# Maths Formulas for Physics

Maths formulas for physics:

Physics is possible only with the help of mathematics. We need to do a lot of mathematical calculation in Physics.

Mathematics simplifies and helps in solving the problems in physics.

Thus here are some Important mathematics formulas that are needed in physics to solve and simplify any problem:

So Mathematical formulas for physics are:

Geometry:

If  Radius of a circle = r then:

it’s circumference = $2 \pi r$

and it’s area = $\pi r^2$

If radius and height of a right circular cylinder are r and h respectively then:

it’s area = $2 \pi r^2 + 2 \pi r h$

And it’s volume = $\pi r^2 h$

Area of a triangle with base “a” and height “h” is : $\dfrac{1}{2} a h$

Mathematical Signs and Symbols:

$=$ :- equals

$\approx$ :- Equals approximately

$\sim$ :- is the order of magnitude of

$\equiv$ :- is identical to , is defined as

$>$ :- is greater than

$\gg$ :- is much greater than

$<$ :- is less than

$\ll$ :- is much less than

$\ge$ :- is greater than or equal to , or is no less than

$\le$ :- is less than or equal to , or is no more than

$\pm$ :- plus or minus

$\propto$ :- is proportional to

$\sum$ :- the sum of

$x_{avg}$ :- the average value of x

if $ax^2 + bx + c =0$ is a quadratic equation

then it’s roots or “x” is: $\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Trigonometric Functions of angle $\theta$ :

With reference to the figure following:

trigonometric functions

a> $\sin \theta = \dfrac{y}{r}$ & $\cos \theta = \dfrac{x}{r}$

b> $\tan \theta = \dfrac{y}{x}$ & $\cot \theta = \dfrac{x}{y}$

c> $\sec \theta = \dfrac{r}{x}$ & $\csc \theta = \dfrac{r}{y}$

Trigonometric Identities:

a.  $\sin (90 - \theta) = \cos \theta$

b. $\cos (90- \theta) = \sin \theta$

c. $\dfrac{sin \theta}{\cos \theta} = \tan \theta$

d. $\sin ^2 \theta + \cos ^2 \theta = 1$

e. $\sec ^2 \theta - \tan ^2 \theta = 1$

f. $\csc ^2 \theta - \cot ^2 \theta = 1$

g. $\sin 2 \theta = 2 . \sin \theta . \cos \theta$

h. $\cos 2 \theta = \cos ^2 \theta - \sin ^2 \theta = 2 \cos ^2 \theta -1 = 1 - 2 \sin ^2 \theta$

i. $\sin ( \alpha \pm \beta ) = \sin \alpha . \cos \beta \pm \cos \alpha . \sin \beta$

j. $\cos ( \alpha \pm \beta ) = \cos \alpha . \cos \beta \mp \sin \alpha . \sin \beta$

k. $\tan ( \alpha \pm \beta ) = \dfrac{ \tan \alpha \pm \tan \beta }{1 \mp \tan \alpha . \tan \beta}$

l. $\sin \alpha \pm \sin \beta = 2 \sin \frac{1}{2} ( \alpha \pm \beta ) . \cos \frac{1}{2} ( \alpha \mp \beta )$

m. $\cos \alpha + \cos \beta = 2 \cos \frac{1}{2} ( \alpha + \beta ) . \cos \frac{1}{2} ( \alpha - \beta )$

n. $\cos \alpha - \cos \beta = -2 \sin \frac{1}{2} ( \alpha + \beta ) . \sin \frac{1}{2} ( \alpha - \beta )$

Pythagorean theorem:

Referring to the image below:

pythagorean theorem

In the right angled triangle:

$h^2 = p^2 + b^2$

Triangles:

In the following triangle:

triangle

Angles are A , B , C and their corresponding opposite sides are : a , b ,c

Then:

$A + B + C = 180^o$

Sine law:

$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$

Cosine law:

$c^2 = a^2 + b^2 - 2ab \cos c$

, $b^2 = a^2 + c^2 - 2ac \cos b$

& $a^2 = b^2 + c^2 - 2bc \cos a$

Binomial Theorem:

$(1 + x)^n = 1 + \dfrac{nx}{1!} + \dfrac{n(n - 1).x^2}{2!} + \cdots (x^2 < 1)$

Exponential Expansion:

$e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots$

Logarithmic Expansion:

$\ln (1 + x) = x - \frac{1}{2} x^2 + \frac{1}{3} x^3 - \cdots (|x| < 1)$

Trigonometric Expansions:

Note: All $\theta$ are in radians.

a. $\sin \theta = \theta - \dfrac{\theta ^3}{3!} + \dfrac{\theta ^5}{5!} - \cdots$

b. $\cos \theta = 1 - \dfrac{\theta ^2}{2!} + \dfrac{\theta ^4}{4!} - \cdots$

c. $\tan \theta = \theta + \dfrac{\theta ^3}{3!} + \dfrac{2 \theta ^5}{15!} + \cdots$

Cramer’s Rule:

Two simultaneous equations in unknown x and y,

$a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$

Have the solutions:

$x = \dfrac{\begin{vmatrix}c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = \dfrac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}$

& $y = \dfrac{\begin{vmatrix}a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = \dfrac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}$

Product of Vectors:

If $\hat{i}$ , $\hat{j}$ & $\hat{k}$ be unit vectors in the x , y and z directions , Then:

$\hat{i} . \hat{i} = \hat{j} . \hat{j} = \hat{k} . \hat{k} = 1$

, $\hat{i} . \hat{j} = \hat{j} . \hat{k} = \hat{k} . \hat{i} = o$

And:

$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$

& $\hat{i} \times \hat{j} = \hat{k}$ , $\hat{j} \times \hat{k} = \hat{i}$ , $\hat{k} \times \hat{i} = \hat{j}$

If $\theta$ is the smaller angle between two vectors $\overrightarrow{a}$ & $\overrightarrow{b}$ then:

$\overrightarrow{a} . \overrightarrow{b} = a_x b_x + a_y b_y + a_z b_z = ab \cos \theta$

And :

$\overrightarrow{a} \times \overrightarrow{b} = -\overrightarrow{b} \times \overrightarrow{a} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$

$= (a_y b_z - a_z b_y) \hat{i} + (a_z b_x - a_x b_z) \hat{j} + (a_x b_y - a_y b_x) \hat{k}$

And $| \overrightarrow{a} \times \overrightarrow{b} | = ab \sin \theta$

Derivatives and Integrals:

The letter “u” & “v” used in formulas following stands for any functions of “x”.

And “a” and “m” are constants.

And it should be noted that although it is not specified in the following formulas , to each indefinite integrals should be added an arbitrary constant of integration.

Derivative formulas:

a. $\dfrac{d}{dx} x = 1$

b. $\dfrac{d}{dx}au = a . \dfrac{d}{dx}u$

c. $\dfrac{d}{dx} (u \pm v) = \dfrac{d}{dx} u \pm \dfrac{d}{dx} v$

d. $\dfrac{d}{dx}x^m = m . x^{m-1}$

e. $\dfrac{d}{dx} \ln x = \dfrac{1}{x}$

f. $\dfrac{d}{dx} (u.v) = v . \dfrac{d}{dx} u + u . \dfrac{d}{dx} v$

g. $\dfrac{d}{dx} e^x = e^x$

h. $\dfrac{d}{dx} \sin x = \cos x$

i. $\dfrac{d}{dx} \cos x = - \sin x$

j. $\dfrac{d}{dx} \tan x = \sec ^2 x$

k. $\dfrac{d}{dx} \sec x = \sec x . \tan x$

l. $\dfrac{d}{dx} \cot x = - \csc ^2 x$

m. $\dfrac{d}{dx} \csc x = - \csc x . \cot x$

n. $\dfrac{d}{dx} e^u = e^u . \dfrac{d}{dx} u$

o. $\dfrac{d}{dx} \sin u = \cos u . \dfrac{d}{dx} u$

p. $\dfrac{d}{dx} \cos u = - \sin u . \dfrac{d}{dx} u$

Integral formulas:

a. $\int dx = x$

b. $\int au dx = a \int u dx$

c. $\int (u + v) dx = \int u dx + \int v dx$

d. $\int x^m dx = \dfrac{x^{m+1}}{m+1} (m \neq -1)$

e. $\int \frac{dx}{x} = \ln |x|$

f. $\int u \frac{dv}{dx} . dx = \int u . dv = uv - \int v . du$

g. $\int e^x dx = e^x$

h. $\int \sin x . dx = - \cos x$

i. $\int \cos x . dx = \sin x$

j. $\int \tan x . dx = \ln | \sec x |$

k. $\int \sin ^2 x . dx = \frac{1}{2} x - \frac{1}{4} \sin 2x$

k. $\int e^{-ax} . dx = - \frac{1}{a} e^{-ax}$

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