The ambiguous case arises when using the Sine Rule to find an angle in a triangle. It occurs because the Sin function is symmetric about 90°, so that When we solve forthere is an acute solution,and an obtuse solution, Example: Find the angle A in the triangle below. The Sine Rule states Then This is […]

## The Difference Between an Expression, an Equation and a Formula

Many students get confused between anything in algebra involving‘s. Anything without an equals sign is an expression. are all expressions. The aim is always to simplify expressions. Anything with an equals sign containing x’s and no other letters, or containing more than two letters with no letter being the subject is an equation. are all […]

## Simple Interest

If you put money into a bank account, the bank will normally pay interest on the money in your account. If the interest is simple interest, then the yearly amount of interest will be calculated on the original deposit. This is unlike compound interest, for which the amount of interest is calculated on the balance […]

## Tessellations

Tessellating a plane is to cover it with shapes leaving no gaps. The shapes may all be the same, or there may be several shapes. Each shape may be rotated, reflected or translated to fit somewhere else leaving no gaps. The shapes tessellating the planes below are all identical, but have been rotated, translated or […]

## Directed Numbers

Directed numbers are inequalities that describe a set of numbers. The inequalities may be represented on a number line. The diagram below represents the inequalityThe filled in black circle atindicates that 2 satisfies the inequality. The diagram below represents the inequalityThe white filled in circle atindicates thatdoes not satisfy the inequality. The above examples illustrate […]

## Drawing Three Dimensional Solids With a Constant Cross Section

Three dimensional solids with a constant cross section can be easily drawn by drawing a copy of the cross section in the background and joining up corresponding vertices. To draw a three dimensional solid with the cross section below: Draw a copy of the cross section in the background: Now join up the vertices. The […]

## Probability Tables

Some probabilities are especially suitable to calculate from tables. If we have two events with a range of possible outcomes then we can display these in a table and use the table to calculate probabilities. Suppose for example that we have two six sided fair dice. Each dice is equally likely to score anything from […]

## Vectors I

A vector is the difference between two points. If two points areandthen we can write the difference(when written in a column between brackets) asThe arrow above means that we are going from the pointto the pointIf we swapandthen we are going fromtoand writeThis is shown below. We can also write points as pairs of coordinates and […]

## Parts of a Circle

Labels for the different parts of a circle are given below Radius – the radius of a circle is the distance from the centre labelled O to the circumference. The radius is labelled The circumference of a circle is the perimeter, and is labelled Diameter – diameter of a circle is the length of a […]

## Proof That a Line From a Point to the Centre of a Circle Bisects the Angle Between Two Tangents Drawn From That Point

The theorem is illustrated below. Proof: Construct the trianglesandby drawing radii as below. since both are radii of the circle andis common to both. Further, angle since these are between a tangent and a radius. From Pythagoras theorem,soand the triangles have three equal lengths so are congruent and angleandbisects