Proving multiple angle formulae for hyperbolic trigonometric formulae is very similar to proving multiple angle formulae for ordinary trigonometric functions. For instance, to prove the formula we use the formulae and Takeand To prove the formula we use the formulae and Takeand Other formulae can be similarly proved.

## Summary of The Rules for Finding Partial Fractions

separates into separates into In general, if a denominator of degree n factorises completely, it will separate into the sum of n terms, each of which is of the form separates into separates into In general if there is a power of a linear factor in the denominator there will be denominators consisting of ascending […]

## Inverting the Hyperbolic Trigonometric Formulae

We often have to find analytic expressions for the trigonometric formulaeThe method is illustrated in the following examples. Example: Find an expression for Ifthen Multiply both sides byto obtain Now multiply both sides by 2 obtainingand subtractfrom both sides to obtain This is a quadratic expression inso we can solve it using the ordinary quadratic […]

## Integrating Expressions Involving Reciprocals of Sums of Trigonometric Functions Using the Substitution

To be able to use this method we have to expressandin terms of (1) (2) (3) (4) To evaluate any integral such asorwe make the substitutionand use (1), (2), (3) and (4) above to express everything in terms of t then integrate. Example: Find Sub (1) and (4)

## Finding the Intrinsic Equation of a Curve from the Curvature or Radius of Curvature

Intrinsic coordinates label a point on a curve by the length along the curve from a fixed point, often the origin. The curvature of a point on the curve can be written asso if we have an expression for the curvatureor the radius of curvaturewe can find an the intrinsic equation of the curve in […]

## Finding Equations of Asymptotes

The easiest asymptotes to find are those which result from the factorisation of the denominator. Ifthen there are asymptotes at An asymptote that results from the highest power ofin the numerator being one larger than the highest power ofin the denominator or vice versa is also easily found. – find the equations of the asymptotes. […]

## Solving Hyperbolic Trigonometric Equations (2)

An equation of the form(1) may have none, one or two solutions, unlike for the equivalent ordinary trigonometric equation which may have many solutions. We can solve equations of form (1) by substituting and Then multiply though byand we have a quadratic equation inwhich we solve by the normal method of substitution and factorisation or […]

## Hyperbolae

The above equations are referred to as the canonical form of the hyperbolic equation: both equations are transformations of the simple form Instead of being defined by an equation, a hyperbola can be described by it’s geometrical properties. We draw two vertical linesandand plot two points, called fociandas shown above. The locus of the parabola […]

## Ellipses

An ellipse may be defined geometrically in several different ways. One of these is that if the sum of the lengths of tow lines drawn from two points is made to eaqual a constant, the locus generates an ellipse. We may also define the ellipse by an equation:(x^-x-0)^2 over a^2 +(y-y-0)^2 over b^2 =1. The […]

## Derivation of the Curvature Formulae

If we have an arc of a circle, the length of the arc isthe radius of the circle isand the angle subtended by the arc isthenand the radiusThe curvature of the circle isWe can generalise this idea to find the curvature of a curve at any point on the curve by taking the limit asand […]