Mathematics

Overview

Engineering reference data for Mathematics in mathematics.

Key Formulas

Quadratic Formula

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

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Trigonometric Laws

Law of Sines

For any triangle with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$:

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

Law of Cosines

Relates one side of a triangle to the other two sides and the included angle:

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

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

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

Law of Tangents

$\frac{a - b}{a + b} = \frac{\tan\left(\frac{A - B}{2}\right)}{\tan\left(\frac{A + B}{2}\right)}$

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Geometric Shapes

Areas of Common Figures

Circle Properties

• Circumference: $C = 2\pi r = \pi d$
• Diameter: $d = 2r$
• Chord length for $n$ equal segments: $L = 2r\sin\left(\frac{\pi}{n}\right)$

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Number Systems

Binary (Base-2)

Uses digits: 0, 1

Binary to Decimal conversion: $d_n = \sum_{i=0}^{n} b_i \times 2^i$

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Logarithms

Common Logarithm (Base 10)

$\log_{10}(x) \quad \text{where } x > 0$

Natural Logarithm (Base $e$)

$\ln(x) = \log_e(x) \quad \text{where } e \approx 2.71828$

Logarithm Rules

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Complex Numbers

A complex number is expressed as:

$z = a + bi$

where $a$ is the real part, $b$ is the imaginary part, and $i = \sqrt{-1}$.

Modulus: $|z| = \sqrt{a^2 + b^2}$

Conjugate: $\bar{z} = a - bi$

Euler's Formula: $e^{i\theta} = \cos\theta + i\sin\theta$

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Series and Sequences

Arithmetic Series

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1 + (n-1)d)$

Geometric Series

$S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1)$

Summation of Integers

$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

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Hyperbolic Functions

Identity: $\cosh^2 x - \sinh^2 x = 1$

References

• Original Source