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
Square root of any negative real number
where and are real numbers
Modulus of a complex number
Argument of a complex number
The argument of is the angle between the positive x-axis and a line drawn between the origin and the point in the complex plane (see [1])
In general, if and ,
Complex conjugates
If and then , with the proviso that may have to be added to, or subtracted from, if is outside the permitted range for .
If and then , with the same proviso regarding the size of the angle .
Equating real and imaginary parts
Coordinate geometry on Argand diagrams
If the complex number is represented by the point , and the complex number is represented by the point in an Argand diagram, then , and is the angle between and the positive direction of the x-axis.
Loci on Argand diagrams
represents a circle with centre and radius
represents a circle with centre and radius
represents a straight line — the perpendicular bisector of the line joining the points and
represents the half line through inclined at an angle to the positive direction of
represents the half line through the point inclined at an angle to the positive direction of
De Moivre's theorem and its applications
De Moivre's theorem
De Moivre's theorem for integral n
The cube roots of unity
The cube roots of unity are , and , where
and the non-real roots are
The nth roots of unity
The equation has roots
The roots of zn = α where α is a non-real number
The equation , where , has roots
Hyperbolic functions
Definitions of hyperbolic functions
Hyperbolic identities
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
Integration of hyperbolic functions
Inverse hyperbolic functions
Derivatives of inverse hyperbolic functions
Integrals which integrate to inverse hyperbolic functions
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
Further reading
The AQA's free textbook [2]