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Expand sin and cos of sums and differences, use the tan and cot compound-angle formulae, convert between sums and products, and apply the double and triple angle identities for sine, cosine and tangent.
Read articleDefine all six trigonometric functions with the unit circle, prove and use sin² x + cos² x = 1, find the sign of each function in every quadrant, evaluate allied angles, and state domains, ranges and periods with their graphs.
Read articleRepresent positive and negative angles in standard position, convert between degree and radian measure with pi radians equal to 180 degrees, and use theta = l/r to find arc lengths, radii and angles at the centre.
Read articleDecide whether a relation is a function, learn the domain, range and graph of the identity, constant, polynomial, rational, modulus, signum and greatest-integer functions, find domains and ranges from formulas, and combine real functions.
Read articleForm Cartesian products and count their elements, find unknowns from equal ordered pairs, write relations in roster, arrow-diagram and set-builder form, and find the domain, co-domain and range of a relation and the number of relations.
Read articleChoose a universal set and draw Venn diagrams, find unions, intersections and differences, identify disjoint sets, take complements and verify De Morgan's laws, and use the laws of set algebra to simplify and prove identities.
Read articleWrite sets in roster and set-builder form, decide whether a collection is well defined, classify empty, finite and infinite sets, test equality, count subsets with 2^n, and describe subsets of real numbers as intervals.
Read articleLearn what random experiments, sample spaces and events are, calculate the probability of a simple event, use P(not A) = 1 - P(A) with impossible and sure events, and solve problems on cards, coloured balls and numbered tickets.
Read articleFind the median of raw data and the median class of grouped data, find the mode and modal class, work out quartiles and the inter-quartile range, and find a missing frequency or value from a given mean or median.
Read articleFind the mean of raw data and of a frequency table, then find the mean of grouped data by the direct method, the short-cut method and the step-deviation method — and see why all three give the same answer.
Read articleDraw histograms for grouped data, estimate the mode with the diagonal construction, build a cumulative frequency table and less-than ogive, and read the median, quartiles and counts above or below a value from the curve.
Read articleDefine and mark angles of elevation and depression, find heights and distances with one right triangle, solve two-triangle problems on towers, and work out river widths, distances between boats and the height of a pole on a building.
Read articleProve identities with fractions by taking the LCM or rationalising, eliminate theta from a pair of equations, use complementary-angle relations, and read four-figure trigonometric tables in both directions.
Read articleSimplify expressions with sin² A + cos² A = 1, use 1 + tan² A = sec² A and 1 + cot² A = cosec² A, prove identities by writing everything in sine and cosine, and evaluate expressions at the standard angles.
Read articleFind the surface area and volume of combined solids, use melting and recasting to find new dimensions or counts, work with hollow cylinders and spheres, and calculate how far water rises or falls when a solid is immersed.
Read articleFind the curved surface area, total surface area and volume of cylinders, cones, spheres and hemispheres, use slant height correctly, and work out the cost of painting, plastering or covering a solid.
Read articleConstruct a pair of tangents from an external point, circumscribe a circle about a triangle with perpendicular bisectors, inscribe a circle with angle bisectors, and draw both circles on a regular hexagon, checking every radius by calculation.
Read articleUse the intersecting chords theorem for chords meeting inside and outside a circle, find tangent lengths with PT² = PA × PB, and solve incircle and circumcircle problems using equal tangents.
Read articleUse the fact that a tangent is perpendicular to the radius, find lengths with equal tangents from an external point, handle circles that touch each other, and apply the alternate segment theorem to angles between a tangent and a chord.
Read articleUse the fact that opposite angles of a cyclic quadrilateral are supplementary, apply the exterior angle property, prove that four points are concyclic, and solve riders that mix cyclic properties with parallel lines and equal chords.
Read articleUse the theorem that the angle at the centre is double the angle at the circumference, find equal angles in the same segment, apply the right angle in a semicircle, and solve mixed problems with isosceles triangles and angle sums.
Read articleLearn what a locus is and describe loci in words, use the three standard loci, construct a perpendicular bisector and an angle bisector with ruler and compasses, and mark the points that satisfy two conditions at once.
Read articleUse the theorem on areas of similar triangles, find area ratios from medians, altitudes and perimeters, convert lengths, areas and volumes with map and model scales, and solve enlargement and reduction problems with a scale factor.
Read articleLearn the SSS, SAS and AA conditions for similar triangles, tell similarity from congruency, use the Basic Proportionality Theorem and its converse, and find unknown sides from the ratio of corresponding sides.
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