CheatSheetsHero

plevel-danek / Equations

November 04, 2025 21:31
equations
formulas
mathematics
physics
science
Download PDF
Missing something?

Equations

A concise reference for fundamental equations and formulas across various mathematical and scientific domains. From basic algebra to advanced calculus and physics, this cheat sheet provides a quick guide to key concepts and their applications.

Algebraic Formulas

Basic Identities

Square of a Sum:

(a + b)2 = a2 + 2ab + b^2

Square of a Difference:

(a - b)2 = a2 - 2ab + b^2

Difference of Squares:

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

Cube of a Sum:

(a + b)3 = a3 + 3a2b + 3ab2 + b^3

Cube of a Difference:

(a - b)3 = a3 - 3a2b + 3ab2 - b^3

Sum of Cubes:

a3 + b3 = (a + b)(a2 - ab + b2)

Difference of Cubes:

a3 - b3 = (a - b)(a2 + ab + b2)

Quadratic Formula

For a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant:

\Delta = b^2 - 4ac
  • If \Delta > 0, there are two distinct real roots.
  • If \Delta = 0, there is exactly one real root.
  • If \Delta < 0, there are no real roots (two complex conjugate roots).

Logarithms and Exponents

Product Rule:

Quotient Rule:

Power Rule:

Change of Base:

Exponential Growth:

Exponential Decay:

Geometric Formulas

Area Formulas

Square:

Rectangle:

Triangle:

Circle:

Trapezoid:

Parallelogram:

Volume Formulas

Cube:

Rectangular Prism:

Sphere:

Cylinder:

Cone:

Pyramid:

Surface Area Formulas

Cube:

Rectangular Prism:

Sphere:

Cylinder:

Cone:

Trigonometric Formulas

Basic Trigonometric Functions

Sine:

Cosine:

Tangent:

Cosecant:

Secant:

Cotangent:

Pythagorean Identities

Angle Sum and Difference Formulas

Sine Sum:

Sine Difference:

Cosine Sum:

Cosine Difference:

Tangent Sum:

Tangent Difference:

Calculus Formulas

Basic Derivatives

Power Rule:

Constant Multiple Rule:

Sum Rule:

Product Rule:

Quotient Rule:

Chain Rule:

Basic Integrals

Power Rule:

Integral of 1/x:

Integral of e^x:

Integral of sin(x):

Integral of cos(x):

Fundamental Theorem of Calculus

Part 1: If f is continuous on [a, b], then the function F defined by

F(x) = \int_a^x f(t) dt \quad a \leq x \leq b

is an antiderivative of f, that is, F'(x) = f(x).

Part 2: If f is continuous on [a, b], then

\int_a^b f(x) dx = F(b) - F(a)

where F is any antiderivative of f, that is, F' = f.

Equations Cheat Sheet

Algebraic Equations

Linear Equation:
ax + b = 0
Solve for x:
x = -\frac{b}{a}

Quadratic Equation:
ax^2 + bx + c = 0
Quadratic Formula:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

System of Linear Equations (2 variables):
a_1x + b_1y = c_1
a_2x + b_2y = c_2
Solve using substitution, elimination, or matrices.

Factoring:
a^2 - b^2 = (a + b)(a - b)
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2

Polynomial Equations:
Use factoring, synthetic division, or numerical methods to find roots.

Exponential Equations:
a^x = b
Solve using logarithms:
x = \frac{\log(b)}{\log(a)}

Logarithmic Equations:
\log_a(x) = b
Convert to exponential form:
x = a^b

Absolute Value Equations:
|x| = a
Then:
x = a \text{ or } x = -a

Rational Equations:
Clear denominators by multiplying all terms by the least common denominator.

Calculus Equations

Derivative Rules:
Power Rule: \frac{d}{dx}x^n = nx^{n-1}
Constant Rule: \frac{d}{dx}c = 0
Product Rule: \frac{d}{dx}(uv) = u'v + uv'
Quotient Rule: \frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2}
Chain Rule: \frac{d}{dx}f(g(x)) = f'(g(x))g'(x)

Integral Rules:
Power Rule: \int x^n dx = \frac{x^{n+1}}{n+1} + C
Exponential: \int e^x dx = e^x + C
Trigonometric: \int \sin(x) dx = -\cos(x) + C
\int \cos(x) dx = \sin(x) + C

Fundamental Theorem of Calculus:
\int_a^b f(x) dx = F(b) - F(a)
where F'(x) = f(x)

Limits:
\lim_{x \to a} f(x)
Evaluate by direct substitution, factoring, rationalizing, or L’Hôpital’s Rule.

L’Hôpital’s Rule:
If \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\infty}{\infty}
then
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Taylor Series:
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Integration by Parts:
\int u dv = uv - \int v du

Partial Derivatives:
\frac{\partial f}{\partial x}
Derivative of f with respect to x while holding other variables constant.

Gradient:
\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)

Trigonometric Equations

Basic Identities:
\sin^2(\theta) + \cos^2(\theta) = 1
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}

Pythagorean Identities:
1 + \tan^2(\theta) = \sec^2(\theta)
1 + \cot^2(\theta) = \csc^2(\theta)

Double Angle Formulas:
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}

Sum and Difference Formulas:
\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)
\cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b)
\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)}

Solving Trigonometric Equations:
Isolate the trigonometric function, then use inverse trigonometric functions to find solutions.
Consider the period and general solutions.

Law of Sines:
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Law of Cosines:
a^2 = b^2 + c^2 - 2bc \cos(A)
b^2 = a^2 + c^2 - 2ac \cos(B)
c^2 = a^2 + b^2 - 2ab \cos(C)

Half Angle Formulas:
\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}
\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)}

Product-to-Sum Formulas:
\sin(a)\cos(b) = \frac{1}{2}[\sin(a+b) + \sin(a-b)]
\cos(a)\sin(b) = \frac{1}{2}[\sin(a+b) - \sin(a-b)]
\cos(a)\cos(b) = \frac{1}{2}[\cos(a+b) + \cos(a-b)]
\sin(a)\sin(b) = \frac{1}{2}[\cos(a-b) - \cos(a+b)]

Physics Equations

Kinematics:
v = v_0 + at
x = x_0 + v_0t + \frac{1}{2}at^2
v^2 = v_0^2 + 2a(x - x_0)

Newton’s Second Law:
F = ma

Work and Energy:
W = Fd\cos(\theta)
KE = \frac{1}{2}mv^2
PE = mgh

Momentum:
p = mv

Impulse:
J = \Delta p = F\Delta t

Gravitational Force:
F = G\frac{m_1m_2}{r^2}

Ohm’s Law:
V = IR

Power (Electrical):
P = IV = I^2R = \frac{V^2}{R}

Wave Equation:
v = f\lambda

Ideal Gas Law:
PV = nRT

Geometric Equations

Area of a Circle:
A = \pi r^2

Circumference of a Circle:
C = 2\pi r

Area of a Triangle:
A = \frac{1}{2}bh

Pythagorean Theorem:
a^2 + b^2 = c^2

Volume of a Sphere:
V = \frac{4}{3}\pi r^3

Surface Area of a Sphere:
A = 4\pi r^2

Volume of a Cylinder:
V = \pi r^2 h

Surface Area of a Cylinder:
A = 2\pi r h + 2\pi r^2

Area of a Rectangle:
A = lw

Perimeter of a Rectangle:
P = 2l + 2w

Financial Equations

Simple Interest:
I = PRT
Where:

  • I = Interest
  • P = Principal
  • R = Rate
  • T = Time

Compound Interest:
A = P(1 + \frac{r}{n})^{nt}
Where:

  • A = Amount
  • P = Principal
  • r = interest rate
  • n = number of times interest applied per time period
  • t = number of time periods elapsed

Present Value:
PV = \frac{FV}{(1 + r)^n}
Where:

  • PV = Present Value
  • FV = Future Value
  • r = Discount rate
  • n = Number of periods

Future Value:
FV = PV(1 + r)^n
Where:

  • FV = Future Value
  • PV = Present Value
  • r = Interest rate
  • n = Number of periods

Annuity (Future Value):
FV = P \times \frac{((1 + r)^n - 1)}{r}
Where:

  • FV = Future Value
  • P = Periodic Payment
  • r = Interest Rate
  • n = Number of periods

Annuity (Present Value):
PV = P \times \frac{(1 - (1 + r)^{-n})}{r}
Where:

  • PV = Present Value
  • P = Periodic Payment
  • r = Interest Rate
  • n = Number of periods

Mortgage Payment:
M = P \frac{r(1+r)^n}{(1+r)^n - 1}
Where:

  • M = Monthly Payment
  • P = Principal Loan Amount
  • r = Monthly Interest Rate
  • n = Number of Months

Return on Investment (ROI):
ROI = \frac{Net Profit}{Cost of Investment} \times 100

Statistics Equations

Mean:
\mu = \frac{\sum_{i=1}^{n} x_i}{n}

Variance:
\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}

Standard Deviation:
\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}}

Z-Score:
z = \frac{x - \mu}{\sigma}

Correlation Coefficient:
r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}}

Regression Equation:
y = a + bx
where
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
and
a = \bar{y} - b\bar{x}

Probability:
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

Conditional Probability:
P(A|B) = \frac{P(A \cap B)}{P(B)}

Bayes’ Theorem:
P(A|B) = \frac{P(B|A)P(A)}{P(B)}