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.

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

$z=x+iy$

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 [1])

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

$\arg z=\theta$

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

Polar form of a complex number

$x+iy=z=|z|e^{i\theta }=\left(\sqrt{x^2+y^2}\right)e^{i\theta }$

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

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

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

Addition, subtraction and multiplication of complex numbers of the form x + iy

In general, if $z_1=a_1+i{b}_1$ and $z_2=a_2+i{b}_2$,

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

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

$z_1{z}_2=a_1{a}_2-b_1{b}_2+i(a_2{b}_1+a_1{b}_2)$

Complex conjugates

$\text{If }z=x+iy\text{, then }{z}^{*}=x-iy$

$z{z}^{*}=|z{|}^2$

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 $z_1=(r_1,{\theta }_1)$ and $z_2=(r_2,{\theta }_2)$ then $z_1{z}_2=(r_1{r}_2,{\theta }_1+{\theta }_2)$, 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 $z_1=(r_1,{\theta }_1)$ and $z_2=(r_2,{\theta }_2)$ then $\frac{z_1}{z_2}=(\frac{r_1}{r_2},{\theta }_1-{\theta }_2)$, with the same proviso regarding the size of the angle ${\theta }_1-{\theta }_2$.

Equating real and imaginary parts

$\text{If }a+ib=c+id\text{, where }a\text{, }b\text{, }c\text{ and }d\text{ are real, then }a=c\text{ and }b=d$

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 $z_2$

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

$\text{arg}(z-z_1)=\alpha$ 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

${(\cos \theta +i\sin \theta )}^n=\cos n\theta +i\sin n\theta$

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

$\text{If }z=r(\cos \theta +i\sin \theta )\text{, }$

$\text{then }z=r{e}^{i\theta }$

$\text{and }{z}^n={\left(r{e}^{i\theta }\right)}^n=r^n{e}^{ni\theta }$

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

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

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

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

The roots of zⁿ = α where α is a non-real number

The equation $z^n=\alpha$, where $\alpha =r{e}^{i\theta }$, has roots

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

Hyperbolic functions

Definitions of hyperbolic functions

$\sinh x=\frac{e^x-e^{-x}}{2}$

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

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

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

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

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

Hyperbolic identities

${\cosh }^{2}x-{\sinh }^{2}x=1$

$1-{\tanh }^{2}x={\mathrm{sech}}^{2}x$

${\coth }^{2}x-1={\mathrm{cosech}}^{2}x$

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

$\sinh 2x=2\sinh x\cosh y$

$\begin{array}{cc}\cosh 2x& ={\cosh }^{2}x+{\sinh }^{2}x\\ & =2{\cosh }^{2}x-1\\ & =1+2{\sinh }^{2}x\end{array}$

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

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

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

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

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

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

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

Integration of hyperbolic functions

$\int \sinh xdx=\cosh x+c$

$\int \cosh xdx=\sinh x+c$

$\int {\mathrm{sech}}^{2}xdx=\tanh x+c$

$\int \tanh xdx=\ln \cosh x+c$

$\int \coth xdx=\ln \sinh x+c$

Inverse hyperbolic functions

Logarithmic form of inverse hyperbolic functions

${\sinh }^{-1}x=\ln (x+\sqrt{x^2+1})$

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

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

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

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

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

Further reading

The AQA's free textbook [2]