further maths core pure

• osborne’s rule

  • using the regular trigonometric identities, change each one to a hyperbolic, then where ever you have a product of two sines, negate that term.
• differentiating:

  coshx → sinhx

  sinhx → coshx
• to differentiate an inverse hyperbolic function (e.g. y = arcoshx):

  1. rearrange from y= into x=
  2. differentiate implicitly with respect to x
  3. rearrange into dy/dx=
  4. use an identity to get the equations in terms of x
  5. check signs by drawing the graphs and comparing the direction of the gradient
• if you’re given an integral with a quadratic on the denominator that’s square rooted, you will need to complete the square in order to get it in one of the inverse hyperbolic integral forms
• when creating partial fractions with a quadratic term in the denominator that won’t factorise, the numerator should be linear
• when integrating a polynomial over a polynomial, its often necessary to split it into partial fractions
• formula for the integrating factor

  IF = e ^ ∫ P(x) dx

  to use the integrating factor, make sure the coefficient of the second derivative is 1, if it’s not, then divide by this coefficient
• what is the formula for IFFY SQUIFF?

  I.F * y = ∫(Q(x) * I.F) dx
• reverse product rule:

  this method works when the coefficient of y is the derivative of the coefficient of dy/dx
• how do you solve differential equations using the reverse product rule?

  integrate both sides to eliminate dy/dx then rearrange to make it y=
• cosx in terms of e^ix:

  cosx=(e^[ix] + e^[−ix])/2
• sinx in terms of e^ix:

  sinx=(e^[ix] - e^[−ix])/2i
• solving problems using e^iθ

  1. Write out a 'C' and an 'S' summation equation, where C is in terms of cosθ and S is in terms of sinθ

  2. Add them into C+iS, and factorise as much as possible

  3. Use de moivre's theorem to simplify, then use (e^iθ = cosθ + isinθ) where possible

  4. See if your summation is also a maclaurin expansion

  5. Look for previous parts of the equation to substitute into the equation

  6. Depending on whether you're looking for C or S, take the real or imaginary part of your final equation
• integrating regular inverse trig functions

  1. set the function equal to y
  2. rearrange to equal x
  3. differentiate
  4. reciprocate
  5. use an identity to make it in terms of x
  6. work out the domain and range where this is valid
• using arcsinx as an example of integrating inverse trig functions:

  1. set y=arcsinx
  2. rearrange into x=siny
  3. differentiate to dx/dy=cosy
  4. reciprocate into dy/dx=1/cosy
  5. use an identity to remove the trigonometric function and get the equation in terms of x, the example would use cos^2y+sin^2y=1, which you would have to rearrange to cosy=√(1-sin^2y)
  6. you can then substitute in x as siny, to leave you with dy/dx=1/√(1-x^2)
• what is de moivre's theorem as an equation?

  $(r(\cos(\theta) +$$i\sin(\theta))^n =$$r^n(\cos(n\theta) +$$i\sin(n\theta))$
• if you have (cosx - isinx)^n, you can manipulate it into (cos-x + isin-x)^n

  if you have (sinx + icosx)^n you can manipulate it by multiplying it all by i, then using the steps above to make isinx positive
• • an imaginary number is…

  in the form ‘bi’ where b is any real number.
• • a complex number is…

  in the form a+bi where a and b are any real number.
• • a complex conjugate is…

  if a+bi = z, then a-bi = z*
• in an argand diagram, the modulus of z (a complex number), is it’s distance from the origin
• in an argand diagram, the argument of z, or arg(z), is the anti-clockwise rotation (in radians) from the positive real axis (usually the x-axis)
• the argument of z^2 is twice the argument of z
• the modulus of z^2 is equal to |z|^2
• adding or subtracting two complex numbers works in the same way as adding vectors, you add the real parts and add the imaginary parts separately
• multiplying two complex numbers involves multiplying their magnitudes together and adding their arguments
• dividing two complex numbers involves dividing their magnitudes by one another and subtracting their arguments
• what is the matrix transformation x degrees about the x axis?

  [1 0 0 ]

  [0 cosx -sinx ]

  [0 sinx cosx ]
• what is the matrix transformation x degrees about the y-axis?

  [cosx 0 sinx ]

  [0 1 0 ]

  [-sinx 0 cosx ]
• what is the matrix transformation x degrees about the z-axis?

  [cosx -sinx 0 ]

  [sinx cosx 0 ]

  [0 1 0 ]
• for a process that does transformation A then transformation B then transformation C, the combined transformation is CBA
• on an argand diagram, |z-z'| = r is a...

  circle with a centre z' and radius r
• on an argand diagram, |z-z'| = |z-z"| is...

  the perpendicular bisector of z' and z"
• on an argand diagram, arg(z-z') = x is a...

  half line starting at z', with an angle to the real axis of x
• z^n - z^-n =

  $2i\sin(n\theta)$
• z^n + z^-n =

  $2\cos(n\theta)$
• if a question asks for the least value of |z|, where z is a variable complex number, like a perpendicular bisector, circle or half line, it wants you to find the shortest distance from the origin
• the maclaurin series is based upon the idea that if you take an infinite polynomial, you can approximate it by using differentiation to find the coefficients for each of the terms
• to use the maclaurin series:

  1. differentiate your function as many times as necessary
  2. substitute in 0 as x
  3. put each term into the formula
• when using the maclaurin series standard formula:

  if your f’(x) result forms any kind of binomial expansion (e.g. when differentiating arcsinx which becomes 1/√(1-x^2)), you can use the regular binomial expansion formula for f'(x) and then integrate it to get f(x) as an infinite series.