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3 Geometry and trigonometry

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  1. 3.1 Geometry and trigonometry - SL content

    1. SL 3.1Three-dimensional geometry

      • Use distance and midpoint in three-dimensional space. • Use volume and surface area of 3D solids and combinations. • Identify right-angled triangles in 3D objects to find lengths and angles.

    2. SL 3.2Triangle trigonometry

      • Use sine, cosine and tangent in right-angled triangles. • Use sine rule, cosine rule and area formula 1/2 ab sin C. • Ambiguous sine-rule case is not included at SL.

    3. SL 3.3Applications of trigonometry

      • Apply right and non-right triangle trigonometry, including Pythagoras' theorem. • Use elevation/depression, bearings and labelled diagrams from written statements.

    4. SL 3.4Arc length and sector area

      • Find circle arc length and sector area. • Radians are not required at SL.

    5. SL 3.5Perpendicular bisectors

      • Find equations of perpendicular bisectors. • Use two points, or a line segment equation and midpoint.

    6. SL 3.6Voronoi diagrams

      • Use sites, vertices, edges and cells in Voronoi diagrams. • Add a site, use nearest-neighbour interpolation and solve toxic-waste-dump style problems. • Find boundaries, closest sites and region areas; applications include planning, disease, ecology and meteorology.

  2. 3.2 Geometry and trigonometry - AHL content

    1. AHL 3.7 (HL)—Radian measure

      • Define radians and convert between degrees and radians. • Use radians for arc length and sector area.

    2. AHL 3.8 (HL)—Unit circle and trigonometric equations

      • Define sin theta and cos theta using the unit circle; tan theta=sin theta/cos theta. • Use cos^2 theta + sin^2 theta = 1 and ambiguous sine-rule case. • Construct sin x and cos x graphs from the unit circle; solve trig equations graphically on finite intervals.

    3. AHL 3.9 (HL)—Matrix transformations and fractals

      • Use matrices for 2D transformations: reflections, stretches, enlargements, translations and rotations. • Compose transformations and interpret determinant as area scale factor. • Use iterative techniques to generate fractals.

    4. AHL 3.10 (HL)—Vectors

      • Use vectors/scalars, directed line segments, unit vectors and base vectors i, j, k. • Use components, column representation, vector algebra, magnitude, normalization and resultants.

    5. AHL 3.11 (HL)—Vector equations of lines

      • Use vector equation r=a+lambda b in 2D and 3D. • Convert vector line equations to parametric form.

    6. AHL 3.12 (HL)—Vector kinematics

      • Model constant-velocity linear motion in 2D/3D with r=r0+vt. • Find positions, intersections, paths, closest approach times and distances. • Include variable velocity in 2D; projectile and circular motion are special cases.

    7. AHL 3.13 (HL)—Scalar and vector products

      • Use scalar product to find angles and test perpendicular vectors. • Use vector product, right-hand rule and |v x w| for parallelogram/triangle area. • Resolve components parallel and perpendicular to another vector.

    8. AHL 3.14 (HL)—Graph theory basics

      • Use graph terminology: vertices, edges, adjacent items, degree, simple/complete/weighted graphs. • Represent real-world structures as weighted or unweighted graphs. • Know connected, strongly connected, directed graphs, in/out degree, subgraphs and trees.

    9. AHL 3.15 (HL)—Adjacency matrices and transition matrices

      • Use adjacency matrices to count k-length walks using powers of a matrix. • Use weighted adjacency tables. • Construct transition matrices for strongly connected directed/undirected graphs.

    10. AHL 3.16 (HL)—Graph algorithms

      • Use walks, trails, paths, circuits, cycles, Eulerian trails/circuits and Hamiltonian paths/cycles. • Use Kruskal's and Prim's algorithms for minimum spanning trees. • Solve Chinese postman problems for weighted graphs with up to four odd vertices. • Use nearest-neighbour and deleted-vertex algorithms for travelling salesman bounds.