image y in set B. We write f : A ? B, where
f (x) = y.
? A function f : X ? Y is one-one (or injective) if
f(x
1
) = f(x
2
) ? x
1
 = x
2
 ? x
1
, x
2
 ? X.
? A function f : X ? Y is onto (or surjective) if given any
y ? Y, ? x ? X such that f(x) = y.
? Many-One Function :
A function f : A ? B is called many- one, if two or more different elements of A have the
same f-image in B.
? Into function :
A function f : A ? B is into if there exist at least one element in B which is not the f -
image of any element in A.
? Many One-Onto function :
A function f : A ? R is said to be many one- onto if f is onto but not one-one.
? Many One-Into function :
A function is said to be many one-into if it is neither one-one nor onto.
? A function f : X ? Y is invertible if and only if f is one-one and onto.
Trigonometric Functions and Equations
? General Solution of the equation sin? = 0:
when sin? = 0
? = n? : n ? I i.e. n = 0, ± 1, ±2...........
General solution of the equation cos? = 0 :
when cos? = 0
? = (2n + 1)?/2, n ? I. i.e. n = 0, ±1, +2.......
General solution of the equation tan? = 0:
General solution of tan? = 0 is ? = n?; n ? I
? General solution of the equation
(a) sin? = sin? : ? = n? + (?1)
n
? ; n ? I
(b) sin? = k, where ?1 ? k ? 1.
? = n? + (?1)
n
?, where n ? I and ? = sin
?1
k
(c) cos? = cos? : ? = 2n? ± ?, n ? I
(d) cos? = k, where ?1 < k < 1.
? = 2n? ± ?, where n ? I and ? = cos
?1
 k
(e) tan? = tan? : ? = n? + ? ; n ? I
(f) tan? = k, ? = n? + ?, where n ? I and ? = tan
?1
k
(g) sin
2
? = sin
2
? : ? = n? ± ?; n ? I
(h) cos
2
? = cos
2
? : ? = n? ± ? ; n ? I
(i) tan
2
? = tan
2
? : ? = n? ± ? ; n ? I
  JEE  Mathematics Imporatant  Formulas
                
  
? sin? + sin (? + ?) + sin (? +2?) +........ to n terms
? ?
n 1 n
sin sin
2 2
; 2n
sin / 2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
?
? cos? + cos (? + ?) + cos (? +2?) +........ to n terms
n 1 n
cos sin
2 2
; 2n
sin
2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
? ? ?
? ?
? ?
? tan 
B C b c A
cot
2 b c 2
? ? ? ? ? ? ? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ?
? sin
A (s b)(s c)
2 bc
? ? ? ?
?
? ?
? ?
? tan
A (s b)(s c)
2 s(s a)
? ? ? ?
?
? ?
?
? ?
? R = 
a b c
2sin A 2sinB 2sinC
? ?
? R = 
abc
4?
? r = 4R sin
A B C
.sin .sin
2 2 2
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? a = c cos B + b cos C
? Maximum value of a sin ? + b cos ? = 
2 2
a b ? and minimum value of a sin ? + b cos
? = ?
2 2
a b ?
Inverse Trigonometric Functions
? Properties of inverse trigonometric function
? tan
?1
 x + tan
?1
 y = 
1
1
1
x y
tan , if xy 1
1 xy
if x 0,y 0 x y
tan ,
and xy 1 1 xy
if x 0, y 0 x y
tan ,
and xy 1 1 xy
?
?
?
? ? ? ?
?
? ? ?
?
? ?
?
?
? ? ? ? ? ?
??
? ? ?
? ?
? ?
?
?
? ? ? ? ?
?
???
? ?
? ? ?
? ? ?
Page 3


                
  
Relations and Functions
? A relation R from a set A to a set B is a subset of the cartesian product A × B obtained
by describing a relationship between the first element x and the second element y of the
ordered pairs in A × B.
? Function : A function f from a set A to a set B is a specific type of relation for which every
element x of set A has one and only one image y in set B. We write f : A ? B, where
f (x) = y.
? A function f : X ? Y is one-one (or injective) if
f(x
1
) = f(x
2
) ? x
1
 = x
2
 ? x
1
, x
2
 ? X.
? A function f : X ? Y is onto (or surjective) if given any
y ? Y, ? x ? X such that f(x) = y.
? Many-One Function :
A function f : A ? B is called many- one, if two or more different elements of A have the
same f-image in B.
? Into function :
A function f : A ? B is into if there exist at least one element in B which is not the f -
image of any element in A.
? Many One-Onto function :
A function f : A ? R is said to be many one- onto if f is onto but not one-one.
? Many One-Into function :
A function is said to be many one-into if it is neither one-one nor onto.
? A function f : X ? Y is invertible if and only if f is one-one and onto.
Trigonometric Functions and Equations
? General Solution of the equation sin? = 0:
when sin? = 0
? = n? : n ? I i.e. n = 0, ± 1, ±2...........
General solution of the equation cos? = 0 :
when cos? = 0
? = (2n + 1)?/2, n ? I. i.e. n = 0, ±1, +2.......
General solution of the equation tan? = 0:
General solution of tan? = 0 is ? = n?; n ? I
? General solution of the equation
(a) sin? = sin? : ? = n? + (?1)
n
? ; n ? I
(b) sin? = k, where ?1 ? k ? 1.
? = n? + (?1)
n
?, where n ? I and ? = sin
?1
k
(c) cos? = cos? : ? = 2n? ± ?, n ? I
(d) cos? = k, where ?1 < k < 1.
? = 2n? ± ?, where n ? I and ? = cos
?1
 k
(e) tan? = tan? : ? = n? + ? ; n ? I
(f) tan? = k, ? = n? + ?, where n ? I and ? = tan
?1
k
(g) sin
2
? = sin
2
? : ? = n? ± ?; n ? I
(h) cos
2
? = cos
2
? : ? = n? ± ? ; n ? I
(i) tan
2
? = tan
2
? : ? = n? ± ? ; n ? I
  JEE  Mathematics Imporatant  Formulas
                
  
? sin? + sin (? + ?) + sin (? +2?) +........ to n terms
? ?
n 1 n
sin sin
2 2
; 2n
sin / 2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
?
? cos? + cos (? + ?) + cos (? +2?) +........ to n terms
n 1 n
cos sin
2 2
; 2n
sin
2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
? ? ?
? ?
? ?
? tan 
B C b c A
cot
2 b c 2
? ? ? ? ? ? ? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ?
? sin
A (s b)(s c)
2 bc
? ? ? ?
?
? ?
? ?
? tan
A (s b)(s c)
2 s(s a)
? ? ? ?
?
? ?
?
? ?
? R = 
a b c
2sin A 2sinB 2sinC
? ?
? R = 
abc
4?
? r = 4R sin
A B C
.sin .sin
2 2 2
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? a = c cos B + b cos C
? Maximum value of a sin ? + b cos ? = 
2 2
a b ? and minimum value of a sin ? + b cos
? = ?
2 2
a b ?
Inverse Trigonometric Functions
? Properties of inverse trigonometric function
? tan
?1
 x + tan
?1
 y = 
1
1
1
x y
tan , if xy 1
1 xy
if x 0,y 0 x y
tan ,
and xy 1 1 xy
if x 0, y 0 x y
tan ,
and xy 1 1 xy
?
?
?
? ? ? ?
?
? ? ?
?
? ?
?
?
? ? ? ? ? ?
??
? ? ?
? ?
? ?
?
?
? ? ? ? ?
?
???
? ?
? ? ?
? ? ?
                
  
? tan
?1
 x ? tan
?1
 y = 
1
1
1
x y
tan , if xy 1
1 xy
x y
tan , if x 0, y 0 and xy 1
1 xy
x y
tan , if x 0, y 0 and xy 1
1 xy
?
?
?
? ? ? ?
??
? ? ?
?
? ?
?
?
? ? ? ?
?? ? ? ??
? ? ?
?
? ?
?
?
? ? ?
?
??? ? ? ??
? ?
? ?
? ? ?
? sin
?1
 x + sin
?1
 y = 
2 2
1 2 2
2 2
1 2 2
2 2
1 2 2 2 2
if 1 x, y 1 and x y 1
sin {x 1 y y 1 x } ,
or if xy 0 and x y 1
if 0 x, y 1
sin {x 1 y y 1 x } ,
and x y 1
sin {x 1 y y 1 x } , if 1 x, y 0 and x y 1
?
?
?
? ? ? ? ? ?
? ? ?
?
? ? ?
?
?
? ?
?
?? ? ? ?
?
? ?
?
?
??? ? ? ? ? ? ? ? ?
?
?
?
? cos
?1
 x + cos
?1
 y = 
1 2 2
1 2 2
cos {xy 1 x 1 y } , if 1 x, y 1 and x y 0
2 cos {xy 1 x 1 y }, if 1 x, y 1 and x y 0
?
?
?
? ? ? ? ? ? ? ?
?
?
?? ? ? ? ? ? ? ? ? ?
?
1 2
1 1 2
1 2
1 1
sin (2x 1 x ) , if x
2 2
1
2sin x sin (2x 1 x ) , if x 1
2
1
sin (2x 1 x ) , if 1 x
2
?
? ?
?
?
? ? ? ?
?
?
?
? ?? ? ? ?
?
?
?
??? ? ? ? ??
?
?
1
2
1 1
2
1
2
2x
tan , if 1 x 1
1 x
2x
2tan x tan , if x 1
1 x
2x
tan , if x 1
1 x
?
? ?
?
? ? ?
? ? ?
? ? ?
?
? ?
?
?
? ? ?
? ?? ?
?
? ?
?
? ?
?
?
? ?
??? ?? ?
? ?
? ? ? ? ?