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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$s=\int _{x_{1}}^{x_{2}}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx=\int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt$

$S=2\pi \int _{x_{1}}^{x_{2}}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx=2\pi \int _{t_{1}}^{t_{2}}y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt$

Further reading

The AQA's free textbook [2]