Mathematical Formulas Cheatsheet

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Algebra Quadratic Equation For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Discriminant: $\Delta = b^2 - 4ac$ Factoring & Expanding $(a+b)^2 = a^2 + 2ab + b^2$ $(a-b)^2 = a^2 - 2ab + b^2$ $(a+b)(a-b) = a^2 - b^2$ $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ $a^3+b^3 = (a+b)(a^2-ab+b^2)$ $a^3-b^3 = (a-b)(a^2+ab+b^2)$ Exponents $a^m \cdot a^n = a^{m+n}$ $\frac{a^m}{a^n} = a^{m-n}$ $(a^m)^n = a^{mn}$ $(ab)^n = a^n b^n$ $(\frac{a}{b})^n = \frac{a^n}{b^n}$ $a^0 = 1$ (for $a \neq 0$) $a^{-n} = \frac{1}{a^n}$ $a^{m/n} = \sqrt[n]{a^m}$ Logarithms $y = \log_b x \iff b^y = x$ $\log_b (MN) = \log_b M + \log_b N$ $\log_b (\frac{M}{N}) = \log_b M - \log_b N$ $\log_b (M^p) = p \log_b M$ $\log_b b = 1$, $\log_b 1 = 0$ Change of Base: $\log_b x = \frac{\log_c x}{\log_c b}$ Natural Logarithm: $\ln x = \log_e x$ Geometry & Mensuration 2D Shapes Rectangle: Area $A = lw$, Perimeter $P = 2(l+w)$ Square: Area $A = s^2$, Perimeter $P = 4s$ Triangle: Area $A = \frac{1}{2}bh$ Circle: Area $A = \pi r^2$, Circumference $C = 2\pi r = \pi d$ Trapezoid: Area $A = \frac{1}{2}(b_1+b_2)h$ Parallelogram: Area $A = bh$ 3D Shapes Cube: Volume $V = s^3$, Surface Area $SA = 6s^2$ Rectangular Prism: Volume $V = lwh$, Surface Area $SA = 2(lw+lh+wh)$ Cylinder: Volume $V = \pi r^2 h$, Surface Area $SA = 2\pi r^2 + 2\pi rh$ Cone: Volume $V = \frac{1}{3}\pi r^2 h$, Surface Area $SA = \pi r^2 + \pi r l$ ($l$ is slant height) Sphere: Volume $V = \frac{4}{3}\pi r^3$, Surface Area $SA = 4\pi r^2$ Pythagorean Theorem $a^2 + b^2 = c^2$ (for a right triangle with legs $a, b$ and hypotenuse $c$) Trigonometry Basic Ratios (SOH CAH TOA) $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin \theta}{\cos \theta}$ $\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta}$ Identities $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ $\sin(2\theta) = 2 \sin \theta \cos \theta$ $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$ Law of Sines & Cosines Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$ Calculus Derivatives Constant Rule: $\frac{d}{dx}(c) = 0$ Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c f'(x)$ Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$ Product Rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$ Quotient Rule: $\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$ $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(a^x) = a^x \ln a$ $\frac{d}{dx}(\ln x) = \frac{1}{x}$ $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ Integrals (Antiderivatives) $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$) $\int \frac{1}{x} dx = \ln|x| + C$ $\int e^x dx = e^x + C$ $\int a^x dx = \frac{a^x}{\ln a} + C$ $\int \sin x dx = -\cos x + C$ $\int \cos x dx = \sin x + C$ $\int \sec^2 x dx = \tan x + C$ Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x)=f(x)$ Statistics & Probability Mean, Median, Mode Mean (Arithmetic): $\bar{x} = \frac{\sum x_i}{n}$ Median: Middle value when data is ordered Mode: Most frequent value Variance & Standard Deviation Variance: $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ (population), $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ (sample) Standard Deviation: $\sigma = \sqrt{\sigma^2}$ (population), $s = \sqrt{s^2}$ (sample) Probability $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ For independent events: $P(A \text{ and } B) = P(A) \cdot P(B)$ Conditional Probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ Factorial: $n! = n \times (n-1) \times \dots \times 1$ Permutations: $P(n, k) = \frac{n!}{(n-k)!}$ Combinations: $C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$