There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we choose or have been given: Given we have chosen the anglewe label the sides for this angle. Opposite = is the side opposite the angle Hypotenuse =is the […]

## Lapping Racing Drivers

Good – or more dangerous – racing drivers are faster than other racing drivers. Towards the end of a race, the slower drivers may find themselves being lapped by other racing drivers who have completed one more circuit than they have. If the speeds of the racing drivers areandwithon a circuit ofmiles, how long will […]

## Solving Cosine Equations for More Than One Solution

To solve the equationinvert both sides, obtaining This is not the only the solution. There are an infinite number of solutions. Typically though, we want solutions in the range 0° – 360°. We can use the cosine curve below to find the solutions. Draw the horizontal line y=0.4 (because the equation to solve is cos x […]

## Rules of Inequalities

In order to solve inequalities, we have to simplify them. In simplifying inequalities, we have to observe four rules. 1. If we add or subtract the same number to both sides of an inequality, the inequality sign is maintained, i.e.and Example: Ifthen 2. If we multiply or divide both sides by a positive number the […]

## Solving Exponential Simultaneous Equations Algebraically

If some quantity is increasing decreasing by the same proportion or factor in each time interval, then it is said to vary exponentially. A good example is money left in a bank which attracts compound interest at the same rate each year. If the rate of interest is 5%, then at the end of every […]

## Proof of Pythagoras Theorem

The above diagrams represent rearrangements of sets of shapes. The blue triangles are all right angled, with the right angles at the corners of the yellow square, and congruent so all have the same area and the yellow squares are congruent so have the same area. We can find the area of the square on […]

## Speeds – There and Back

Travelling to some place and back again might involve travelling twice the distance of only travelling there, but might well not mean taking twice as long. Going in one direction might mean travelling uphill or going against the wind or travelling in rush hour, meaning that this part of the total journey takes longer. To […]

## Solving Equations Involving Algebraic Fractions

An algebraic fraction is any expression with fraction which include x terms in numerator and/or denominator. If this expression is equal to something then typically we can solve it by reararranging the fraction into a linear or quadratic equation and solving. For example, ifthen we can multiply both sides byto clear the fraction, obtaining Expanding […]

## Problem Solving II

More advanced problem solving questions involve simultaneous equations or quadratics. Example: The height of a rectangle is 2 cm longer than the base. If the area is 48 cm2 find the length of the base. If the base isthe height is The area isand equating this to 48 givesExpanding the brackets and subtracting 48 gives […]

## Advanced Questions Involving Differences of Squares

Suppose we want to find all the integers satisfying is a difference of squares and factorises asand 7 factorises asorsoorororWe may equate factors on either side to give either andorand The first of these gives the simultaneous equations The solution isand The second of these gives the simultaneous equation The solution is The first of […]