Unit Overview

Geometry and trigonometry

IB Mathematics Applications and Interpretation HL· 5 min read 📊 15-20% of overall exam

1. Unit at a glance

This unit follows a logical learning arc from foundational 2D geometry up to advanced 3D spatial reasoning and applied geometric modeling. We begin with coordinate geometry and triangle trigonometry, build to trigonometric functions and equation solving, then introduce vectors, finishing with the applied topic of Voronoi diagrams.

Concepts from this unit appear frequently in both multiple-choice and extended response questions, often in context-rich problems linking geometry to real-world scenarios like navigation, resource management, and periodic trend modeling.

All sub-topics in this unit are listed below:

2D and 3D coordinate geometry

Calculate distances, midpoints, and gradients between points in 2D and 3D space.

★★⏱ 8 min

Right triangle trigonometry

Use SOHCAHTOA to find unknown sides and angles in right triangles.

★⏱ 6 min

Non-right triangles: Law of Sines, Law of Cosines, area

Solve for unknown sides, angles and area in any general triangle.

★★⏱ 8 min

Unit circle, radian measure, trigonometric identities

Convert between radians and degrees, and use core Pythagorean identities.

★★★⏱ 10 min

Graphs of trigonometric functions

Sketch and interpret transformed sine, cosine and tangent graphs for modeling.

★★★⏱ 10 min

Solving trigonometric equations

Solve linear and quadratic trigonometric equations over a specified domain.

★★★★⏱ 12 min

Vectors: basics, operations, dot product

Learn vector notation, operations and use dot product to calculate angles between vectors.

★★★⏱ 12 min

Vector equation of a line, intersections

Write vector equations of lines and find intersection points of lines.

★★★★⏱ 10 min

Vector Applications to Kinematics

Model straight-line motion with time-dependent position vectors, and calculate speed and points of intersection.

★★★★⏱ 8 min

Voronoi diagrams: basic construction

Construct Voronoi diagrams using the perpendicular bisector method.

★★★⏱ 9 min

Voronoi diagrams: application problems

Solve nearest neighbor and largest empty circle optimization problems.

★★★★⏱ 10 min

2. Common Pitfalls

Wrong move:

Stopping at the first solution when solving trigonometric equations

Why:

Trigonometric functions are periodic, so equations almost always have multiple solutions over a given domain.

Correct move:

Use periodicity to find all valid solutions that fall within the specified interval.

Wrong move:

Accepting intersection points without verifying they lie on all the given lines

Why:

Solving the system of equations can produce extraneous solutions for skew or parallel lines.

Correct move:

Always substitute the calculated intersection back into all original equations to confirm it is valid.

Wrong move:

Miscalculating the maximum distance for largest empty circle problems on Voronoi diagrams

Why:

Students often forget that the center of the largest empty circle can lie on the edge of the diagram, not just at a Voronoi vertex.

Correct move:

Check both Voronoi vertices and boundary edges for the maximum empty circle center.

3. Quick Reference Cheatsheet

Concept / Formula	Description
Distance between 3D points: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$	Calculate straight-line distance between any two points in 3D space
Law of Cosines: $c^{2} = a^{2} + b^{2} - 2 ab cos C$	Find an unknown side or angle in any non-right triangle
Area of any triangle: $A r e a = \frac{1}{2} ab sin C$	Calculate area when you know two sides and the included angle
Pythagorean Identity: $sin^{2} θ + cos^{2} θ = 1$	Simplify expressions and solve trigonometric equations
Dot product: $a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$	Calculate the angle between two vectors and check for perpendicularity
Vector equation of a line: $r = a + t d$	Parametric description of a line with direction vector $d$ through point $a$
Voronoi diagram core property	Any point inside a cell is closer to the cell's site than any other site in the diagram

What's Next

Begin your study of this unit with the first sub-topic on 2D and 3D coordinate geometry to build your foundational knowledge. Work through each sub-topic in order, as later concepts build on earlier ones. Once you complete all sub-topics in this unit, you can move on to the first sub-topic of the next unit on statistics.