Maths formulae ESAT prep

Sphere Volume Formula

4/3πr³

Sphere Surface Area Formula

4πr²

Sin 2A =

2sinAcosA

Cos 2A =

2 cos² A - 1

= 1 - 2 sin² A

= cos² A - sin² A

Tan 2A =

(2 tan A) / (1 - tan² A)

Area of a triangle (not 90°)

1/2 ab sinC

Cosine rule

a² = b² + c² - 2bcCosA

Trig identity with sec

1 + tan²(x) = sec²(x)

Trig Identity with cot

1 + cot²(x) = cosec² (x)

Arc length (radians)

rθ

Sector area (radians)

1/2 r²θ

Sin(A±B) =

sinAcosB ± cosAsinB

Cos(A±B) =

cosAcosB ∓ sinAsinB

Tan(A±B)=

(tanA ± tan B) / 1 ∓ tanAtanB

R addition formula

Effective resistance in series

Effective resistance in parallel

Power =

P = E/t (J/s)

resistance based on dimensions

ρ = resistivity

A = cross sectional area

L = length

Voltage =

Charge (coloumbs) =

Work done =

force x distance,

wave speed formula

𝑣 = 𝑓λ

E =

E = VIt

Diffraction grating formula

𝑛λ = 𝑑sinθ

n = order of maxima

d = slit separation

θ = angle between light and horizontal

fringe spacing formula

w = λD/s

D = distance from slits to screen

s = separation of slits

Frictional force for object in equilibrium

R = normal contact force

Frictional force at point of sliding

Centripetal force

Moment =

Moment = Force × Perpendicular Distance

Momentum=

Momentum = mass × velocity

p = mv

Impulse =

change in momentum

FΔt = Δp = Δ(mv)

Stress =

Strain =

Young's modulus =

(F x L) /(A x ΔL)

Hooke's Law

F = kΔL

k =

Elastic potential energy

E = ½ FΔL = ½ k ΔL²

Radioactivity amount

The final activity, A

after n half-lives,

of a sample with initial activity A₀

Number of undecayed nuclei

Amount of undecayed nuclei left, N

after n half-lives,

of a sample with initial number of undecayed nuclei N₀

Term in arithmetic sequence

Sum of arithmetic sequence

term in geometric sequence

sum of geometric sequence

converging sequence sum

exterior angle

360/n

Factorisation of (a³ - b³) =

(a - b)(a² + ab + b²)

Factorisation of (a³ + b³) =

(a + b)(a² - ab + b²)

fence post formula

b - a + 1

Finding a value in Pascal's triangle

n = highest power

k = power to find constant for

shortest distance between a point and a line

trapezium rule formula