MATHS FORMULAS

In an x-y Cartesian coordinate system,

equation of the circle is:

(x-a)2 + (y-b)2 = r2

sin 0° = 0

cos 0° = 1 tan 0° = 0

sin 30° = 1/2 cos 30° = (√3)/2

tan 30° = 1/(√3) sin 45° = 1/(√2)

cos 45° = 1/(√2) tan 45° = 1

sin 60° = (√3)/2 cos 60° = 1/2

tan 60° = √3 sin 90° = 1

cos 90° = 0 tan 90° = ∞

sin = 1/cosec cos = 1/sec

tan = 1/cot sin/cos = tan

sin2 + cos2 = 1 1 + tan2 = sec2

1 + cot2 = cosec2 sin( 90° - ) = cos

cos( 90° - ) = sin tan( 90° - ) = cot

sin( 90° + ) = cos cos( 90° + ) = - sin

tan( 90° + ) = - cot sin( 180° - ) = sin

cos( 180° - ) = - cos tan( 180° - ) = - tan

sin( 180° + ) = - sin cos( 180° + ) = - cos

tan( 180° + ) = tan Pythagorean Theorem:

The square of the length of the

hypotenuse of a right triangle is

equal to the sum of the squares of

the legs

a2 = b2 + c2

a,b - two sides of the triangle

connected by the right angle

c - hypotenuse of the triangle

circumference of a circle circumference of a circle = 2 . π . r

where,

π = PI = 22/7

r = radius of circle

Area of Triangle Area of Triangle = (1/2) . b . h

where,

h = height of triangle

b = the length of the base of triangle

Area of rectangle Area of rectangle = l . b

where,

l = length of rectangle

b = width of rectangle

Area of circle Area of circle = π . r 2

where,

π = PI = 22/7

r = radius of circle

Area of trapezoid Area of trapezoid = (1/2) . (height). (base

one + base two)

Area of Ellipse Area of Ellipse = π . r1 . r2

where,

r1 = major radius

r2 = minor radius

Area of Cylinder (surface area) Area of Cylinder (surface area) =

2 . π . r . h

where,

r = radius of cylinder

h = length of cylinder

Area of Cone (surface area) Area of Cone (surface area) =

π . r . l

where,

r = radius of cone

l = length of side of the cone

Volume of cylinder Volume of cylinder = π . r 2 . h

where,

π = PI = 22/7

r = radius of cylinder

h = length of cylinder

Volume of sphere Volume of sphere = (4/3) . π . r 3

where,

π = PI = 22/7

r = radius of sphere

Volume of Cone Volume of Cone = (1/3) . π . r 2 . h

where,

π = PI = 22/7

r = radius of Cone

h = height of cone

sin(-x) = -sin(x)

cosec(-x) = -cosec(x) cos(-x) = cos(x)

sec(-x) = sec(x) tan(-x) = -tan(x)

cot(-x) = -cot(x) sin( + ) = sin cos + cos

sin

sin( - ) = sin cos - cos sin cos( + ) = cos cos - sin sin

cos( - ) = cos cos + sin sin tan( + ) =

(tan + tan) (1 - tan tan)

tan( - ) = (tan - tan) (1 + tan tan)

sin2 = 2 sin cos

cos2 = cos2 - sin2 = 2cos2 - 1 = 1 -

2sin2 tan2 =

(2 tan) (1 - tan2)

sin3 = 3sin - 4sin3 cos3 = 4cos3 - 3cos

For any triangle ABC with side lengths

a,b,c

Law of Cosines c2 = a2 + b2 - 2 a b cos C

b2 = c2 + a2 - 2 c a cos B

a2 = b2 + c2 - 2 b c cos A

Law of Sines sinA/a = sinB/b = sinC/c

cos θ cos β = cos(θ - β) + cos(θ + β) 2

sin θ sin β = cos(θ - β) - cos(θ + β)

2

sin θ cos β = sin(θ + β) + sin(θ - β) 2

sin3 θ = 3sin θ - sin 3θ 4

cos3 θ = 3cos θ + cos 3θ 4

sin3 θ . cos3 θ = 3sin 2θ - sin 6θ 32

sin4 θ = 3 - 4 cos 2θ + cos 4θ 8

cos4 θ = 3 + 4 cos 2θ + cos 4θ 8

sin4 θ . cos4 θ = 3 - 4 cos 4θ + cos 8θ 128

sin5 θ = 10 sin θ - 5 sin 3θ + sin 5θ 16

cos5 θ = 10 cos θ + 5 cos 3θ + cos 5θ 16

sin5 θ . cos5 θ = 10 sin 2θ - 5 sin 6θ + sin 10θ 512

Quadratic Equation For the equation: a x 2 + b x + c = 0 the value of x will be

x = - b ± √ (b2 - 4 a c) 2 a

(a + b)2 = a2 + 2 a b + b2 (a - b)2 = a2 - 2 a b + b2

(a + b) . (a - b) = (a2 - b2) Arithmetic progression:

Arithmetic progression is a

sequence of numbers such that the

difference of any two successive

members of the sequence is a

constant.

For example: Suppose a1, a2, a3, a4,

...... , an-1, an are in sequence of

arithmetic progression

Then the first term of an arithmetic

series is a1 and assume that the

common difference of successive

members is d, then the nth term of

the sequence is:

an = a1 + (n - 1).d

The sum of all the components of

an arithmetic series is:

Sn = a1 + a2 + a3 + ....... + an-1 + an

i.e. Sn = (n).(a1 + an)/2

Geometric progression: geometric

progression OR geometric series is a

sequence of numbers such that the quotient

of any two successive members of the

sequence is a constant. The ratio of two

successive number is called common ratio.

The constant ratio must not be equal to 0.

Example of geometric series :

ar1, ar2, ar3, . ....... ., arn-1, arn

The nth term of the geometric series can be

defined as:

an = a r(n - 1)

r is called common ratio and n must be

greater than 0:

Logarithms: logb 1 = 0

logb b = 1

logb(X . Y) = logbX + logbY

logb(X / Y) = logbX - logbY

logb(Xn) = n . logbX

logmn . lognm = 1

dC = 0

dx

d(Cu) = C

du dx dx

d(u + v) =

du +

dv dx dx dx

d(u . v) = u

dv + v

du dx dx dx

d ( u

) = v (du/dx) - u (dv/dx)

dx v v2

d ( u n ) = n u n-1

du dx dx

d ( x n ) = n x n-1

dx

d ( C u ) = C u ln(C)

du dx dx

d ( e u ) = e u

du dx dx

d( ln(u)) =

1 du dx u dx

d( sin(u)) = cos(u)

du dx dx

d( cos(u)) = - sin(u)

du dx dx

d( tan(u)) = sec2(u)

du dx dx

d( cosec(u)) = - cosec(u) . cot(u)

du dx dx

d( sec(u)) = sec(u) . tan(u)

du dx dx

d( cot(u)) = - cosec2(u)

du dx dx

d(sin-1u) =

1 du

dx

dx

d(cos-1u) =

-1 du

dx

dx

d(tan-1u) =

1 du dx 1 + u2 dx

d(cot-1u) =

- 1 du dx 1 + u2 dx

d(sec-1u) =

1 du

dx |u| dx

d(cosec-1u) =

-1 du

dx |u| dx

Chemistry formulas

Ideal Gas law

PV = nRT

n = number of moles

R = universal gas constant = 8.3145

J/mol K

Combined Gas law

P1V1 =

P2V2

T1 T2

Boyle's law

P1V1 = P2V2

Charles law

V1 =

V2

T1 T2

Gay-Lussac law

P1 =

P2

T1 T2

Diffusion: Rate at which two gases mix

Graham's law of diffusion

The rate of diffusion of a gas is inversely

proportional to the square root of their density

or the molar mass of the gas.

Effusion: Rate at which a gas escapes

thru pin hole

Graham's law of effusion

The rate of effusion of a gas is inversely

proportional to the square root of either

the density or the molar mass of the gas.

Solution: Solution is a homogeneous mixture

of two or more substances.

Solute is a substance that is dissolved in the

solution.

Solvent is the substance that dissolves the

solute. Solvent is present in greater amount.

Concentration is the ratio of solute and

solvent.

Concentration can be measured using

molarity, molality and mole fraction.

Molarity (M) = moles of solute liters of solution

Molality (m) = moles of solute

kg of solution

Unit of Molarity (M) : mol/L : moles per litre

Unit of Molality (M) : mol/kg : moles per kg

Mole fraction: Mole fraction of a

component in solution is the number of

moles of that component divided by the

total number of moles of all

components in the solution. Mole-fraction (Xa)= molesa

Dilution: Siluting a solution means adding

more solvent in solution without the addition of

more solute.

MiVi = MfVf

molesa + molesb .... Mi: Molarity of solution before diluting.

Vi: Volume of solution before diluting.

Mf: Molarity of solution after diluting.

Vf: Volume of solution after diluting.

Mole: Mole is the amount of substance

that contains same number of particles

as there are atoms in Carbon-12. One

mole of substance is Avogadro's

number (i.e. 6.023 x 1023).

One mole of gas has volume of 22.4 liter at

STP.

Relation between moles and grams

1 mole = molecular weight of substance

in grams.

Ionization Enthalpy: It is the energy needed to

remove an electron from an atom or molecule

(i.e from low state to n=∞). It is always

endothermic (i.e. positive).

OR

Ionization energy: energy needed to remove an

electron from an atom

Henderson-Hasselbalch equation:

where

[A-]: Concentration of conjugate base

[HA]: concentration of the acid

OR

pH = pKa + log10 [A-]

[HA]

pH = pKa + log10 [Conjugate Base]

[Acid]

teacherone.com

PHYSICS FORMULAS

Density is mass per unit volume

Density = mass / volume

velocity = displacement /

time

Force = rate of change of momentum Momentum = mass .

velocity

Power is rate of work done

Power = work / time

Unit of power is watt

Potential energy (P) PE = m.g.h

m = mass

g = acceleration due to gravity

(9.81m/s2)

h = height

Kinetic energy (P) P = (1/2).m.v2

m = mass

v = velocity

Gravity (Force due to gravity) Fg : Force of attraction

G : Gravitational constant

M1 : Mass of first object

M2 : Mass of second object

Fg = G M1 M2

r2

Acceleration due to

gravity at a depth 'd' from

earth surface is :

gd = g(1- d

) R

Acceleration due to gravity at height 'h'

from earth surface is :

h is very much smaller than R

gh = g(1- 2h

) R

Escape velocity Escape velocity from a

body of mass M and

radius r is

For example if you want

to calculate the escape

verlocity of sa object from

earth then,

M is dmass of earth

r is radius of earth

OPTICS

Index of refraction n = c/v

n - index of refraction

c - velocity of light in a vacuum

v - velocity of light in the given

Under constant

acceleration linear

motion v = final velocity

u = intitial velocity

a = acceleration

t = time taken to reach

velocity v from u

material

s = displacement

v = u + a t

s = ut + (1/2)a t 2

s = vt - (1/2)a t 2

v2 = u2 + 2 a s

Friction force (kinetic friction) When the object is moving then

Friction is defined as :

Ff = μ Fn

where

Ff = Friction force, μ= cofficient of

friction

Fn = Normal force

Linear Momentum Momentum = mass x

velocity

Capillary action

The height to which the liquid can be

lifted is given by:

h = 2γcosθ

ρgr

γ: liquid-air surface

tension(T)(T=energy/area)

θ: contact angle

ρ: density of liquid

g: acceleration due to gravity

r: is radius of tube

Simple harmonic motion Simple harmonic motion

is defined by:

d2x/dt2 = - k x

Time period of pendulum

Waves

f = 1

T

ω = 2 π

T

v = f . λ

where

ω = Angular frequency,

T=Time period, v = Speed

of wave, λ=wavelength

Doppler effect Relationship between

observed frequency f and emitted Resonance of a string frequency = f = nv

frequency f0:

where,

v=velocity of wave

vs=velocity of source. It is positive if

source of wave is moving away from

observer. It is negative if source of

wave is moving towards observer.

f = f0( v

) v + vs

where,

L: length of the string

n = 1, 2, 3...

2L

Resonance of a open tube of

air(approximate)

where,

L: length of the cylinder

n = 1, 2, 3...

v = speed of sound

Approximate frequency = f = nv

2L

Resonance of a open

tube of air(accurate)

where,

L: length of the cylinder

n: 1, 2, 3...

v: speed of sound

d:diameter of the

resonance tube

frequency = f = nv

2(L+0.8D)

Resonance of a closed tube of

air(approximate)

where,

L: length of the cylinder

n = 1, 2, 3...

v = speed of sound

Approximate frequency = f = nv

4L

Resonance of a closed

tube of air(accurate)

where,

L: length of the cylinder

n: 1, 2, 3...

v: speed of sound

d:diameter of the

resonance tube

frequency = f = nv

4(L+0.8D)

intensity of sound

intensity of sound =

Sound Power

area

Bragg's law nλ = 2d sinθ

where

I=intensity of interest in Wm-2

I0=intensity of interest in 10-12Wm-2

intensity of sound in decibel= 10log10 I

I0

dB = 10log10 I

I0

where

n = integer (based upon

order)

λ = wavelength

d = distance between the

planes

θ = angle between the

surface and the ray

de Broglie equation

where

p = momentum

λ = wavelength

h = Planck's constant

v = velocity

λ = h =

h

p mv

Relation between energy

and frequency E = hν

where

E = Energy

h = Planck's constant

ν = frequency

Davisson and Germer experiment

where

e = charge of electron

m = mass of electron

V = potential difference between the

plates thru which the electron pass

λ = wavelength

λ = h

Centripetal Force (F)

F = m v2

= m ω2 r r

Circular motion formula

v = ω r

Centripetal acceleration (a) = v2

r

Torque (it measures how

the force acting on the

object can rotate the

object) Torque is cross product of

radius and Force

Torque = (Force) X

(Moment arm) X sin θ

T = F L sin θ

whete θ = angle between

force and moment arm

Forces of gravitation F = G (m1.m2)/r

2

where G is constant. G = 6.67E - 11 N

m2 / kg2

Stefan-Boltzmann Law The energy radiated by a

blackbody radiator per

second = P

P = AσT4

where,

σ = Stefan-Boltzmann

constant

σ = 5.6703 × 10-8

watt/m2K4

Efficiency of Carnot cycle

η = 1 - Tc

Th

Ideal gas law P V = n R T

P = Pressure (Pa i.e.

Pascal)

V = Volume (m3)

n = number of of gas (in

moles)

R = gas constant (

8.314472 .m3.Pa.K-1mol-1]

)

T = Temperatue ( in

Kelvin [K])

Boyles law (for ideal gas) P1 V1 = P2V2

T (temperature is constant)

Charles law (for ideal

gas) V1

= V2

T1 T2

P (pressure is constant)

Translational kinetic energy K per

gas molecule (average molecular

kinetic energy:)

K = 3 k T

2

Internal energy of

monoatomic gas

K = 3 n R T

2

k = 1.38066 x 10-23 J/K Boltzmanns

constant

n = number of of gas (in

moles)

R = gas constant (

8.314472 .m3.Pa.K-1mol-1]

)

Root mean square speed of gas

V2rms =

3 k T

m

k = 1.38066 x 10-23 J/K Boltzmanns

constant

m = mass of gas

Ratio of specific heat (γ)

γ = Cp

Cv

Cp = specific heat capacity

of the gas in a constant

pressure process

Cv = specific heat capacity

of the gas in a constant

volume process

Internal entergy of ideal gas

Internal entergy of ideal gas (U) = cv

nRT

In Adiabatic process no

heat is gained or lost by

the system.

Under adiabetic condition

PVγ = Constant

TVγ-1 = Constant

where γ is ratio of specific

heat.

γ = Cp

Cv

Boltzmann constant (k)

k = R

Na

R = gas constant

Na = Avogadro's number.

Speed of the sound in gas

R = gas constant(8.314

J/mol K)

T = the absolute

temperature

M = the molecular weight

of the gas (kg/mol)

γ = adiabatic constant =

cp/cv

Capillary action The height to which the liquid can be

lifted is given by

h=height of the liquid lifted

T=surface tension

r=radius of capillary tube

h= 2T

ρrg

Resistance of a wire

R = ρL

A

ρ = rsistivity

L = length of the wire

A = cross-sectional area of

the wire

Ohm's law V = I . R

V = voltage applied

R = Resistance

I = current

Electric power (P) = (voltage applied)

x (current) P = V . I = I2 . R

V = voltage applied

R = Resistance

I = current

Resistor combination

If resistors are in series

then equivalent

resistance will be Req = R1 + R2 + R3 + . . . .

. . + Rn

If resistors are in

parallel then equivalent

resistance will be 1/Req = 1/R1 + 1/R2 + 1/R3

+ . . . . . . + 1/Rn

In AC circuit average power is :

Pavg = VrmsIrms cosφ

where,

Pavg = Average Power

Vrms = rms value of voltage

Irms = rms value of current

In AC circuit

Instantaneous power is :

PInstantaneous = VmIm sinωt

sin(ωt-φ)

where,

PInstantaneous =

Instantaneous Power

Vm = Instantaneous

voltage

Im = Instantaneous current

Capacitors Q = C.V

where

Q = charge on the capacitor

C = capacitance of the capacitor

V = voltage applied to the capacitor

Total capacitance (Ceq)

for PARALLEL

Capacitor

Combinations: Ceq = C1 + C2 + C3 + . . . .

. . + Cn

Total capacitance (Ceq)

for SERIES Capacitor

Combinations: 1/Ceq = 1/C1 + 1/C2 + 1/C3

+ . . . . . . + 1/Cn

Parallel Plate Capacitor

C = κ ε0 A

d

where

C = [Farad (F)]

κ = dielectric constant

A = Area of plate

d = distance between the plate

ε0 = permittivity of free space (8.85 X

Cylindrical Capacitor

C = 2 π κ ε0 L

ln (b/a)

where

C = [Farad (F)]

κ = dielectric constant

L = length of cylinder [m]

a = outer radius of

conductor [m]

10-12 C2/N m2) b = inner radius of

conductor [m]

ε0 = permittivity of free

space (8.85 X 10-12 C2/N

m2)

Spherical Capacitor

C = 4 π κ ε0 a b

b - a

where

C = [Farad (F)]

κ = dielectric constant

a = outer radius of conductor [m]

b = inner radius of conductor [m]

ε0 = permittivity of free space (8.85 X

10-12 C2/N m2)

Magnetic force acting on

a charge q moving with

velocity v F = q v B sin θ

where

F = force acting on charge

q (Newton)

q = charge (C)

v = velocity (m/sec2)

B = magnetic field

θ = angle between V

(velocity) and B (magnetic

field)

Force on a wire in magnetic field (B)

F = B I l sin θ where

F = force acting on wire

(Newton)

I = Current (Ampere)

l = length of wire (m)

B = magnetic field

θ = angle between I (current) and B

(magnetic field)

In an RC circuit (Resistor-

Capacitor), the time

constant (in seconds) is:

τ = RC

R = Resistance in Ω

C = Capacitance in in

farads.

In an RL circuit (Resistor-inductor ),

the time constant (in seconds) is:

τ = L/R

R = Resistance in Ω

C = Inductance in henries

Self inductance of a

solenoid = L = μn2LA

n = number of turns per

unit length

L = length of the solenoid.

Mutual inductance of two solenoid two

long thin solenoids, one wound on top

of the other

M = μ0N1N2LA

N1 = total number of turns per unit

length for first solenoid

N2 = number of turns per unit length for

second solenoid

A = cross-sectional area

L = length of the solenoid.

Energy stored in capacitor

E = 1 C V 2

2

Coulomb's Law Like charges repel, unlike charges

attract.

F = k (q1 . q2)/r2

where k is constant. k = 1/(4 π ε0) ≈ 9 x

109 N.m2/C2

q1 = charge on one body

q2 = charge on the other body

r = distance between them

Calculator based upon Coulomb's Law

Ohm's law

V = IR

where

V = voltage

I = current

R = Resistence

Electric Field around a point charge

(q) E = k ( q/r2 )

where k is constant. k = 1/(4 π ε0) ≈ 9 x

109 N.m2/C2

q = point charge

r = distance from point charge (q)

Electric field due to thin

infinite sheet

E = σ

2 ε0

where

E = Electric field (N/C)

σ = charge per unit area

C/m2

ε0 = 8.85 X 10-12 C2/N m2

Electric field due to thick infinite

sheet

E = σ

ε0

where

E = Electric field (N/C)

σ = charge per unit area C/m2

ε0 = 8.85 X 10-12 C2/N m2

Magnetic Field around a

wire (B) when r is

greater than the radius

of the wire.

B = μ0 I

2 π r

where

I = current

r = distance from wire

and r ≥ Radius of the wire

Magnetic Field around a wire (B)

when r is less than the radius of the

wire.

B = μ0 I r

2 π R2

where

I = current

R = radius of wire

r = distance from wire

and r ≤ Radius of the wire (R)

Magnetic Field At the

center of an arc

B = μ0 I φ

4 π r

where

I = current

r = radius from the center

of the wire

Bohr's model

L = nh

2 π

where

L = angular momentum

n = principal quantum number =

1,2,3,...n

h = Planck's constant.

Emitting

Photons(Rydberg

Formula)

where

n1 < n2

E0 = 13.6 eV

Ephoton = E0( 1

- 1

) n1

2 n22

Half life of radioactive element

t1/2 = ln(2)

λ

Average life of

radioactive element

τ = 1

λ