basic maths (1)

MATHEMATICAL PHYSICS
ALGEBRA
COMMON IDENTITIES
(i) (a + b)2 = a2 + 2ab + b2 = (a – b)2 + 4ab
(ii) (a – b)2 = a2 – 2ab + b2 = (a + b)2 – 4ab
(iii) a2 – b2 = (a + b)(a – b)
(iv) (a + b)3 = a3 + 3a2b + 3ab2 + b3 = a3 + b3 + 3ab(a + b)
(v) (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
(vi) a3 + b3 = (a + b)(a2 – ab + b2) = (a + b)3 – 3ab(a + b)
(vii) a3 – b3 = (a – b)(a2 + ab + b3) = (a – b)3 + 3ab(a – b)
(viii) (a + b)2 + (a – b)2 = 2(a2 + b2)
(ix) (a + b)2 – (a – b)2 = 4ab
(x) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
QUADRATIC EQUATION
An algebraic equation of second degree having the form ax2 + bx + c = 0 is called
quadratic equation. Here ‘a’ is called the coefficient of x2, ‘b’ is called the coefficient of
x and c is a constant term.
General solution of the quadratic equation is given by
24
2
bb ac
xa
− −
=
Clearly, there are two roots of the equation:
22
12
12
44
and
22
b b ac b b ac
xx
aa
b
xx a
−+ − −− −
==
+=−
In general, sum of roots =
2
Coefficient of
Coefficient of
x
x
−
Again,
12
c
xx a =
In general, product of roots =
2
Constant
Coefficient of x
BINOMIAL THEOREM
If | x | < 1 i.e., x lies between – 1 and + 1, then
23
( 1) ( 1)( 2)
(1 ) 1 ......
2! 3!
nnn nn n
x nx x x
−−−
+=++ + +
where n is any number which may be positive, negative, integer or a fraction.
Here, 2 ! = 2  1, 3 ! = 3  2  1, …...., n ! = n(n – 1)(n – 2)….  3  2  1
If n is a positive integer, then the expansion will have (n + 1) terms.

If n is a negative integer or a fraction, then the number of terms in the expansion will
be infinite.
When | x | << 1, then only the first two terms of the expansion are significant. The
second and higher order terms can be neglected. In this case, the expansions shall reduce
to the following simplified forms:
(1 + x)n = 1 + nx
(1 + x)–n = 1 – nx
(1 – x)n = 1 – nx
(1 – x)–n = 1 + nx
DEFINITION OF LOGARITHM
Logarithm of a number with respect to a given base is the power to which the base must
be raised to represent that number.
For example : 1000 = 103
 log10 1000 = 3
In general, if N = ax
then loga N = x
It follows that loga 1 = 0 ( aº = 1)
and loga a = 1 ( a1 = a)
FOUR STANDARD FORMULAE OF LOGARITHM
These four formulae are commonly used. We are mentioning them without proof.
(i) loga mn = loga m + loga n …..(product formula)
(ii)
loga
m
n
= loga m – loga n …..(quotient formula)
(iii) loga mn = n loga m ……(power formula)
(iv) loga m = logb m  loga b ….(base change formula)
Note that the product formula, quotient formulae & power formula in log can be
extended to any number of quantities.
TWO SYSTEMS OF LOGARITHM
The two systems of Logarithm in common use are:
(i) Neperian log or Natural log. Here the base is e, where
111
1 ....
1! 2! 3!
e=+ + + +
e = 2.718(approx).
In all theoretical calculations, we use Natural logarithm
(ii) Common Logarithm. Here, the base of the log is 10. In all numerical
calculations, we use common log.
We can convert Natural log into common log, as follows:
loge m = log10 m  loge 10
As loge 10 = 2.3026 log10 10
= 2.3026  1 = 2.3026
10
[101]log =
loge m = 2.3026 log10 m

TRIGONOMETRY
SYSTEMS OF MEASUREMENT OF AN ANGLE
(i) Sexagesimal system. In this system,
1 right angle = 90º (degree)
o
160'=
(minute)
1' 60''=
(second)
(ii) Centesimal system. In this system
1 right angle
100g
=
(grade)
1100'
g=
(minute)
'
1100''=
(second)
(iii) Circular system. In this system, the unit of angle is radian
One radian is the angle subtended at the centre of a circle by an arc whose
length is equal to the radius of the circle.
If l is the length of an arc and  is the angle subtended at the centre of the
circle as shown in figure, then
Arc radian
Radius
l
r
==
Angle subtended at the centre of the circle is
Circumference 2 =2 radian
Radius
r
r
==
 radian = 180º = 200g
1 radian =
o
57 16'22'' 63 63'64''
g
=
TRIGONOMETRICAL RATIOS
In right angled OMP, of figure, OMP = 90º and POM = . We can define the
trigonometric ratios as follows:
perpendicular
sine = sin
hypotenuse
PM
OP
==
base
cosine = cos
hypotenuse
OM
OP
==
perpendicular
tangent = tan
base
PM
OM
==
hypotenuse
cosecant = cosec
perpendicular
OP
PM
==
hypotenuse
secant = sec
base
OP
OM
==
base
cotangent = cot
perpendicular
OM
PM
==

FUNDAMENTAL TRIGONOMETRIC RELATIONS
(i)
111
cosec = , sec , cot
sin cos tan
==
(ii)
sin cos
tan , cot
cos sin
==
(iii)
22 22 2 2
sin cos 1, 1 tan sec , 1 cot cosec +=+= +=
SIGNS OF T-RATIOS AND LIMITS OF THE VALUE OF T-RATIOS
In order to understand the signs of T-ratios, let us first understand the rule for signs of
the sides of the triangle OMP.
Keeping in mind this sign convention and the definitions of T-ratios, we shall find
that:
(a) In the first quadrant, all the T-ratios are positive.
(b) In the second quadrant, only the sine and (its reciprocal) cosecant are +ve.
(c) In the third quadrant, only the tangent and (its reciprocal) cotangent are +ve.
(d) In the fourth quadrant, only the cosine and (its reciprocal) secant are +ve.
Let us now mention a few facts about the limits to the values of T-ratios.
(i) The maximum, i.e., greatest value of sin  or cos  is 1. The minimum, i.e.,
least value of sin  or cos  is – 1.
(ii) sec  and cosec  cannot be numerically less than one.
(iii) tan  and cot  can have any numerical values.
T-RATIOS OF ALLIED ANGLES