How the geometry and trigonometry topic fits together, from distances, solids and triangles to radians, circular functions and identities and, at HL, vectors, lines and planes in three dimensions.
Each one has its own page once something is published for it.
Geometry is the study of shape, size and position in space; trigonometry links the sides of triangles to their angles and then extends those links to periodic behaviour of every kind. Together they measure the world: the volume of a building, the height of a mountain, the course of a ship, the length of an arc, the rise and fall of a tide. This topic builds the tools for these measurements and the functions that describe repeating patterns.
The circular functions and are transformed like any function in Topic 2, and calculus in Topic 5 differentiates and integrates them, using radians throughout. At HL, vectors connect to the complex numbers and systems of equations of Topic 1.
The topic begins with measurement in two and three dimensions, moves from triangles to the circle, and then develops trigonometry as a set of functions. The HL subtopics extend the functions and identities, then introduce vectors.
Subtopic | What it covers |
|---|---|
3.1 Distance, midpoint, volume and surface area | Distance and midpoint in 3D, volume and surface area of pyramids, cones, spheres and composite solids, angles inside solids |
3.2 Trigonometry in triangles | Sine, cosine and tangent in right-angled triangles, the sine rule, the cosine rule and |
3.3 Applications of trigonometry | Angles of elevation and depression, bearings, surveying and problems in 3D |
3.4 Radians, arcs and sectors | Radian measure, arc length and sector area |
3.5 The unit circle | Sine and cosine of any angle, exact values, the ambiguous case of the sine rule |
3.6 Trigonometric identities | , double angle identities, proving identities |
3.7 Circular functions | Graphs of , and , amplitude, period and sinusoidal models |
3.8 Solving trigonometric equations | All solutions in an interval, graphically and analytically, including quadratic types |
3.9 Reciprocal and inverse trigonometric functions (HL) | , , , their identities, and , , |
3.10 Compound angle identities (HL) | , , and |
3.11 Symmetry of trigonometric graphs (HL) | Odd and even properties, translations between graphs, using symmetry to solve |
3.12 Vectors (HL) | Position and displacement vectors, components, magnitude, unit vectors, geometric proof |
3.13 The scalar product (HL) | , the angle between vectors, perpendicular vectors |
3.14 Vector equations of lines (HL) | Vector, parametric and Cartesian forms, angles between lines, simple kinematics |
3.15 Coincident, parallel, intersecting and skew lines (HL) | Classifying pairs of lines, intersections and shortest distances |
3.16 The vector product (HL) | , its properties, areas of parallelograms and triangles |
3.17 Vector equations of planes (HL) | Vector, normal and Cartesian forms of a plane |
3.18 Intersections of lines and planes (HL) | Lines meeting planes, two and three planes, angles between lines and planes |
Subtopics 3.1 to 3.3 work with fixed shapes: every angle is between and , and the main tools are Pythagoras's theorem, the trigonometric ratios, the sine rule and the cosine rule. Subtopic 3.4 introduces the radian, the natural unit for angles in the rest of the course. The unit circle in 3.5 then defines sine and cosine for any angle, which turns them into the periodic functions of 3.7 and leads to the identities of 3.6 and the equations of 3.8.
The HL subtopics follow two strands. Subtopics 3.9 to 3.11 extend the trigonometric functions and identities, preparing for the calculus of these functions in Topic 5. Subtopics 3.12 to 3.18 develop vectors step by step, from the definition of a vector to the intersections of lines and planes, and give a second, algebraic approach to the three-dimensional geometry of 3.1.
Space and quantity. Distance, area, volume and angle are the basic measurements of space. Many three-dimensional problems are solved by finding a flat triangle inside a solid.
Relationships. The sine rule, the cosine rule and the trigonometric ratios express fixed relationships between the sides and angles of a triangle, which is why a few measurements determine all the others.
Representation. The same angle can be written in degrees or radians, the same point by coordinates or by a position vector, and the same line by a vector, parametric or Cartesian equation.
Generalization. The unit circle generalizes sine and cosine from acute angles in triangles to all real numbers, and the cosine rule generalizes Pythagoras's theorem to every triangle.
Modelling and change. Sinusoidal functions model quantities that repeat in cycles, such as tides, daylight hours and the height of a seat on a Ferris wheel.
SL students study subtopics 3.1 to 3.8. HL students study all of these and the ten HL subtopics, 3.9 to 3.18. The SL subtopics are the foundation for HL: the double angle identities of 3.6 are special cases of the compound angle identities of 3.10, and the angles inside solids in 3.1 return in 3.18 as angles between lines and planes found with vectors.
The entries of the mathematics: analysis and approaches formula booklet that 3 uses. Each opens in the explained booklet.
And 38 more in the formula booklet.
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