A Level Maths Equations

Pure

 

Arithmetic Series

S_{n} = \dfrac{1}{2}n(a + l)\\

 

S_{n} = \dfrac{1}{2}n(2a + (n - l)d)\\

 

Geometric Series

S_{n} = \dfrac{a(1 - r^n)}{1 - r}\\

 

S_{\infty} = \dfrac{a}{1 - r} for |r\ < 1\\

 

Binomial Expansion

(a + b)^n = a^n +  ^nC_1 a^{n-1} b +  ^nC_2 a^{n-2} b^2 + ... +   ^nC_r a^{n-r} b^r + ... + b^n

where (n \in \N) and ^nC_r = \begin{pmatrix}n\\r\end{pmatrix} = \dfrac{n!}{r (n-r)!}

(1 + x)^n = 1 + nx + \dfrac{n(n - 1)}{1\times2} x^2 + ... + \dfrac{n(n - 1) ... (n - r + 1)}{1\times2\times...\times r} + ...

for |x| < 1, n \in \R

 

Curved Surface Area 

Surface area of sphere = 4\pi r^2

Area of curved surface of cone = \pi r \times  slant height

 

Exponentials and Logarithms

\log_a (x) \dfrac{\log_b (x)}{\log_b (a)}\\

 

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

 

Exponentials and Logarithms

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

 

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

 

\tan(A\pm B)\equiv\dfrac{\tan(A)\pm\tan(B)}{1\mp\tan(A)\tan(B)}        (A\pm B\neq(K+\dfrac{1}{2})\pi)\\

 

\sin(A)\pm\sin(B)=2\sin(\dfrac{A\pm B}{2})\cos(\dfrac{A\mp B}{2})\\

 

\cos(A)+\cos(B)=2\cos(\dfrac{A+B}{2})\cos(\dfrac{A-B}{2})\\

 

\cos(A)-\cos(B)=-2\sin(\dfrac{A+B}{2})\sin(\dfrac{A-B}{2})\\

 

Small Angle Approximations

\sin(\theta)\approx\theta\\

 

\cos(\theta)\approx 1-\dfrac{1}{2}\theta ^2\\

 

\tan(\theta)\approx\theta\\

 

Differentiation

First principles: f'(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}\\

 

Quotient rule: \dfrac{d}{dx}\left(\dfrac{u(x)}{v(x)}\right)=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}

 

\textbf f(x)	\textbf f'(x)
\tan(kx)	k\sec^2(kx)
\sec(kx)	k\sec(kx)\tan(kx)
\cot(kx)	-k\cosec^2(kx)
\cosec(kx)	-k\cosec(kx)\cot(kx)

 

Integration

\int u\dfrac{dv}{dx}dx = uv - \int v\dfrac{du}{dx}dx\\

 

\textbf f(x)	\int \textbf f(x) dx
\sec^2(kx)	\dfrac{1}{k}\tan(kx) + c
\tan(kx)	\dfrac{1}{k}\ln |\sec(kx)|+ c
\cot(kx)	\dfrac{1}{k}\ln |\sin(kx)|+ c
\cosec(kx)	-\dfrac{1}{k}\ln |\cosec(kx)|+\cot(kx)| + c \dfrac{1}{k}\ln |\tan(\dfrac{1}{2}kx)| + c
\sec(kx)	-\dfrac{1}{k}\ln |\sec(kx)|+\tan(kx)| + c \dfrac{1}{k}\ln |\tan(\dfrac{1}{2}kx + \dfrac{\pi}{4})| + c

 

Numerical Methods

Trapezium rule: \int_{b}{a}y dx = \dfrac{1}{2} h (y_{0} + y_{n} + 2(y_{1} + y_{2} +...+ y_{n-1})) where h=\dfrac{b-a}{h}

 

Newton-Raphson iteration for solving f(x)=0 is x_{n+1}=x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}