# Geometry & Trigonometry

> IB Mathematics: Analysis and Approaches Higher Level · IB AA HL
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u3-overview/
> Weight: 18-20% of total IB exam

This unit covers core trigonometry, coordinate geometry, and vector geometry for IB AA HL, with HL-only extensions to 3D vector planes, forming a foundational toolkit for pure and applied math problems.

**Prerequisites:** [Basic algebraic manipulation and coordinate geometry fundamentals](https://www.owlsprep.com/study/ib-math-aa-hl-u1-overview/); [Function properties and graphing from the previous unit](https://www.owlsprep.com/study/ib-math-aa-hl-u2-overview/)

## Learning objectives

- Model 2D and 3D spatial problems using trigonometry and vector methods
- Solve linear and quadratic trigonometric equations and apply core trigonometric identities
- Calculate angles, distances, and intersections between lines and planes in 3D space (HL)
- Prove geometric properties using coordinate, vector, congruence, and similarity approaches
- Apply triangle trigonometry rules to solve applied and abstract geometric problems

## Unit at a glance

This unit builds incrementally from basic trigonometric definitions up to advanced 3D vector geometry for HL. We start with angle measurement and triangle trigonometry, then move to trigonometric equations and identities before shifting to coordinate and vector geometry for 2D and 3D problems.

Core content is accessible to all AA HL students, with HL-only extensions clearly marked. The 13 sub-topics below build on each other, so we recommend working through them in order to master all concepts for the exam.

Full list of sub-topics in this unit:
- [Radians and the unit circle](https://www.owlsprep.com/study/ib-math-aa-hl-u3-radians-and-the-unit-circle/) — Learn radian angle measurement and define trigonometric functions using the unit circle.
- [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-right-triangle-trigonometry/) — Define sine, cosine, and tangent for right triangles and solve for unknown sides and angles.
- [Non-right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-non-right-triangle-trigonometry/) — Apply the sine rule, cosine rule, and area formula to any non-right triangle.
- [Trigonometric equations](https://www.owlsprep.com/study/ib-math-aa-hl-u3-trigonometric-equations/) — Solve linear and quadratic trigonometric equations over specified domains.
- [Compound and double angle identities](https://www.owlsprep.com/study/ib-math-aa-hl-u3-compound-and-double-angle-identities/) — Use addition and double angle formulas to simplify expressions and solve equations.
- [2D coordinate geometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-2d-coordinate-geometry/) — Explore lines, circles, and geometric properties on the 2D coordinate plane.
- [3D coordinate geometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-3d-coordinate-geometry/) — Extend coordinate concepts to 3-dimensional coordinate systems.
- [Vector fundamentals and dot product](https://www.owlsprep.com/study/ib-math-aa-hl-u3-vector-fundamentals-and-dot-product/) — Learn vector notation, operations, and the dot product for angles and magnitudes.
- [Cross product of vectors (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u3-cross-product-of-vectors/) — Calculate cross products and use them to find areas and normals to planes.
- [Vector equations of lines and planes (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u3-vector-equations-of-lines-and/) — Write and interpret vector forms for lines and planes in 3D space.
- [Intersections and angles between lines and planes (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u3-intersections-and-angles-between-lines/) — Calculate intersection points/lines and angles between lines and planes.
- [Distance from point to plane (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u3-distance-from-point-to-plane/) — Use vector formulas to find the shortest distance from a point to a plane.
- [Congruence and similarity](https://www.owlsprep.com/study/ib-math-aa-hl-u3-congruence-and-similarity/) — Prove geometric properties using congruence and similarity criteria for triangles.

## Common pitfalls

- **Wrong:** Forgetting all solutions when solving trigonometric equations over large domains
  - Why it fails: Trigonometric functions are periodic, so multiple solutions usually exist
  - Correct: Always check your solution interval and list all valid angles
- **Wrong:** Mixing up degrees and radians in calculator calculations
  - Why it fails: Most IB trigonometry questions require radians, but calculators default to degrees
  - Correct: Confirm your calculator mode before starting any calculation
- **Wrong:** Omitting magnitude terms when using the dot product to find angles
  - Why it fails: The formula relies on normalized vectors to get the correct cosine value
  - Correct: Always divide by the product of the vector magnitudes

## Cheatsheet

| Concept | Key Formula |
| --- | --- |
| Sine Rule | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ |
| Cosine Rule | $c^2 = a^2 + b^2 - 2ab \cos C$ |
| Compound Sine Identity | $\sin(A+B) = \sin A \cos B + \cos A \sin B$ |
| Double Angle Cosine | $\cos 2\theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$ |
| Dot Product | $\mathbf{a} \cdot \mathbf{b} = \|a\|\|b\|\cos \theta = a_1b_1 + a_2b_2 + a_3b_3$ |
| Cross Product Magnitude | $\|\mathbf{a} \times \mathbf{b}\| = \|a\|\|b\|\sin \theta$ (area of parallelogram) |
| Vector Plane Equation | $\mathbf{r} \cdot \mathbf{n} = d$, where $\mathbf{n}$ is the unit normal |
| Distance Point to Plane | $\frac{\|ax_0 + by_0 + cz_0 + d\|}{\sqrt{a^2 + b^2 + c^2}}$ |

## What's next

Start with the first sub-topic of this unit to build your foundational trigonometry knowledge, working through each sub-topic in order as concepts build incrementally. HL-only content is clearly marked, so you can focus on core content first if needed. After completing this full unit, you will move on to the next unit covering calculus fundamentals, starting with differentiation.

- [Radians and the unit circle (first sub-topic of this unit)](https://www.owlsprep.com/study/ib-math-aa-hl-u3-radians-and-the-unit-circle/)
- [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-right-triangle-trigonometry/)
- [Non-right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-non-right-triangle-trigonometry/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-aa-hl-u3-overview/