Given a straight line plotted on a graph we can estimate the values of the gradient and either using a point on the line or by estimating the y – intercept, we can find the equation of the line in the form The most convenient form in which to analyse the relationship between two variables […]

## Proof of the Sine Rule

The Sine Rule states that for the triangle labelled as below, To prove it, start by dropping a perpendicular from a vertex to cut the opposite side at right angles, say fromto b. Label this perpendicular Thenso(1) Now draw a line fromto cut the sideat right angles. soCombine this with (1) to obtain

## Possible Factors of Polynomials

When looking at possible factors of a polynomialit is necessary to look at the coefficients of the highest and lowest powers of If a factor of the polynomialisthenmust be a factor of the coefficient of the highest power ofandmust be a factor of the lowest power ofFor example ifthen possible factors are We can however […]

## Problems involving the Binomial Expansion

Sometimes we have to go just a little bit further than expand a bracket using the binomial theorem. We may have two brackets multiplied together, both brackets raised to a power. We may have to expand the brackets up to the term infor example. We do this by expanding each bracket using the binomial theorem, […]

## Using Discriminants to Find the Number of Roots of a Quadratic Curve

In general a quadratic curve may have one, two or no roots, as shown. For the curve the number of roots depends entirely on the discriminant Ifthere are no roots. Ifthere is one root. Ifthere are two roots. We are typically asked: Find the set of values offor which the curve has no roots: always, […]

## Tangents and Normals

Tangents Given a curvewe can find the equation of the tangent toat a point whereby Finding Differentiating f(x) and substituting x=p to find Usingto find the equation of the tangent. Example: Find the equation of the tangent to the curve Multiply the brackets out to getand differentiate to get and now find Normals Give a […]

## Solving Exponential Equations

The simplest sort of exponential equation is of the formBy inspectionbecause However not all equations can be solved by inspection even if they are of the simple form given above. The equationhas no simple integer solutions. This equation can only be solving by taking the log of both sides: and then using the power rule […]

## Integration and Area

Integration is equivalent to finding the area under a curve The area The numbers a and b are called limits, and we say, “find the area by integratingbetween and”. Example: Find the shaded area for the graphshown. Remember that when we integration any polynomial exceptwe ALWAYS add one to the power and divide by the […]

## Solving Multiple Angle Trigonometric Equations

It is quite easy to find an answer for the equationfor example. We finddegrees. However this is not the only answer. Because of the periodicity of the cosine curve, repeating every 360o and symmetry of the cosine curve aboutetc, the following are also solution: Suppose instead we have to solve the equation Now we can […]

## Sequences and Limits

Sequences can be defined in various ways, in iterative and closed form. We have so far analysed geometric and arithmetic series. The terms of an arithmetic series always tend to(we say the series does not tend to a limit): The terms of a geometric series may tend to neither or 0 depending on the values […]