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 coefficents of the polynomials are real.

Complex numbers

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 and its applications

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 n^th 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

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

Hyperbolic functions

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

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

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

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

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

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

Inverse hyperbolic functions

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

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

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

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

Arc length and area of surface of revolution

Calculation of the arc length of a curve and the area of a surface using Cartesian or parametric coordinates

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

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