WJEC A Level Formula Sheet

Laws of logarithms$$\log_a x + \log_a y = \log_a(xy)$$ $$\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$$ $$k\log_a x = \log_a(x^k)$$

Change of base$$\log_a x = \frac{\log_b x}{\log_b a}$$

Exponential growth / decay$$y = Ae^{kt} \quad \text{or} \quad y = Ae^{-kt}$$

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

Discriminant$$\Delta = b^2 - 4ac$$

Completing the square$$ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$$

Arithmetic series$$n\text{th term: } a + (n-1)d$$ $$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$$

Geometric series$$n\text{th term: } ar^{n-1}$$ $$S_n = \frac{a(1-r^n)}{1-r}, \quad r \neq 1$$ $$S_\infty = \frac{a}{1-r}, \quad |r| < 1$$

Binomial expansion (positive integer $n$)$$(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + b^n$$

Summation formulas$$\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$$ $$\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$$ $$\sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4}$$

Pythagorean identities$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \cosec^2\theta$$

Sine rule$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

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

Area of triangle$$\text{Area} = \frac{1}{2}ab\sin C$$

Compound angle formulas$$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$$ $$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$$ $$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$

Double angle formulas$$\sin 2A = 2\sin A\cos A$$ $$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$ $$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$$

Small angle approximations (radians)$$\sin x \approx x, \quad \cos x \approx 1 - \frac{x^2}{2}, \quad \tan x \approx x$$

Differentiation — standard results$$(x^n)' = nx^{n-1}$$ $$(e^x)' = e^x, \quad (e^{kx})' = ke^{kx}$$ $$(\ln x)' = \frac{1}{x}, \quad (\ln|x|)' = \frac{1}{x}$$ $$(\sin x)' = \cos x, \quad (\cos x)' = -\sin x, \quad (\tan x)' = \sec^2 x$$

Product rule$$(uv)' = u'v + uv'$$

Quotient rule$$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$

Chain rule$$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$

Integration — standard results$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + c \quad (n \neq -1)$$ $$\int e^{kx}\,dx = \frac{1}{k}e^{kx} + c$$ $$\int \frac{1}{x}\,dx = \ln|x| + c$$ $$\int \sin kx\,dx = -\frac{1}{k}\cos kx + c$$ $$\int \cos kx\,dx = \frac{1}{k}\sin kx + c$$

Integration by parts$$\int u\,dv = uv - \int v\,du$$

Trapezium rule$$\int_a^b y\,dx \approx \frac{h}{2}\left[y_0 + 2(y_1 + y_2 + \cdots + y_{n-1}) + y_n\right]$$

Newton-Raphson$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Scalar (dot) product$$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta$$

Vector equation of a line$$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$$

Angle between two vectors$$\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$$

Binomial distribution$$P(X = r) = \binom{n}{r}p^r(1-p)^{n-r}$$ $$E(X) = np, \quad \text{Var}(X) = np(1-p)$$

Normal distribution — standardisation$$Z = \frac{X - \mu}{\sigma}$$

Product moment correlation coefficient (pmcc)$$r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$$

Spearman's rank correlation$$r_s = 1 - \frac{6\sum d^2}{n(n^2-1)}$$

SUVAT equations (constant acceleration)$$v = u + at$$ $$s = ut + \frac{1}{2}at^2$$ $$v^2 = u^2 + 2as$$ $$s = \frac{1}{2}(u+v)t$$ $$s = vt - \frac{1}{2}at^2$$

Newton's second law$$\mathbf{F} = m\mathbf{a}$$

Weight$$W = mg$$

Limiting friction$$F_{\max} = \mu R$$

Projectile motion$$x = (u\cos\alpha)t, \quad y = (u\sin\alpha)t - \frac{1}{2}gt^2$$

Centripetal acceleration$$a = \frac{v^2}{r} = r\omega^2$$

Moment of a force$$\text{Moment} = Fd$$ (perpendicular distance)

Modulus and argument$$|z| = \sqrt{x^2 + y^2}, \quad \arg z = \tan^{-1}\left(\frac{y}{x}\right)$$ $$z = x + iy = r(\cos\theta + i\sin\theta) = re^{i\theta}$$

De Moivre's theorem$$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$$ $$r^n(\cos\theta + i\sin\theta)^n = r^n(\cos n\theta + i\sin n\theta)$$

$n$th roots of complex number$$z^{1/n} = r^{1/n}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right], \quad k = 0, 1, \ldots, n-1$$

Roots of unity$$\omega^k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right), \quad k = 0, 1, \ldots, n-1$$

Loci on Argand diagram$$|z - a| = r \text{ — circle}$$ $$|z - a| = |z - b| \text{ — perpendicular bisector}$$ $$\arg(z - a) = \theta \text{ — half-line}$$

Definitions$$\sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x}$$

Fundamental identity$$\cosh^2 x - \sinh^2 x = 1$$

Inverse hyperbolic functions$$\text{arsinh } x = \ln(x + \sqrt{x^2 + 1})$$ $$\text{arcosh } x = \ln(x + \sqrt{x^2 - 1}), \quad x \geq 1$$ $$\text{artanh } x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right), \quad |x| < 1$$

Derivatives$$\frac{d}{dx}(\sinh x) = \cosh x, \quad \frac{d}{dx}(\cosh x) = \sinh x, \quad \frac{d}{dx}(\tanh x) = \sech^2 x$$

$2\times 2$ inverse$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Rotation matrix (2D)$$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

Reflection matrices$$x\text{-axis: } \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad y\text{-axis: } \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \quad y=x\text{: } \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Enlargement (scale factor $k$)$$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$$

Characteristic equation$$\det(A - \lambda I) = 0$$

Diagonalisation$$A = PDP^{-1}, \quad A^n = PD^nP^{-1}$$

Conversions$$x = r\cos\theta, \quad y = r\sin\theta$$ $$r^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x}$$

Area enclosed by polar curve$$\text{Area} = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta$$

Arc length (polar)$$s = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta$$

Arc length (Cartesian)$$s = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$$

Arc length (parametric)$$s = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$$

Surface area of revolution (about $x$-axis)$$S = 2\pi\int_a^b y\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$$

Mean value of a function$$\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx$$

Integration — hyperbolic & inverse functions$$\int \frac{1}{\sqrt{a^2+x^2}}\,dx = \text{arsinh}\left(\frac{x}{a}\right) + c$$ $$\int \frac{1}{\sqrt{x^2-a^2}}\,dx = \text{arcosh}\left(\frac{x}{a}\right) + c$$ $$\int \frac{1}{a^2+x^2}\,dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + c$$ $$\int \frac{1}{a^2-x^2}\,dx = \frac{1}{a}\text{artanh}\left(\frac{x}{a}\right) + c$$

Integrating factor (first-order linear)$$\frac{dy}{dx} + P(x)y = Q(x), \quad \text{I.F.} = e^{\int P(x)\,dx}$$

Second-order homogeneous — auxiliary equation$$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$$ $$am^2 + bm + c = 0$$

Complementary function cases$$\text{Real distinct roots } m_1, m_2:\; y = Ae^{m_1 x} + Be^{m_2 x}$$ $$\text{Repeated root } m:\; y = (A + Bx)e^{mx}$$ $$\text{Complex roots } \alpha \pm i\beta:\; y = e^{\alpha x}(A\cos\beta x + B\sin\beta x)$$

Particular integral — standard trial forms$$\text{Constant: } y = k$$ $$\text{Polynomial degree } n\text{: } y = \text{polynomial degree } n$$ $$\text{Exponential } e^{mx}\text{: } y = ke^{mx}$$ $$\text{Trigonometric } \cos\omega x \text{ or } \sin\omega x\text{: } y = a\cos\omega x + b\sin\omega x$$

Poisson distribution$$P(X = r) = \frac{e^{-\lambda}\lambda^r}{r!}, \quad r = 0, 1, 2, \ldots$$ $$E(X) = \lambda, \quad \text{Var}(X) = \lambda$$

Geometric distribution$$P(X = x) = (1-p)^{x-1}p, \quad x = 1, 2, 3, \ldots$$ $$E(X) = \frac{1}{p}, \quad \text{Var}(X) = \frac{1-p}{p^2}$$

Negative binomial distribution$$P(X = x) = \binom{x-1}{r-1}p^r(1-p)^{x-r}, \quad x = r, r+1, r+2, \ldots$$ $$E(X) = \frac{r}{p}, \quad \text{Var}(X) = \frac{r(1-p)}{p^2}$$

Chi-squared test statistic$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$

Expected frequency (contingency table)$$E_{ij} = \frac{(\text{row } i \text{ total}) \times (\text{column } j \text{ total})}{\text{grand total}}$$

Impulse and momentum$$\mathbf{I} = m\mathbf{v} - m\mathbf{u} = \Delta(m\mathbf{v})$$

Conservation of momentum$$m_1\mathbf{u}_1 + m_2\mathbf{u}_2 = m_1\mathbf{v}_1 + m_2\mathbf{v}_2$$

Newton's law of restitution$$e = \frac{\text{speed of separation}}{\text{speed of approach}}$$

Kinetic energy$$KE = \frac{1}{2}mv^2$$

Gravitational potential energy$$GPE = mgh$$

Elastic potential energy$$EPE = \frac{1}{2}\frac{\lambda x^2}{l}$$

Power$$P = Fv$$

Work-energy principle$$\text{Total work done} = \Delta KE$$

Vector product$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$

Scalar triple product$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$

Equation of a plane$$\mathbf{r} \cdot \mathbf{n} = k \quad \text{or} \quad ax + by + cz = d$$

Shortest distance from point to plane$$d = \frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$$