From the triangle hiding inside a cuboid to the graph theory that decides a postal route. The widest topic on the course, and the one where drawing the diagram is most often the mark.
The first three sub-topics are stated in both guides as content common to Analysis and Applications, so those pages serve both courses and are not duplicated. From 3.4 onwards this is Applications only.
3.1 · Three-dimensional geometry shared Distance and midpoint in three dimensions, volume and surface area of solids, and the angle between a line and a plane. Three triangles on one cuboid, where the two most people draw are both too big. 3.2 · The sine and cosine rules shared Right-angled ratios, the sine rule, the cosine rule and the area of a triangle. The ambiguous case is not set at Standard Level. The sine rule hands back the wrong angle, and the angle sum is the only thing that tells you. 3.3 · Applications, including bearings shared Right and non-right angled trigonometry in context, with Pythagoras, bearings, elevation and depression. The angle at the turn is neither bearing, and the wrong one lands the boat 1.04 km short. 3.4 · Arcs and sectors Length of an arc and area of a sector. Degrees only: radians are not required at Standard Level on this course. Forget the fraction on the arc and you get the number that is the area. 3.5 · Perpendicular bisectors The equation of a perpendicular bisector from two points, or from a line segment and its midpoint. Every point on it is equidistant from both, which is the check and the definition. 3.6 · Voronoi diagrams Sites, vertices, edges and cells. Adding a site, nearest neighbour interpolation, and the toxic waste dump problem. The middle is 4.00 km from the nearest depot. The right answer is 4.33.Ten more sub-topics that only Higher Level students take. They sit on top of everything above. The run from 3.10 to 3.13 builds in order, and so does 3.14 to 3.16: a graph becomes an adjacency matrix, the matrix answers questions about walks, and the algorithms then solve real routing problems.
Radian measure is assumed on Higher Level papers unless a question says otherwise, which is why 3.7 comes first.
3.7 · Radians HL The definition of a radian, converting to and from degrees, and arc length and sector area with the fraction already folded in. Put 60 where π/3 belongs and the sector becomes nine and a half whole circles. 3.8 · The unit circle HL Cosine and sine as the coordinates of a point, the Pythagorean identity, and the ambiguous case that Standard Level left out. a = 7, b = 10, A = 40° gives two real triangles, third sides 10.43 and 4.89. 3.9 · Transformations with matrices HL Reflections, rotations, stretches and enlargements written as matrices, compositions, and what the determinant does and does not measure. Rotate-then-reflect and reflect-then-rotate both have determinant −1. 3.10 · Vectors and scalars HL Components, base vectors, magnitude, unit vectors, position vectors and the vector between two points. A 3 and a 4 combine to anything from 1 to 7, and to 7 exactly once. 3.11 · The vector equation of a line HL r = a + tb in two and three dimensions, the parameter, and testing whether a point lies on a line. (1,2) + t(3,4) and (4,6) + s(6,8) share no number and are the same line. 3.12 · Vectors and kinematics HL Position, velocity and speed, linear motion at constant velocity, and the time and distance of closest approach. The obvious moment gives 3.33; the closest they ever get is 3.16, at t = 3. 3.13 · The scalar and vector products HL The dot product and the angle between two vectors or lines, the cross product, and the area its magnitude measures. A dot product of 0 is the perpendicularity result, not a failed sum. 3.14 · Graph theory HL Vertices, edges, degree, simple and complete and weighted graphs, directed graphs, subgraphs, trees and cycles. Degrees 3, 3, 3, 2, 2, 2 sum to 15, so no such graph exists at all. 3.15 · Adjacency matrices HL Writing a graph as a matrix, counting walks of length k with the kth power, weighted tables and transition matrices. A has a zero diagonal and A² does not, and both are correct. 3.16 · Route algorithms HL Trails, paths, circuits and cycles. Eulerian and Hamiltonian routes, Kruskal and Prim, the Chinese postman, and the salesman's two bounds. Every step the cheapest available, and the tour is still one too long.Sub-topics 3.1 to 3.3 above are yours too, word for word, because both guides state them as common content. Analysis then continues with its own 3.4 onwards, including the unit circle, identities and vectors at Higher Level. Those pages live at the Analysis index.
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