Formulae:
Pythagoras theorem:
the square of the hypotenuse of a right angle triangle is equal to the sum of the squares on
the other two sides.
a 2  b2  c 2
Indices
1. a m  a n  a mn
2. a m  a n  a mn
3.
a 
m n
 a mn
m
n
4. a  n a m
1
5. a n  n
a
6. a 0  1 , a  0.
Logarithm
1. log(ab)  log a  log b
a
2. log( )  log a  log b
b
3. log a n  n log a
4. log1  0
Trigonometry
Identities:
1
cos 
1
2. csc  
sin 
1
cos 

3. cot  
tan  sin 
sin 
4. tan  
cos 
1. sec  
5. sin 2   cos2   1
6. 1  tan 2   sec2 
7. 1  cot 2   cos ec2
Double angle:
1. sin 2  2 sin  cos 
2. cos 2  cos2   sin 2 
 2 cos2   1
 1  2 sin 2 
2 tan 
3. tan 2 
1  tan 2 
Sum and difference of ratio:


1.
sin   sin   2 sin
cos
2
2


2.
sin   sin   2 cos
sin
2
2


3.
cos   cos   2 cos
cos
2
2


4.
cos   cos   2 sin
sin
2
2
Product of ratio:
1. 2 sin  cos   sin(  )  sin(  )
2. 2 cos  cos   cos(  )  cos(  )
3. 2 sin  sin   cos(  )  cos(  )
Half angle:
1
1. tan   t
2
1
t
2. sin  
2
1 t2
1
1
3. cos  
2
1 t2
Differentiation:
y
c
cFn
sin F
cos F
tan F
sec F
csc F
cot F
dy
dx
0
F is a function of x ,c is a constant.
y
c  nF n 1
dF
dx
dF
 sin F 
dx
cos F 
dF
dx
F ' sec F tan F
sec 2 F 
F ' csc F cot F
 F ' csc 2 F
dy
dx
sin1 F
F'
cos1 F
1 F 2
F'
tan 1 F
cot 1 F
1 F 2
F'
1 F 2
 F'
1 F 2
sec1 F
F'
csc1 F
F F 2 1
 F'
sinh 1 F
F F 2 1
F'
cosh1 F
1 F 2
F'
F 2 1
dF
dx
1 dF

F dx
eF
ln F
sinh F
dF
dx
dF
sinh F 
dx
dF
sec h 2 F 
dx
tanh F
2.
3.
sec h 1F
coth 1 F
cosh F 
cosh F
1.
tanh1 F
eF 
Product rule: if u and v are function of x ,then
d (uv )
dv
du
.
u
v
dx
dx
dx
Quotient rule: if u and v are function of x ,then
u
du
dv
d( ) v
u
v  dx
dx
2
dx
v
dy dy du
Chain rule: If y{u ( x)} ,then


dx du dx
If u  f ( x, y, z) ,then
du u dx u dy u dz



dt x dt y dt z dt
Integration:
Integration by parts:  udv  uv   vdu
b
Area under a curve, A   ydx
a
b
Volume formed, V    y 2 dx
a
Mean, M 
b
1
ydx
b  a a
Root mean square value, (rms) 2 
b
1
y 2 dx
b  a a
Complex number
1. z1 z 2  r1 r2 [cos(1   2 )  i sin(1   2 )]
z
r
2. 1  1 [cos(1   2 )  i sin(1   2 )]
z2 r2
3. DeMoivre’s theorem: r (cos   i sin )  r n (cos n  i sin n)
n
in
4. e  cos n  i sin n
F'
1 F 2
 F'
F 1 F 2
 F'
1 F 2
 in
 cos n  i sin n
5. e
1
6. z   2 cos 
z
1
7. z n  n  2 cos n
z
1
8. z   2i sin 
z
1
9. z n  n  2i sin n
z
Hyperbolic function:
e  e 
1. cosh  
2

e  e
2. sinh  
2

e  e
3. tanh    
e e
Vectors




If P  ( a i  b j  c k ) then

magnitude of P is  r  a 2  b 2  c 2
a
b
c
Direction cosine of is given by: l  , m  , n  and l 2  m2  n 2  1
r
r
r


 
 
Dot product of vector a and vector b is ( a . b )  a b cos 
Angle between two vectors is cos   ll 'mm'nn'
For perpendicular vectors, ll ' mm' nn'  0

A straight line through A parallel the free vector b is















r  a   b where r  x i  y j  z k , b  b1 i  b2 j  b3 k and a  a1 i  a2 j  a3 k
x  a1 y  a2 z  a3
Vector equation in Cartesian form is


b1
b2
b3
Series
Arithmetic series: a  (a  d )  (a  2d )  (a  3d )    
un  a  (n  1)d
n
S  [ 2 a  ( n  1) d ]
2
Geometric series : a  ar  ar 2  ar 3     
un  ar n1
Sn 
If
a(1  r n )
1 r
r  1, S  
a
1 r
Power of natural numbers:
n
n(n  1)
1.  r 
2
r 1
n
r
2.
r 1
2

n(n  1)(2n  1)
6
 n( n  1) 
r3  


 2 
r 1
n
3.
2
Maclaurin’s series:
x2
x3
f ( x)  f (0)  xf ' (0) 
f " (0) 
f ' ' ' (0)     
2!
3!
Taylor’s series for one variable:
h2
h3
f ( x  h)  f ( x)  hf ' ( x) 
f " ( x) 
f ' ' ' ( x)     
2!
3!
Taylor’s series of two variables:
f ( x, y ) near the point ( xo , yo ) is
f ( x, y )  f ( x0 , y0 )  ( x  x0 ) f x ( x0 , y0 )  ( y  y0 ) f y ( x0 , y0 )
1
( x  x0 ) 2 f xx ( x0 , y0 )  2( x  x0 )( y  y0 ) f xy ( x0 , y0 )  ( y  y0 ) 2 f yy ( x0 , y0 )  ...  (1)
2!

Binomial series:
( a  b ) n  a n  na n 1b 

n ( n  1) n 2 2 n ( n  1)( n  2) n 3 3
a b 
a b      b n
2!
3!
Standard series:
x3 x5 x7
sin x  x        
3! 5! 7!
x2 x4 x6
cos x  1        
2! 4! 6!
x 2 x3 x 4
x
e 1 x         
2! 3! 4!