What are Standard Mathematical Relations for Physics?
Math is an integral part of science. Without mathematical calculations, the study of science cannot be completed. In basic study of physics, chemistry or engineering topics, we need to apply some standard mathematical relations to get a desired result.
Some standard mathematical relations are summarized here.
Standard Mathematical Relations in Algebra
Some standard mathematical relations in algebra which are commonly used in physics and engineering calculations are summarized below.
| 1 | \quad \left ( a + b \right )^2 = a^2 + 2ab + b^2 |
| 2 | \quad \left ( a - b \right )^2 = a^2 - 2ab + b^2 |
| 3 | \quad a^2 - b^2 = \left ( a + b \right ) \left ( a - b \right ) |
| 4 | \quad \left ( a + b \right )^3 = a^3 + 3a^2b + 3ab^2 + b^3 |
| 5 | \quad \left ( a + b \right )^3 = a^3 + b^3 + 3ab \left ( a + b \right ) |
| 6 | \quad \left ( a - b \right )^3 = a^3 - 3a^2b + 3ab^2 - b^3 |
| 7 | \quad \left ( a + b \right )^3 = a^3 - b^3 - 3ab \left ( a - b \right ) |
| 8 | \quad a^3 + b^3 = \left ( a + b \right ) \left ( a^2 - ab + b^2\right ) |
| 9 | \quad a^3 + b^3 = \left ( a + b \right )^3 - 3ab \left ( a + b\right ) |
| 10 | \quad a^3 - b^3 = \left ( a - b \right ) \left ( a^2 + ab + b^2\right ) |
| 11 | \quad a^3 + b^3 = \left ( a - b \right )^3 + 3ab \left ( a - b\right ) |
| 12 | \quad \left ( a + b + c \right )^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2 ac |
Quadratic Equation
An equation of second degree is called a quadratic equation. It is in the form of \left ( ax^2 + bx + c = 0 \right ) . Then roots of the quadratic equation will be –
x = \frac { - b \pm \sqrt {b^2 -4ac} }{ 2a }
By solving this equation we will get two values (solutions) for variable ( x ) . One of the values is acceptable depending upon the feasibility of given conditions in the problem.
Binomial Theorem
If ( n ) is any integer (either positive or negative) or a fraction and ( x ) is any real number, then –
\left ( 1 + x \right )^n = 1 + xn + x^2 \left [ \frac { n ( n - 1 ) }{ 2! } \right ] + x^3 \left [ \frac { n ( n - 1 )( n - 2 ) }{ 3! } \right ] + \ .....
Where, \quad 2! = 2 \times 1 = 2, \quad 3! = 3 \times 2 \times 1 = 6 etc.
In general, \quad n! = n \times ( n - 1 ) \times ( n - 2 ) \ ...... 3 \times 2 \times 1
If, \quad x << 1 then \left ( 1 + x \right )^n = 1 + nx
Logarithms
| 1 | \quad ( N = a^x ) Then by definition of logarithm \quad \log_a {N} = x |
| 2 | \quad \log_a mn = \log_a m + \log_a n |
| 3 | \quad \log_a { \frac {m}{n}} = \log_a m - \log_a n |
| 4 | \quad \log_a m^n = n \log_a m |
| 4 | \quad \log_a m = \log_b m \times \log_a b |
Standard Mathematical Relations in Trigonometry
| 1 | \quad \sin ( - \theta ) = - \sin \theta | 2 | \quad \cosec ( - \theta ) = - \cosec \theta |
| 3 | \quad \cos ( - \theta ) = \cos \theta | 4 | \quad \sec ( - \theta ) = \sec \theta |
| 5 | \quad \tan ( - \theta ) = - \tan \theta | 6 | \quad \cot ( - \theta ) = - \cot \theta |
Ratio of angles in First Quadrant
| 1 | \quad \sin ( 90 - \theta ) = \cos \theta | 2 | \quad \cosec ( 90 - \theta ) = \sec \theta |
| 3 | \quad \cos ( 90 - \theta ) = \sin \theta | 4 | \quad \sec ( 90 - \theta ) = \cosec \theta |
| 5 | \quad \tan ( 90 - \theta ) = \cot \theta | 6 | \quad \cot ( 90 - \theta ) = \tan \theta |
Ratio of angles in Second Quadrant
| 1 | \quad \sin ( 90 + \theta ) = \cos \theta | 2 | \quad \cosec ( 90 + \theta ) = \sec \theta |
| 3 | \quad \cos ( 90 + \theta ) = - \sin \theta | 4 | \quad \sec ( 90 + \theta ) = - \cosec \theta |
| 5 | \quad \tan ( 90 + \theta ) = - \cot \theta | 6 | \quad \cot ( 90 + \theta ) = - \tan \theta |
| 7 | \quad \sin ( 180 - \theta ) = \sin \theta | 8 | \quad \cosec ( 180 - \theta ) = \cosec \theta |
| 9 | \quad \cos ( 180 - \theta ) = - \cos \theta | 10 | \quad \sec ( 180 - \theta ) = - \sec \theta |
| 11 | \quad \tan ( 180 - \theta ) = - \tan \theta | 12 | \quad \cot ( 180 - \theta ) = - \cot \theta |
Ratio of angles in Third Quadrant
| 1 | \quad \sin ( 180 + \theta ) = - \sin \theta | 2 | \quad \cosec ( 180 + \theta ) = - \cosec \theta |
| 3 | \quad \cos ( 180 + \theta ) = - \cos \theta | 4 | \quad \sec ( 180 + \theta ) = - \sec \theta |
| 5 | \quad \tan ( 180 + \theta ) = \tan \theta | 6 | \quad \cot ( 180 + \theta ) = \cot \theta |
| 7 | \quad \sin ( 270 - \theta ) = - \cos \theta | 8 | \quad \cosec ( 270 - \theta ) = - \sec \theta |
| 9 | \quad \cos ( 270 - \theta ) = - \sin \theta | 10 | \quad \sec ( 270 - \theta ) = - \cosec \theta |
| 11 | \quad \tan ( 270 - \theta ) = \cot \theta | 12 | \quad \cot ( 270 - \theta ) = \tan \theta |
Ratio of angles in Fourth Quadrant
| 1 | \quad \sin ( 270 + \theta ) = - \cos \theta | 2 | \quad \cosec ( 270 + \theta ) = - \sec \theta |
| 3 | \quad \cos ( 270 + \theta ) = \sin \theta | 4 | \quad \sec ( 270 + \theta ) = \cosec \theta |
| 5 | \quad \tan ( 270 + \theta ) = - \cot \theta | 6 | \quad \cot ( 270 + \theta ) = - \tan \theta |
| 7 | \quad \sin ( 360 - \theta ) = - \sin \theta | 8 | \quad \cosec ( 360 - \theta ) = - \cosec \theta |
| 9 | \quad \cos ( 360 - \theta ) = \cos \theta | 10 | \quad \sec ( 360 - \theta ) = \sec \theta |
| 11 | \quad \tan ( 360 - \theta ) = - \tan \theta | 12 | \quad \cot ( 360 - \theta ) = - \cot \theta |
Tricks to remember Trigonometrical Ratios
Angles can be represented in the form of [ \theta = ( n \pi + x ) ] . Where ( n ) is an integer and may take the value of 0, \ 1, \ 2, \ 3 \ \& \ 4 ) .
Following thumb rules is followed –
- In the ratio if ( n = 0, \ 2, \ 4 ) ( even number ), then ratio of the angle will not change.
- In the ratio if ( n = 1, \ 3 ) ( odd number ), then ratio of the angle will change. ( \sin x ) will change to ( \cos x ) and ( \cos x ) will change to ( \sin x ) and ( \tan x ) will change to ( \cot x ) . In similar manner angles ( \cosec x ), \ ( \sec x ) and ( \cot x ) are also change accordingly.
- Sign of ratio ( \pm ) will depend upon the position of the angle in the quadrant as shown in figure.
EXAMPLE –
- \sin ( - \theta ) = \sin ( 0 \times \pi - \theta ) = \sin \theta ( Ratio not changed because n is even. Plus sign used because it falls in first quadrant. In first quadrant all angles are positive.)
- \cos ( 180 \degree - \theta ) = \cos ( 2 \times \pi - \theta ) = - \cos \theta ( Ratio not changed because n is even. Minus sign is used because it falls in second quadrant. In second quadrant only ( \sin x \ \& \ \cosec x ) will be positive. All other angles will be negative.)
- \sin ( 270 \degree + \theta ) = \sin ( 3 \times \pi + \theta ) = - \cos \theta ( Ratio changed because n is odd. Minus sign is used because it falls in fourth quadrant. In fourth quadrant only ( \cos x \ \& \ \sec x ) will be positive. All other angles will be negative.)
Trigonometrical Identities
| 1 | \quad \sin ( A + B ) = \sin A \ \cos B + \cos A \ \sin B |
| 2 | \quad \sin ( A - B ) = \sin A \ \cos B - \cos A \ \sin B |
| 3 | \quad \cos ( A + B ) = \cos A \ \cos B - \sin A \ \sin B |
| 4 | \quad \cos ( A - B ) = \cos A \ \cos B + \sin A \ \sin B |
| 5 | \quad \tan ( A + B ) = \left ( \frac {\tan A + \tan B }{ 1 - \tan A \ \tan B } \right ) |
| 6 | \quad \tan ( A - B ) = \left ( \frac {\tan A - \tan B }{ 1 + \tan A \ \tan B } \right ) |
| 7 | \quad \sin 2A = 2 \sin A \ \cos A = \left ( \frac { 2 \tan A }{ 1 + \tan^2 A } \right ) |
| 8 | \quad \cos 2A = \cos^2 A - \sin^2 A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1 = \left ( \frac { 1 - \tan^2 A }{ 1 + \tan^2 A } \right ) |
| 9 | \quad \tan 2A = \left ( \frac { 2 \tan A }{ 1 - \tan^2 A } \right ) |
| 10 | \quad \sin ( A + B ) + \sin ( A - B ) = 2 \sin A \ \cos B |
| 11 | \quad \sin ( A + B ) - \sin ( A - B ) = 2 \cos A \ \sin B |
| 12 | \quad \cos ( A + B ) + \cos ( A - B ) = 2 \cos A \ \cos B |
| 13 | \quad \cos ( A + B ) - \cos ( A - B ) = - 2 \sin A \ \sin B |
| 14 | \quad \sin C + \sin D = 2 \sin \left ( \frac { C + D }{ 2 } \right ) \ \cos \left ( \frac { C - D }{ 2 } \right ) |
| 15 | \quad \sin C - \sin D = 2 \cos \left ( \frac { C + D }{ 2 } \right ) \ \sin \left ( \frac { C - D }{ 2 } \right ) |
| 16 | \quad \cos C + \cos D = 2 \cos \left ( \frac { C + D }{ 2 } \right ) \ \cos \left ( \frac { C - D }{ 2 } \right ) |
| 17 | \quad \cos C - \cos D = - 2 \sin \left ( \frac { C + D }{ 2 } \right ) \ \sin \left ( \frac { C - D }{ 2 } \right ) |
Values of Trigonometrical Ratios
| Ratio / Angle | 0 \degree | 30 \degree | 45 \degree | 60 \degree | 90 \degree |
| \sin \theta | 0 | \frac {1}{2} | \frac {1}{\sqrt {2}} | \frac {\sqrt {3}}{2} | 1 |
| \cos \theta | 1 | \frac {\sqrt {3}}{2} | \frac {1}{\sqrt {2}} | \frac {1}{2} | 0 |
| \tan \theta | 0 | \frac {1}{\sqrt {3}} | 1 | \sqrt {3} | \infty |
Standard Mathematical Relations in Calculus
Study of calculus has two parts –
- Differential calculus.
- Integral calculus.
1. Differential Calculus
Let, variable ( y ) is a function of variable ( x ) . So we can write –
y = f ( x )
The derivative \left ( \frac { dy }{ dx } \right ) gives the instantaneous rate of change of function ( y ) with respect to variable ( x ) .
Some standard mathematical relations for differential calculus are given below.
| 1 | \quad ( c ) is a constant, then derivative \quad \frac {d}{dx} ( c ) = 0 |
| 2 | \quad ( c ) is a constant, then derivative \quad \frac {d}{dx} ( c \ y ) = c . \frac {dy}{dx} |
| 3 | \quad ( y = x^n ) then derivative \quad \frac {d}{dx} ( x^n ) = n \ x^{( n - 1 )} |
| 4 | \quad ( y = u \pm v ) and ( u ) and ( v ) both are the functions of ( x ) then \quad \frac {dy}{dx} = \left [ \left ( \frac {du}{dx} \right ) \pm \left ( \frac {dv}{dx} \right ) \right ] |
| 5 | \quad ( y = u v ) and ( u ) and ( v ) both are the functions of ( x ) then \quad \frac {dy}{dx} = \left [ u \left ( \frac {dv}{dx} \right ) + v \left ( \frac {du}{dx} \right ) \right ] |
| 6 | \quad ( y = \frac {u}{v} and ( u ) and ( v ) both are the functions of ( x ) then \quad \frac {dy}{dx} = \left [ \frac { v \left ( \frac {du}{dx} \right ) - u \left ( \frac {dv}{dx} \right ) }{v^2} \right ] |
| 7 | \quad ( y ) is a function of ( u ) which itself is a function of ( x ) then \quad \frac {dy}{dx} = \left [ \left ( \frac {dy}{du} \right ) \times \left ( \frac {du}{dx} \right ) \right ] |
| 8 | \quad \frac {d}{dx} ( \log_e {x} = \left ( \frac {1}{x} \right ) |
| 9 | \quad \frac {d}{dx} ( \log_a {x} ) = \left ( \frac {1}{x} \right ) \left ( \log_e {a} \right ) |
| 10 | \quad \frac {d}{dx} ( e^x ) = e^x |
| 11 | \quad \frac {d}{dx} ( a^x ) = a^x \log_e {a} |
| 12 | \quad \frac {d}{dx} ( \sin x ) = \cos x |
| 13 | \quad \frac {d}{dx} ( \cos x ) = - \sin x |
| 14 | \quad \frac {d}{dx} ( \tan x ) = \sec^2 x |
| 15 | \quad \frac {d}{dx} ( \cot x ) = - \cosec^2 x |
| 16 | \quad \frac {d}{dx} ( \sec x ) = \sec^2 x |
| 17 | \quad \frac {d}{dx} ( \cosec x ) = - \cosec x \ \cot x |
2. Integral Calculus
Integration is the reverse process of differentiation. It is the process of finding a function from its derivative.
Let a function is [ f ( x ) ] and its derivative with respect to ( x ) is given as [ f' ( x ) ] . So –
\left ( \frac {d}{dx} \right ) f ( x ) = f' ( x )
Consider that the derivative [ f' ( x ) ] is given. Then we can find the function [ f ( x ) ] by integration.
Therefore, \quad \int f' ( x ) dx = f ( x ) + c
Where, ( c ) is called the constant of integration.
Because, derivative of a constant term is zero i.e. \left [ \frac {d}{dx} ( c ) = 0 \right ] . Hence, the constant of integration ( c ) is always added to the results of integration. The value of this constant is calculated with the given conditions in problem.
Some standard mathematical relations for integral calculus are given below.
| 1 | \quad \int ( x^n ) dx = \left [ \frac { x^{(n + 1)}}{(n + 1)} \right ] + c |
| 2 | \quad \int dx = x + c |
| 3 | \quad \int \left ( \frac {1}{x} \right ) dx = \log_e {x} + c |
| 4 | \quad \int ( \cos x ) dx = \sin x + c |
| 5 | \quad \int ( \sin x ) dx = - \cos x + c |
| 6 | \quad \int ( \sec^2 x ) dx = \tan x + c |
| 7 | \quad \int ( \cosec^2 x ) dx = - \cot x + c |
| 8 | \quad \int ( \sec x )( \tan x ) dx = \sec x + c |
| 9 | \quad \int ( \cosec x )( \cot x ) dx = - \cosec x + c |
| 10 | \quad \int ( ax + b )^n dx = \left [ \frac { ( ax + b )^{( n + 1 )}}{a( n + 1 )} \right ] + c |
| 11 | \quad \int \left ( \frac {1}{( ax + b )} \right ) dx = \left ( \frac {1}{a} \right ) \log_e { ( ax + b ) } + c |
| 12 | \quad \int e^x dx = e^x + c |
| 13 | \quad \int a^x dx = \left [ \frac { a^x }{ \log_e a } \right ] + c = a^x \log_a e + c |
| 14 | \quad \int ( u \pm v \pm w ) dx = \int u dx \pm \int v dx \pm \int w dx + c |