Mathematics Advanced Formula Sheet

Measurement

Length

\displaystyle l = \frac{\theta}{360} \times 2\pi r

Area

\displaystyle A = \frac{\theta}{360} \times \pi r^{2}

\displaystyle A = \frac{h}{2}\,(a + b)

Surface area

\displaystyle A = 2\pi r^{2} + 2\pi r h

\displaystyle A = 4\pi r^{2}

Volume

\displaystyle V = \frac{1}{3} A h

\displaystyle V = \frac{4}{3} \pi r^{3}

Functions

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

\displaystyle \begin{aligned} \text{For } ax^{3} + bx^{2} + cx + d & = 0: \\ \alpha + \beta + \gamma & = -\frac{b}{a} \\ \alpha\beta + \alpha\gamma + \beta\gamma & = \frac{c}{a} \\ \text{and} \hspace{0.5em} \alpha\beta\gamma & = -\frac{d}{a} \end{aligned}

Relations

\displaystyle (x - h)^{2} + (y - k)^{2} = r^{2}

Trigonometric Functions

\displaystyle \sin A = \frac{\mathrm{opp}}{\mathrm{hyp}},\; \cos A = \frac{\mathrm{adj}}{\mathrm{hyp}},\; \tan A = \frac{\mathrm{opp}}{\mathrm{adj}}

\displaystyle A = \frac{1}{2} ab \sin C

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

\displaystyle c^{2} = a^{2} + b^{2} - 2ab \cos C

\displaystyle \cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab}

\displaystyle l = r \theta

\displaystyle A = \frac{1}{2}r^2\theta

Trigonometric identities

\displaystyle \sec A = \frac{1}{\cos A},\; \cos A \ne 0

\displaystyle \cosec A = \frac{1}{\sin A},\; \sin A \ne 0

\displaystyle \cot A = \frac{\cos A}{\sin A},\; \sin A \ne 0

\displaystyle \cos^{2} x + \sin^{2} x = 1

Compound angles

\displaystyle \sin(A + B) = \sin A \cos B + \cos A \sin B

\displaystyle \cos(A + B) = \cos A \cos B - \sin A \sin B

\displaystyle \tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}

\displaystyle \text{If } t = \tan\frac{A}{2} \text{ then: } \\ \begin{aligned} \hphantom{\hspace{2em}}\sin A & = \frac{2t}{1 + t^{2}} \\ \hphantom{\hspace{2em}}\cos A & = \frac{1 - t^{2}}{1 + t^{2}} \\ \hphantom{\hspace{2em}}\tan A & = \frac{2t}{1 - t^{2}} \end{aligned}

\displaystyle \cos A \cos B = \frac{1}{2}\bigl[\cos(A - B) + \cos(A + B)\bigr]

\displaystyle \sin A \sin B = \frac{1}{2}\bigl[\cos(A - B) - \cos(A + B)\bigr]

\displaystyle \sin A \cos B = \frac{1}{2}\bigl[\sin(A + B) + \sin(A - B)\bigr]

\displaystyle \cos A \sin B = \frac{1}{2}\bigl[\sin(A + B) - \sin(A - B)\bigr]

\displaystyle \sin^{2}nx = \frac{1}{2}\bigl(1 - \cos2nx\bigr)

\displaystyle \cos^{2}nx = \frac{1}{2}\bigl(1 + \cos2nx\bigr)

Differential Calculus

Function	Derivative
$\displaystyle y = f(x)^{n}$	$\displaystyle \frac{dy}{dx} = n f'(x)[f(x)]^{n-1}$
$\displaystyle y = uv$	$\displaystyle \frac{dy}{dx} = u\,\frac{dv}{dx} + v\,\frac{du}{dx}$
$\displaystyle y = g(u)\,\text{ where }\, u = f(x)$	$\displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
$\displaystyle y = \dfrac{u}{v}$	$\displaystyle \frac{dy}{dx} = \dfrac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^{2}}$
$\displaystyle y = \sin f(x)$	$\displaystyle \frac{dy}{dx} = f'(x)\cos f(x)$
$\displaystyle y = \cos f(x)$	$\displaystyle \frac{dy}{dx} = -\,f'(x)\sin f(x)$
$\displaystyle y = \tan f(x)$	$\displaystyle \frac{dy}{dx} = f'(x)\sec^{2} f(x)$
$\displaystyle y = e^{f(x)}$	$\displaystyle \frac{dy}{dx} = f'(x)e^{f(x)}$
$\displaystyle y = \ln f(x)$	$\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{f(x)}$
$\displaystyle y = a^{f(x)}$	$\displaystyle \frac{dy}{dx} = (\ln a)\,f'(x)\,a^{f(x)}$
$\displaystyle y = \log_{a} f(x)$	$\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{(\ln a)\,f(x)}$
$\displaystyle y = \sin^{-1} f(x)$	$\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{\sqrt{1 - [f(x)]^{2}}}$
$\displaystyle y = \cos^{-1} f(x)$	$\displaystyle \frac{dy}{dx} = -\,\dfrac{f'(x)}{\sqrt{1 - [f(x)]^{2}}}$
$\displaystyle y = \tan^{-1} f(x)$	$\displaystyle \frac{dy}{dx} = \dfrac{f'(x)}{1 + [f(x)]^{2}}$

Financial Mathematics

\displaystyle A = P(1 + r)^{n}

Sequences and series

\displaystyle T_{n} = a + (n - 1)d

\displaystyle S_{n} = \frac{n}{2}\left[2a + (n - 1)d\right] = \frac{n}{2}(a + l)

\displaystyle T_{n} = a\,r^{n-1}

\displaystyle S_{n} = \frac{a(1 - r^{n})}{1 - r} = \frac{a(r^{n} - 1)}{r - 1},\ \ r \ne 1

\displaystyle S = \frac{a}{1 - r},\ \ |r| < 1

Logarithmic and Exponential Functions

\displaystyle \log_{a} a^{x} = x = a^{\log_{a} x}

\displaystyle \log_{a} x = \frac{\log_{b} x}{\log_{b} a}

\displaystyle a^{x} = e^{x\ln a}

Statistical Analysis

\displaystyle z = \frac{x - \mu}{\sigma}

\displaystyle \begin{aligned} \text{An outlier is a score} \\ \text{less than } & Q_{1} - 1.5 \times IQR \\ \text{or more than } & Q_{3} + 1.5 \times IQR \end{aligned}

Normal distribution

Approximately 68% of scores have z-scores between -1 and 1

Approximately 95% of scores have z-scores between -2 and 2

Approximately 99.7% of scores have z-scores between -3 and 3

\displaystyle E(X) = \mu

\displaystyle \operatorname{Var}(X) = E\big[(X - \mu)^2\big] = E(X^{2}) - \mu^{2}

Probability

\displaystyle P(A \cap B) = P(A)P(B)

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

\displaystyle P(A\mid B) = \frac{P(A \cap B)}{P(B)},\; P(B) \ne 0

Continuous random variables

\displaystyle P(X \le r) = \int_{a}^{r} f(x)\,dx

\displaystyle P(a < X < b) = \int_{a}^{b} f(x)\,dx

Binomial distribution

\displaystyle P(X = r) = {}^{n}C_{r}\, p^{r}(1 - p)^{n-r}

\displaystyle X \sim \operatorname{Bin}(n, p)

\displaystyle \Rightarrow\; P(X = x) = \binom{n}{x} p^{x}(1 - p)^{n-x},\; x = 0, 1, \ldots, n

\displaystyle E(X) = np

\displaystyle \operatorname{Var}(X) = np(1 - p)

Integral Calculus

\displaystyle \int f'(x)[f(x)]^{n}\,dx = \frac{1}{n+1}[f(x)]^{n+1} + c \\ \text{where } n \ne -1

\displaystyle \int f'(x)\sin f(x)\,dx = -\cos f(x) + c

\displaystyle \int f'(x)\cos f(x)\,dx = \sin f(x) + c

\displaystyle \int f'(x)\sec^{2} f(x)\,dx = \tan f(x) + c

\displaystyle \int f'(x)e^{f(x)}\,dx = e^{f(x)} + c

\displaystyle \int \frac{f'(x)}{f(x)}\,dx = \ln\lvert f(x)\rvert + c

\displaystyle \int f'(x)\,a^{f(x)}\,dx = \frac{a^{f(x)}}{\ln a} + c

\displaystyle \int \frac{f'(x)}{\sqrt{a^{2} - [f(x)]^{2}}}\,dx = \sin^{-1} \frac{f(x)}{a} + c

\displaystyle \int \frac{f'(x)}{a^{2} + [f(x)]^{2}}\,dx = \frac{1}{a}\tan^{-1} \frac{f(x)}{a} + c

\displaystyle \int u\,\frac{dv}{dx}\,dx = u v - \int v\,\frac{du}{dx}\,dx

\displaystyle \begin{aligned} & \int_{a}^{b} f(x)\,dx \\[6pt] & \approx \! \frac{b-a}{2n} \! \Bigg\{\begin{aligned} & f(a) + f(b) \, + \\[3pt] & 2\big[ f(x_{1}) \! + \! \dotsb \! + \! f(x_{n-1}) \big] \! \end{aligned}\Bigg\} \\[6pt] & \text{where } a = x_{0} \text{ and } b = x_{n} \end{aligned}

Combinatorics

\displaystyle {}^{n}P_{r} = \dfrac{n!}{(n-r)!}

\displaystyle \binom{n}{r} = {}^{n}C_{r} = \dfrac{n!}{r!(n-r)!}

\displaystyle (x + a)^{n} = x^{n} + \binom{n}{1}x^{n-1}a + \cdots + \binom{n}{r}x^{n-r}a^{r} + \cdots + a^{n}

Vectors

\displaystyle \lvert\underset{\sim}{u}\rvert = \lvert x\,\underset{\sim}{i} + y\,\underset{\sim}{j}\rvert = \sqrt{x^{2} + y^{2}}

\displaystyle \underset{\sim}{u}\!\cdot\!\underset{\sim}{v} = \lvert\underset{\sim}{u}\rvert\,\lvert\underset{\sim}{v}\rvert\cos\theta = x_{1}x_{2} + y_{1}y_{2}, \\\begin{aligned} \text{where }\underset{\sim}{u}=x_{1}\,\underset{\sim}{i}+y_{1}\,\underset{\sim}{j}\\\text{and }\underset{\sim}{v}=x_{2}\,\underset{\sim}{i}+y_{2}\,\underset{\sim}{j}\end{aligned}

\displaystyle r = a + \lambda b

Complex Numbers

\displaystyle z = a + ib = r(\cos\theta + i\sin\theta) = r e^{i\theta}

\displaystyle \big[r(\cos\theta + i\sin\theta)\big]^{n} = r^{n}(\cos n\theta + i\sin n\theta) = r^{n}e^{in\theta}

Mechanics

\displaystyle \frac{d^{2}x}{dt^{2}} = \frac{dv}{dt} = v\,\frac{dv}{dx} = \frac{d}{dx}\!\left(\frac{1}{2}v^{2}\right)

\displaystyle x = a\cos(nt + \alpha) + c

\displaystyle x = a\sin(nt + \alpha) + c

\displaystyle \ddot{x} = -n^{2}(x - c)