Topic 3: Geometry and Trigonometry

Concepts

• Essential understandings

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

• Suggested concepts embedded in this topic

  • Generalization, space, relationships, equivalence, representation,

  • AHL: Quantity, Modelling.

• Content-specific conceptual understandings:

  • The properties of shapes depend 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.

  • Equivalent measurement systems, such as degrees and radians, can be used for angles to facilitate ease of calculation.

  • Different representations of the values of trigonometric relationships, such as exact or approximate, may not be equivalent to one another.

  • The trigonometric functions of angles may be defined on the unit circle, which can visually and algebraically represent the periodic or symmetric nature of their values.

  • AHL: Position and movement can be modelled in three-dimensional space using vectors.

  • AHL: The relationships between algebraic, geometric and vector methods can help us to solve problems and quantify those positions and movements.

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: $\cos C=\frac{a^2+b^2-c^2}{2ab}$ ; $f(x)=\frac{ax+b}{cx^2+dx+e'}$ , and $f(x)=\frac{ax^2+bx+c}{dx+}$
  • 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: radian measure of angles; length of an arc; area of a sector.

• SL 3.5

  • Definition of $\cos \theta, \sin \theta$ in terms of the unit circle.
  • Definition of $\tan \theta$ as $\frac{\sin \theta}{\cos \theta}$ .
  • Exact values of trigonometric ratios of $0, \frac{pi}{6}, \frac{pi}{4}, \frac{pi}{3}, \frac{pi}{2}$ and their multiples.
  • Extension of the sine rule to the ambiguous case.

• SL 3.6

  • The Pythagorean identity $\cos^2 \theta + \sin^2 \theta=1$ . Double angle identities for sine and cosine.
  • The relationship between trigonometric ratios.

• SL 3.7

  • The circular functions $\sin x$ , $\cos x$ , and $\tan x$ ; amplitude, their periodic nature, and their graphs
  • Composite functions of the form $f(x)=a \sin (b(x+c))+d$
  • Transformations.

  • Real-life contexts.

• SL 3.8

  • Solving trigonometric equations in a finite interval, both graphically and analytically.

  • Equations leading to quadratic equations in $\sin x$ , $\cos x$ or $\tan x$

AHL Content

• AHL 3.9

  • Definition of the reciprocal trigonometric ratios $\sec \theta$ , $\csc \theta$ and $\cot \theta$
  • Pythagorean identities: $1 + \tan^2 \theta = \sec^2 \theta$ , $1 + \cot^2 \theta = \csc^2 \theta$
  • The inverse functions $f(x)=\arcsin x, f(x) = \arccos x, f(x) = \arctan x$ ; their domains and ranges; their graphs.
• AHL 3.10

  • Compound angle identities.

  • Double angle identity for $tan$ .
• AHL 3.11

  • Relationships between trigonometric functions and the symmetry properties of their graphs.

• AHL 3.12

  • Concept of a vector; position vectors; displacement vectors.

  • Representation of vectors using directed line segments.

  • Base vectors $i, j, k$
  • Components of a vector: $v=v_1 i + v_2 j + v_3 k$
  • Algebraic and geometric approaches to the following:

    • the sum and difference of two vectors

    • the zero vector $\mathbf{0}$ , the vector $-\mathbf{v}$
    • multiplication by a scalar, $k \mathbf{v}$ , parallel vectors
    • magnitude of a vector, $| \mathbf{v} |$ ; unit vectors, $\frac{\mathbf{v}}{|\mathbf{v}|}$
    • position vectors $\overrightarrow{OA}=\mathbf{a}, \overrightarrow{OB}=\mathbf{b}$
    • displacement vector $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$
  • Proofs of geometrical properties using vectors.

• AHL 3.13

  • The definition of the scalar product of two vectors.

  • The angle between two vectors.

  • Perpendicular vectors; parallel vectors.

• AHL 3.14

  • Vector equation of a line in two and three dimensions: $r=a + \lambda b$
  • The angle between two lines.

  • Simple applications to kinematics.

• AHL 3.15

  • Coincident, parallel, intersecting and skew lines, distinguishing between these cases.

  • Points of intersection.

• AHL 3.16

  • The definition of the vector product of two vectors.

  • Properties of the vector product.

  • Geometric interpretation of $|\mathbf{v} \times \mathbf{w}|$
• AHL 3.17

  • Vector equations of a plane: $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}$ , where $\mathbf{b}$ and $\mathbf{c}$ are non-parallel vectors within the plane.
  • $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$ , where $\mathbf{n}$ is a normal to the plane and a is the position vector of a point on the plane.
  • Cartesian equation of a plane $ax + by + cz = d$ .
• AHL 3.18

  • Intersections of: a line with a plane; two planes; three planes.

  • Angle between: a line and a plane; two planes.