3 Geometry and trigonometry

Introduction

Geometry and trigonometry measure the physical world: the height of a building that cannot be climbed, the volume of a storage tank, the bearing a ship must steer, the region of a city closest to each hospital. At SL the topic works with triangles, three-dimensional solids, circles and the boundaries between nearest points, with every angle in degrees. Almost every problem starts by turning a written description into a labelled diagram, then choosing the right relationship between lengths and angles.

At HL the topic gains more new material than any other and reaches well beyond triangles. Radians give a second way to measure angles, vectors describe position and movement in two and three dimensions, matrices transform shapes, and graph theory models networks such as roads, cables and web links, where the question is often the shortest or cheapest route.

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How the subtopics fit together

The subtopics form five strands. The map of the topic shows each strand as a row, with the SL subtopics shaded and the HL-only subtopics dashed; an arrow means the first subtopic is used in the second.

The triangles strand is the core of SL. Right-angled trigonometry, the sine rule and the cosine rule find any side or angle, and their applications cover angles of elevation and depression and three-figure bearings. Right-angled triangles drawn inside the solids of subtopic 3.1 give their hidden lengths and angles, and the area formula $\frac{1}{2}ab\sin C$ gives the area of a segment of a circle in subtopic 3.4.

The bisectors and Voronoi strand applies coordinate geometry to questions of location, such as which hospital or mobile phone mast is nearest. The perpendicular bisector of the segment joining two points is the line of points equally far from both, so it forms the boundaries of a Voronoi diagram, which splits a plane into the regions nearest to each of a set of sites. At HL, the two remaining strands use vectors and matrices to describe lines, motion and transformations, and graphs of vertices and edges to find efficient routes.

The subtopics

Links to other topics

Perpendicular bisectors rely on the gradients and perpendicular lines of subtopic 2.1. The sinusoidal models of subtopics 2.5 and 2.9 are built on the sine and cosine functions, which subtopic 3.8 extends to angles of any size through the unit circle. Volumes of solids return in subtopic 5.12, where integration finds the volume of a solid formed by rotating a curve.

The HL strands lean on topic 1: transformation matrices use the matrix algebra of subtopic 1.14, and the transition matrix of a graph in subtopic 3.15 becomes a Markov chain in subtopic 4.19. Motion described by vectors in subtopic 3.12 is the two- and three-dimensional version of the calculus kinematics in subtopic 5.13.