## The General Solution of Trigonometric Equations

The basic trigonometric functions – sin, cos, tan – have many symmetries. All areperiodic, repeating everyor 360 degrees. In addition: cos is symmetric aboutsoIn fact cos is symmetric aboutfor any integerso sin is symmetric aboutandso thatand In fact sin is symmetric aboutsofor any integer tan isperiodic soIn factfor any integer These symmetries are illustrated below. […]

## Sketching Curves Involving Quotients of Linear Factors

When attempting to sketch curves of functions of quotients of linear factors, it is best to start by sketching the asymptotes, which imply the behaviour of the function as one coordinate tends to infinity. If the denominator factorises or has real roots, then there is a vertical asymptote for each root of the formEach distinct […]

## Decomposing Transformations II

Matrices representing transformations may be multiplied by each other, representing a composition of transformations. We also need to be able to express a matrix as a product of matrices representing transformations. The transformations are are particularly interested in are enlargements or scalings, rotations and reflections. Suppose then that we have the matrix This matrix has […]

## Transforming and Solving Non – Linear, Non – Homogeneous Differential Equations

Non linear differential equations contain one of more products ofor terms or maybe powers of these terms or even products of powers. They may look more complicated but they can often be transformed into a simpler form and solved using the one of standard techniques of separation of variables, the integrating factor method or the […]

## 2 x 2 Matrices – Sums, Products Determinants and Inverses

Adding Matrices Matrices add in the natural way: Example: Multiplying Matrices Multiplying is a little more complex. Remember that you multiply rows by columns. Example: Determinants of Matrices To find the determinant of a matrix multiply diagonal corners together and subtract. Example: Inverses of Matrices Each entry in the matrix can be divided byin the […]

## Finding the Equation of a Curve After Transformation By a Matrix

Suppose we have a curve which undergoes a linear transformation. The transformation may be represented by a matrixand the curve by a vectorwheremay be a function ofor vice versa, or both are functions of some parameter (I will not deal with this case here). The simplest case is when a line is transformed. To find […]

## Sketching Curves in Polar Coordinates

If you want to sketch a curve expressed in polar formthere are some elementary things to remember. cannot be negative! If some value ofgives a negative value forthe curve is not defined there. is the angle the radius vector makes with the positive– axis in a clockwise sense. Ifis a function ofthen the curve will […]

## Proof By Induction

The simplest proofs by induction have two steps. This is maybe best illustrated with an example. Suppose we wanted to prove that(1) for allW e sayis the proposition thatand we can write this as The first step, writtenoris the statement thatoris true. We can takeas the statementiewhich is true. Now we assume thatare all true. […]

## Finding the Turning Points of Polar Curves

A turning point in cartesian coordinates is the solution toWe cannot form this expression in polar coordinates directly. If in polar coordinates we haveas a function ofthenandso nowandare expressed in terms of the angleand we can use the parametric formula for the gradientto find the gradient at any point. If we want a turning point […]