A-level Mathematics/AQA/MFP2

Roots of polynomials
The relations between the roots and the coefficients of a polynomial equation; the occurrence of the non-real roots in conjugate pairs when the coefficients of the polynomials are real.

Square root of minus one
$$ \sqrt{-1} = i \,\!$$

$$ i^2 = -1 \,\!$$

Square root of any negative real number
$$ \sqrt{-2} = \sqrt{2 \times -1} = \sqrt{2} \times \sqrt{-1} = \sqrt{2} \times i = i\sqrt{2} \,\!$$

$$ \sqrt{-n} = i\sqrt{n} \,\!$$

General form of a complex number
$$ z = x + i y \,\!$$

where $$ x \,\!$$ and $$ y \,\!$$ are real numbers

Modulus of a complex number
$$ |z| = \sqrt{x^2 + y^2} \,\!$$

Argument of a complex number
The argument of $$ z \,\!$$ is the angle between the positive x-axis and a line drawn between the origin and the point in the complex plane (see )

$$ \tan{\theta} = \frac{y}{x} \,\!$$

$$ \arg{z} = \theta \,\!$$

$$ \arg{z} = \tan^{-1}{ \left ( \frac{y}{x} \right ) } \,\!$$

Polar form of a complex number
$$ x+ iy = z = |z|e^{i\theta} = \left ( \sqrt{x^2 + y^2} \right ) e^{i\theta} \,\!$$

$$ e^{i\theta} = \cos{\theta} + i\sin{\theta} \,\!$$

$$ z = |z|e^{i\theta} = |z| \left ( \cos{\theta} + i\sin{\theta} \right ) \,\!$$

$$ e^{i\theta} = \frac{z}{|z|} = \frac{x + iy}{\sqrt{x^2 + y^2}} \,\!$$

Addition, subtraction and multiplication of complex numbers of the form x + iy
In general, if $$ z_1 = a_1 + ib_1 $$ and $$ z_2 = a_2 + ib_2 $$,


 * $$ z_1 + z_2 = (a_1 + a_2) + i(b_1 + b_2) $$


 * $$ z_1 - z_2 = (a_1 - a_2) + i(b_1 - b_2) $$


 * $$ z_1 z_2 = a_1 a_2 - b_1 b_2 + i(a_2 b_1 + a_1 b_2)$$

Complex conjugates
$$ \mbox{If } z = x + iy \mbox{, then } z^* = x - iy \,\!$$

$$ zz^* = |z|^2 \,\!$$

Division of complex numbers of the form x + iy
$$ \frac{z_1}{z_2} = \frac{z_1}{z_2}\frac{z_2^*}{z_2^*} = \frac{z_1 z_2^*}{|z_2|^2} $$

Products and quotients of complex numbers in their polar form
If $$ z_1 = (r_1,\mbox{ } \theta_1) $$ and $$ z_2 = (r_2,\mbox{ } \theta_2) $$ then $$ z_1 z_2 = (r_1 r_2,\mbox{ } \theta_1+\theta_2) $$, with the proviso that $$ 2 \pi $$ may have to be added to, or subtracted from, $$ \theta_1 + \theta_2 $$ if $$ \theta_1 + \theta_2 $$ is outside the permitted range for $$ \theta $$.

If $$ z_1 = (r_1,\mbox{ } \theta_1) $$ and $$ z_2 = (r_2,\mbox{ } \theta_2) $$ then $$ \frac{z_1}{z_2} = \left ( \frac{r_1}{r_2} ,\mbox{ } \theta_1 - \theta_2 \right ) $$, with the same proviso regarding the size of the angle $$ \theta_1 - \theta_2 $$.

Equating real and imaginary parts
$$\mbox{If } a + ib = c + id \mbox{, where } a \mbox{, } b \mbox{, } c \mbox{ and } d \mbox{ are real, then } a = c \mbox{ and } b = d \,\!$$

Coordinate geometry on Argand diagrams
If the complex number $$z_1$$ is represented by the point $$A$$, and the complex number $$z_2$$ is represented by the point $$B$$ in an Argand diagram, then $$|z_2 - z_1| = AB \,\!$$, and $$\arg{(z_2 - z_1)}$$ is the angle between $$\overrightarrow{AB}$$ and the positive direction of the x-axis.

Loci on Argand diagrams
$$|z| = k$$ represents a circle with centre $$O$$ and radius $$k$$

$$|z-z_1| = k$$ represents a circle with centre $$z_1$$ and radius $$k$$

$$|z-z_1| = |z-z_2|$$ represents a straight line — the perpendicular bisector of the line joining the points $$z_1$$ and $$z_2$$

$$\mbox{arg }z = \alpha$$ represents the half line through $$O$$ inclined at an angle $$\alpha$$ to the positive direction of $$Ox$$

$$\mbox{arg}(z-z_1) = \alpha$$ represents the half line through the point $$z_1$$ inclined at an angle $$\alpha$$ to the positive direction of $$Ox$$

De Moivre's theorem
$$\left ( \cos{\theta} + i\sin{\theta} \right )^n = \cos{n\theta} + i\sin{n\theta} \,\!$$

De Moivre's theorem for integral n
$$ z + \frac{1}{z} = 2 \cos{\theta} $$

$$ z - \frac{1}{z} = 2i \sin{\theta} $$

Exponential form of a complex number
$$\mbox{If } z = r(\cos{\theta}+i\sin{\theta})\mbox{, } \,\!$$

$$\mbox{then } z = re^{i\theta} \,\!$$

$$\mbox{and } z^n = \left ( re^{i\theta} \right )^n = r^ne^{ni\theta} \,\!$$

$$\cos{\theta} = \frac{e^{i\theta}+e^{-i\theta}}{2} $$

$$\sin{\theta} = \frac{e^{i\theta}-e^{-i\theta}}{2i} $$

The cube roots of unity
The cube roots of unity are $$1$$, $$w$$ and $$w^2$$, where

$$ w^3 = 1 \,\!$$

$$ 1 + w + w^2 = 0 \,\!$$

and the non-real roots are

$$ \frac{-1 \pm i\sqrt{3}}{2} $$

The nth roots of unity
The equation $$ z^n = 1 $$ has roots

$$ z = e^{\frac{2k \pi i}{n}} \mbox{ where } k = 0,1,2, \dots,(n-1) $$

The roots of zn = α where α is a non-real number
The equation $$ z^n = \alpha $$, where $$ \alpha = re^{i\theta} $$, has roots

$$ z = r^{\frac{1}{n}}e^{\frac{i(\theta+2k\pi)}{n}} \mbox{ where } k = 0,1,2, \dots,(n-1) $$

Definitions of hyperbolic functions
$$ \sinh{x} = \frac{e^x - e^{-x}}{2} $$

$$ \cosh{x} = \frac{e^x + e^{-x}}{2} $$

$$ \tanh{x} = \frac{ \sinh{x} }{ \cosh{x} } $$

$$ \operatorname{cosech}{x} = \frac{1}{ \sinh{x} } $$

$$ \operatorname{sech} = \frac{1}{ \cosh{x} } $$

$$ \coth{x} = \frac{1}{ \tanh{x} } $$

Hyperbolic identities
$$ \cosh^2{x} - \sinh^2{x} = 1 \,\!$$

$$ 1 - \tanh^2{x} = \operatorname{sech}^2{x} \,\!$$

$$ \coth^2{x} - 1 = \operatorname{cosech}^2{x} \,\!$$

Addition formulae
$$ \sinh{(x+y)} = \sinh{x}\cosh{y} + \cosh{x}\sinh{y} \,\!$$

$$ \cosh{(x+y)} = \cosh{x}\cosh{y} + \sinh{x}\sinh{y} \,\!$$

Double angle formulae
$$ \sinh{2x} = 2\sinh{x}\cosh{y} \,\!$$

$$ \begin{align} \cosh{2x} & = \cosh^2{x} + \sinh^2{x} \\ & = 2\cosh^2{x} -1 \\ & = 1 + 2\sinh^2{x} \end{align} \,\!$$

Osborne's rule
Osborne's rule states that:


 * to change a trigonometric function into its corresponding hyperbolic function, where a product of two sines appears, change the sign of the corresponding hyperbolic form

Note that Osborne's rule is an aide mémoire, not a proof.

Differentiation of hyperbolic functions
$$ \frac{d}{dx} \sinh{x} = \cosh{x} $$

$$\frac{d}{dx} \cosh{x} = \sinh{x} $$

$$ \frac{d}{dx} \tanh{x} = \operatorname{sech}^2{x} $$

$$ \frac{d}{dx} \sinh{kx} = k\cosh{kx} $$

$$ \frac{d}{dx} \cosh{kx} = k\sinh{kx} $$

$$\frac{d}{dx} \tanh{kx} = k\operatorname{sech}^2{kx} $$

Integration of hyperbolic functions
$$ \int \sinh{x} \, dx = \cosh{x} + c $$

$$ \int \cosh{x} \, dx = \sinh{x} + c $$

$$ \int \operatorname{sech}^2{x} \, dx = \tanh{x} + c $$

$$ \int \tanh{x} \, dx = \ln{\cosh{x}} + c $$

$$ \int \coth{x} \, dx = \ln{\sinh{x}} + c $$

Logarithmic form of inverse hyperbolic functions
$$ \sinh^{-1}{x} = \ln{\left ( x + \sqrt{x^2 + 1} \right )} $$

$$ \cosh^{-1}{x} = \ln{\left ( x + \sqrt{x^2 - 1} \right )} $$

$$ \tanh^{-1}{x} = \frac{1}{2}\ln{\left ( \frac{1+x}{1-x} \right )} $$

Derivatives of inverse hyperbolic functions
$$\frac{d}{dx} \sinh^{-1}{x} = \frac{1}{\sqrt{1+x^2}}$$

$$\frac{d}{dx} \cosh^{-1}{x} = \frac{1}{\sqrt{x^2-1}}$$

$$\frac{d}{dx} \tanh^{-1}{x} = \frac{1}{1-x^2}$$

$$\frac{d}{dx} \sinh^{-1}{\frac{x}{a}} = \frac{1}{\sqrt{a^2+x^2}}$$

$$\frac{d}{dx} \cosh^{-1}{\frac{x}{a}} = \frac{1}{\sqrt{x^2-a^2}}$$

$$\frac{d}{dx} \tanh^{-1}{\frac{x}{a}} = \frac{1}{a^2-x^2}$$

Integrals which integrate to inverse hyperbolic functions
$$\int \frac{1}{\sqrt{a^2+x^2}} \, dx = \sinh^{-1}{\frac{x}{a}} + c$$

$$\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \cosh^{-1}{\frac{x}{a}} + c$$

$$\int \frac{1}{a^2-x^2} \, dx = \tanh^{-1}{\frac{x}{a}} + c$$

Calculation of the arc length of a curve and the area of a surface using Cartesian or parametric coordinates
$$ s = \int^{x_2}_{x_1} \sqrt{ 1 + \left ( \frac{dy}{dx} \right )^2 } dx = \int^{t_2}_{t_1} \sqrt{ \left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right )^2 } dt $$

$$ S = 2 \pi \int^{x_2}_{x_1} y \sqrt{ 1 + \left ( \frac{dy}{dx} \right )^2 } dx = 2 \pi \int^{t_2}_{t_1} y \sqrt{ \left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right )^2 } dt $$