Finding Multiple Solutions of Trigonometric Equations

In general a trigonometric equation of the formfor example may have more than one solution, and may have in fact an infinite number of solutions. We often have to find the solutions within a certain range eg 0 – 360o ordepending on whether we are working in degrees or radians. All the trigonometric graphs – […]

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Exponentials

An exponential curve is any of the formExamples are given below. Black: Red: Blue: All exponential curveshave the property of passing through the pointwhatever the value ofThere is a very special number, labelledcalled the natural base.is labelled . Typically we have to solve equations of the form: If we substitute we obtain the normal quadratic […]

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Logarithms

There are several “Laws” of Logarithms. They are: The bases seem to cancel in the first expression and to swap with the argument in the second. Ie we eliminate a log by raising the base to the power of both sides. Problems involving logs usually reduce to a simple linear or quadratic equation. Example: Use […]

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Area Between Arcs of Intersecting Circles

The area formed between arcs of intersecting circles, labelled A below, can be though of as the sum of the area of two segments. Draw the chord AB then A is the sum ofandin the diagram below. The formula for the area of a segment subtending an anglein a circle of radius x is Henceandso

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Area Between Arcs of Intersecting Circles

The area formed between arcs of intersecting circles, labelled A below, can be though of as the sum of the area of two segments. Draw the chord AB then A is the sum ofandin the diagram below. The formula for the area of a segment subtending an anglein a circle of radius x is Henceandso

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Logarithms in Base 10

Any number can be written in the formWhenthe number is said to be written in standard form. Powers of 10 are especially important since they are used to estimate orders of magnitude. (Did that man go bankrupt owing millions or tens of millions?). The relationship between powers of 10 and logarithms in base 10 is […]

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Intersecting Circles

If the distance between the centres of two circles is less than the sum of their raddi, the circles will intersect at two points. Suppose we have circlesand The circles have radiiandrespectively. The centres of the circles are atandrespectively. The distance between the centres isso the circles intersect in two points. Expanding the brackets for […]

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Finding the Coefficient of a Power of x in the Binomial Expansion

In the binomial expansion ofthe coefficient ofisThis is the (r+1)th term in the binomial expansion ofin ascending powers of Suppose that a=2 and b=3x^2 . To find the coefficient ofin the binomial expansion ofputthen the coefficient ofis and the coefficient ofis If both a and b are terms in x then things get slightly more […]

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Proof of Factor Theorem

Letbe a polynomial of degreeis a factor ofif and onlydividesor is a factor of Proof Letbe a polynomial and letbe a number. Ifdividesthen the remainder on division ofbyis zero and there is a polynomialsuch thatso thatandis a root of Now assume thatis a root ofso thatPerform long division ofbyto obtain quotientand remainderthen write (1) The […]

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Advanced Geometric Sequence Problems

Some problems involving geometric sequences involve much manipulation. Example: The first three terms of a geometric sequence areandFindthe first term and the common ratio. The common ratio is equal to the second term divided by the first and also equal to the third term divided by the second, hence Cross multiplication gives Expanding both sides […]

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