Topic 3: Geometry and Trigonometry

Concepts

• Essential understandings

  • Geometry and trigonometry allow us to quantify the physical world, enhancing our spatial awareness in two and three dimensions. This branch provides us with the tools for analysis, measurement and transformation of quantities, movements and relationships.

• Suggested concepts embedded in this topic

  • Generalization, space, relationships, systems, representations

  • AHL: Quantity, change.

• Content-specific conceptual understandings:

  • The properties of shapes are highly dependent on the dimension they occupy in space.

  • Volume and surface area of shapes are determined by formulae, or general mathematical relationships or rules expressed using symbols or variables.

  • The relationships between the length of the sides and the size of the angles in a triangle can be used to solve many problems involving position, distance, angles and area.

  • Different representations of trigonometric expressions help to simplify calculations.

  • Systems of equations often, but not always, lead to intersection points.

  • In two dimensions, the Voronoi diagram allows us to navigate, path-find or establish an optimum position.

  • AHL: Different measurement systems can be used for angles to facilitate ease of calculation.

  • AHL: Vectors allow us to determine position, change of position (movement) and force in two and three- dimensional space.

  • AHL: Graph theory algorithms allow us to represent networks and to model complex real-world problems.

  • AHL: Matrices are a form of notation which allow us to show the parameters or quantities of several linear equations simultaneously.

SL Content

• SL 3.1

  • The distance between two points in three- dimensional space, and their midpoint.

  • Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.

  • The size of an angle between two intersecting lines or between a line and a plane.

• SL 3.2

  • Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

  • The sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

  • The cosine rule: $c^2=a^2+b^2-2ab \cos C$

  • $\cos C=\frac{a^2+b^2-c^2}{2ab}$

  • Area of a triangle as $\frac{1}{2} ab \sin C$

• SL 3.3

  • Applications of right and non-right angled trigonometry, including Pythagoras’ theorem.

  • Angles of elevation and depression.

  • Construction of labelled diagrams from written statements.

• SL 3.4

  • The circle: length of an arc; area of a sector.

• SL 3.5

  • Equations of perpendicular bisectors.

• SL 3.6

  • Voronoi diagrams: sites, vertices, edges, cells.

  • Addition of a site to an existing Voronoi diagram.

  • Nearest neighbour interpolation.

  • Applications of the “toxic waste dump” problem.

AHL Content

• AHL 3.7

  • The definition of a radian and conversion between degrees and radians.

  • Using radians to calculate area of sector, length of arc.

• AHL 3.8

  • The definitions of $\cos \theta$ and $\sin \theta$ in terms of the unit circle.

  • The Pythagorean identity: $\cos^2 \theta + \sin^2 \theta = 1$

  • Definition of $\tan \theta$ as $\frac{\sin \theta}{\cos \theta}$

  • Extension of the sine rule to the ambiguous case.

  • Graphical methods of solving trigonometric equations in a finite interval.

• AHL 3.9

  • Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations.

  • Compositions of the above transformations.

  • Geometric interpretation of the determinant of a transformation matrix.

• AHL 3.10

  • Concept of a vector and a scalar.

  • Representation of vectors using directed line segments.

  • Unit vectors; base vectors $i, j, k$ . Components of a vector; column representation; $v= v_1 i + v_2 j + v_3 k$

  • The zero vector $\textbf{0}$ , the vector $-\textbf{v}$ .

  • Position vectors $\overrightarrow{OA}=a$

  • Rescaling and normalizing vectors.

• AHL 3.11

  • Vector equation of a line in two and three dimensions: $r=a+ \lambda b$ , where $b$ is a direction vector of the line.

• AHL 3.12

  • Vector applications to kinematics.

  • Modelling linear motion with constant velocity in two and three dimensions.

  • Motion with variable velocity in two dimensions.

• AHL 3.13

  • Definition and calculation of the scalar product of two vectors.

  • The angle between two vectors; the acute angle between two lines.

  • Definition and calculation of the vector product of two vectors.

  • Geometric interpretation of $|\mathbf{v} \times \mathbf{w}|$ .

  • Components of vectors.

• AHL 3.14

  • Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges. Degree of a vertex.

  • Simple graphs; complete graphs; weighted graphs.

  • Directed graphs; in degree and out degree of a directed graph.

  • Subgraphs; trees.

• AHL 3.15

  • Adjacency matrices. Walks.

  • Number of $k$-length walks (or less than $k$-length walks) between two vertices.

  • Weighted adjacency tables.

  • Construction of the transition matrix for a strongly- connected, undirected or directed graph.

• AHL 3.16

  • Tree and cycle algorithms with undirected graphs.

  • Walks, trails, paths, circuits, cycles.

  • Eulerian trails and circuits. Hamiltonian paths and cycles.

  • Minimum spanning tree (MST) graph algorithms:

  • Kruskal’s and Prim’s algorithms for finding minimum spanning trees.

  • Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once.

  • Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph.

  • Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem.

  • Deleted vertex algorithm for determining a lower bound for the travelling salesman problem.