Comprehensive Mathematics Formula Reference

A complete guide to essential formulas in algebra, calculus, geometry, statistics, and trigonometry.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating them. It forms the basis of advanced math and problem solving.

Algebraic Expressions Formula

An algebraic expression combines variables, constants, and operations.

\( ax^2 + bx + c \)

Where \( a \), \( b \), and \( c \) are constants, and \( x \) is a variable.

Example: \( 3x^2 - 5x + 7 \)

Equation of a Line Formula

The slope-intercept form of a line is:

\( y = mx + b \)
  • \( m \) is the slope
  • \( b \) is the y-intercept

Example: \( y = 2x + 3 \)

Quadratic Formula

The solutions of \( ax^2 + bx + c = 0 \) are:

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

Example: For \( 2x^2 - 5x + 3 = 0 \), the roots are \( x = 1 \) and \( x = \frac{3}{2} \).

Exponents Rules

\( x^a \cdot x^b = x^{a+b} \)

\( \frac{x^a}{x^b} = x^{a-b} \)

\( (x^a)^b = x^{ab} \)

\( x^0 = 1 \)

\( x^{-n} = \frac{1}{x^n} \)

Binary to Decimal Formula
\( N_{10} = \sum_{i=0}^{n-1} b_i \times 2^i \)

Example: \( 1011_2 = 11_{10} \)

Factoring Formulas

\( a^2 - b^2 = (a+b)(a-b) \)

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

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

\( a^3 + b^3 = (a+b)(a^2-ab+b^2) \)

\( a^3 - b^3 = (a-b)(a^2+ab+b^2) \)

Polynomial Formula
\( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

Calculus

Calculus studies change and accumulation. Its two main branches are differential calculus and integral calculus.

Derivative Formula
\( f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \)
Chain Rule
\( [f(g(x))]' = f'(g(x)) \cdot g'(x) \)

Example: \( \frac{d}{dx}[\sin(x^2)] = 2x\cos(x^2) \)

Integral Formulas

\( \int f(x)\,dx = F(x) + C \)

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

\( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

\( \int \frac{1}{x} dx = \ln|x| + C \)

Integration by Parts
\( \int u\,dv = uv - \int v\,du \)
Limit Formula
\( \lim_{x \to a} f(x) = L \)
Arc Length Formula
\( L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx \)
Area Under the Curve
\( A = \int_a^b f(x)\,dx \)
Average Rate of Change
\( \frac{f(b)-f(a)}{b-a} \)
Implicit Differentiation
\( \frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}} \)
Substitution Rule
\( \int f(g(x))g'(x)\,dx = \int f(u)\,du \)
Taylor Series
\( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n \)

Geometry

Geometry focuses on shape, size, dimensions, and the properties of space.

Area of a Circle
\( A = \pi r^2 \)
Pythagorean Theorem
\( a^2 + b^2 = c^2 \)
Volume Formulas

Cube: \( V = s^3 \)

Rectangular Prism: \( V = lwh \)

Sphere: \( V = \frac{4}{3}\pi r^3 \)

Cylinder: \( V = \pi r^2 h \)

Cone: \( V = \frac{1}{3}\pi r^2 h \)

Perimeter Formulas

Square: \( P = 4s \)

Rectangle: \( P = 2(l+w) \)

Circle: \( C = 2\pi r \)

Triangle: \( P = a+b+c \)

Triangle Formula

\( A = \frac{1}{2}bh \)

\( A = \sqrt{s(s-a)(s-b)(s-c)} \)

Surface Area Formulas

Cube: \( SA = 6s^2 \)

Rectangular Prism: \( SA = 2(lw+lh+wh) \)

Sphere: \( SA = 4\pi r^2 \)

Cylinder: \( SA = 2\pi r^2+2\pi rh \)

Statistics

Statistics focuses on collecting, analyzing, interpreting, and presenting data.

Mean, Median, Mode

\( \bar{x} = \frac{\sum x_i}{n} \)

Median = middle value of ordered data

Mode = most frequent value

Standard Deviation

\( \sigma = \sqrt{\frac{\sum (x_i-\mu)^2}{N}} \)

\( s = \sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}} \)

Probability Formulas

\( P(A)=\frac{\text{favorable}}{\text{total}} \)

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

\( P(A\cup B)=P(A)+P(B)-P(A\cap B) \)

\( P(A\cap B)=P(A)\times P(B|A) \)

Normal Distribution
\( f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \)
Coefficient of Determination
\( R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \)
Coefficient of Variation
\( CV = \frac{\sigma}{\mu}\times 100\% \)

Trigonometry

Trigonometry studies the relationships between angles and sides of triangles and their periodic behavior.

Basic Trigonometric Ratios

\( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

\( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

\( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \)

Reciprocal Identities

\( \csc\theta = \frac{1}{\sin\theta} \)

\( \sec\theta = \frac{1}{\cos\theta} \)

\( \cot\theta = \frac{1}{\tan\theta} \)

Pythagorean Identities

\( \sin^2\theta + \cos^2\theta = 1 \)

\( 1 + \tan^2\theta = \sec^2\theta \)

\( 1 + \cot^2\theta = \csc^2\theta \)

Angle Sum and Difference

\( \sin(A \pm B)=\sin A\cos B \pm \cos A\sin B \)

\( \cos(A \pm B)=\cos A\cos B \mp \sin A\sin B \)

\( \tan(A \pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A\tan B} \)

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