# Geometry and trigonometry

> IB Mathematics Applications and Interpretation HL · IB AI HL
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u3-overview/
> Weight: 15-20% of overall exam

This unit covers core geometric and trigonometric concepts tested in both papers of IB AI HL, with applications ranging from modeling periodic motion to spatial resource allocation.

**Prerequisites:** [Foundational algebra and functions (Unit 2)](https://www.owlsprep.com/study/ib-math-ai-hl-u2-overview/)

## Learning objectives

- Apply coordinate, trigonometric and vector methods to solve 2D and 3D geometric problems
- Model periodic real-world phenomena using transformed trigonometric functions
- Construct and interpret Voronoi diagrams to solve spatial optimization problems
- Solve for intersections of lines in 3D space using vector equations

## 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](https://www.owlsprep.com/study/ib-math-ai-hl-u3-2d-and-3d-coordinate-geometry/) — Calculate distances, midpoints, and gradients between points in 2D and 3D space.
- [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-hl-u3-right-triangle-trigonometry/) — Use SOHCAHTOA to find unknown sides and angles in right triangles.
- [Non-right triangles: Law of Sines, Law of Cosines, area](https://www.owlsprep.com/study/ib-math-ai-hl-u3-non-right-triangles-law-of/) — Solve for unknown sides, angles and area in any general triangle.
- [Unit circle, radian measure, trigonometric identities](https://www.owlsprep.com/study/ib-math-ai-hl-u3-unit-circle-radian-measure-trigonometric/) — Convert between radians and degrees, and use core Pythagorean identities.
- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-ai-hl-u3-graphs-of-trigonometric-functions/) — Sketch and interpret transformed sine, cosine and tangent graphs for modeling.
- [Solving trigonometric equations](https://www.owlsprep.com/study/ib-math-ai-hl-u3-solving-trigonometric-equations/) — Solve linear and quadratic trigonometric equations over a specified domain.
- [Vectors: basics, operations, dot product](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vectors-basics-operations-dot-product/) — Learn vector notation, operations and use dot product to calculate angles between vectors.
- [Vector equation of a line, intersections](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vector-equation-of-a-line/) — Write vector equations of lines and find intersection points of lines.
- [Vector Applications to Kinematics](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vector-kinematics/) — Model straight-line motion with time-dependent position vectors, and calculate speed and points of intersection.
- [Voronoi diagrams: basic construction](https://www.owlsprep.com/study/ib-math-ai-hl-u3-voronoi-diagrams-basic-construction/) — Construct Voronoi diagrams using the perpendicular bisector method.
- [Voronoi diagrams: application problems](https://www.owlsprep.com/study/ib-math-ai-hl-u3-voronoi-diagrams-application-problems/) — Solve nearest neighbor and largest empty circle optimization problems.

## Common pitfalls

- **Wrong:** Stopping at the first solution when solving trigonometric equations
  - Why it fails: Trigonometric functions are periodic, so equations almost always have multiple solutions over a given domain.
  - Correct: Use periodicity to find all valid solutions that fall within the specified interval.
- **Wrong:** Accepting intersection points without verifying they lie on all the given lines
  - Why it fails: Solving the system of equations can produce extraneous solutions for skew or parallel lines.
  - Correct: Always substitute the calculated intersection back into all original equations to confirm it is valid.
- **Wrong:** Miscalculating the maximum distance for largest empty circle problems on Voronoi diagrams
  - Why it fails: 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: Check both Voronoi vertices and boundary edges for the maximum empty circle center.

## Cheatsheet

| Concept / Formula | Description |
| --- | --- |
| Distance between 3D points: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ | Calculate straight-line distance between any two points in 3D space |
| Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$ | Find an unknown side or angle in any non-right triangle |
| Area of any triangle: $Area = \frac{1}{2}ab \sin C$ | Calculate area when you know two sides and the included angle |
| Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$ | Simplify expressions and solve trigonometric equations |
| Dot product: $\vec{a} \cdot \vec{b} = \|a\|\|b\| \cos \theta = a_1b_1 + a_2b_2 + a_3b_3$ | Calculate the angle between two vectors and check for perpendicularity |
| Vector equation of a line: $\vec{r} = \vec{a} + t\vec{d}$ | Parametric description of a line with direction vector $\vec{d}$ through point $\vec{a}$ |
| 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.

- [2D and 3D coordinate geometry](https://www.owlsprep.com/study/ib-math-ai-hl-u3-2d-and-3d-coordinate-geometry/)
- [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-hl-u3-right-triangle-trigonometry/)
- [Non-right triangles: Law of Sines, Law of Cosines, area](https://www.owlsprep.com/study/ib-math-ai-hl-u3-non-right-triangles-law-of/)

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