# Geometry & Trigonometry

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u3-overview/
> Weight: 18-21% of overall exam

This core unit covers spatial geometry and trigonometry for IB AA SL, from coordinate systems and triangles to trigonometric functions and identities, and is heavily tested across all exam papers.

**Prerequisites:** Knowledge of basic coordinate geometry and algebraic manipulation; Understanding of linear functions and domain/range of functions

## Learning objectives

- Apply coordinate geometry methods to solve 2D and 3D spatial problems
- Use trigonometric rules and identities to solve problems involving triangles and general angles
- Sketch, interpret and model with graphs of trigonometric functions
- Solve trigonometric equations over specified domains aligned with IB exam requirements

## Unit at a Glance

This unit builds incrementally from basic spatial reasoning up to advanced trigonometric problem-solving. We start with coordinate geometry and core triangle trigonometry, then move to properties of trigonometric functions, identities, and equation solving over specified domains.

Geometry and trigonometry appear across all IB AA SL exam papers, in both multiple choice and extended response questions. Building fluency with key formulas and methods is critical for achieving high scores in this subject.

This unit is split into the following structured sub-topics:
- [2D and 3D coordinate geometry](https://www.owlsprep.com/study/ib-math-aa-sl-u3-2d-and-3d-coordinate-geometry/) — Calculate distances, midpoints, gradients and equations of lines in 2D and 3D space.
- [Right triangle trigonometry and the unit circle](https://www.owlsprep.com/study/ib-math-aa-sl-u3-right-triangle-trigonometry-and-the/) — Define trigonometric ratios for right triangles and extend to any angle using the unit circle.
- [Radians, arc length and sectors](https://www.owlsprep.com/study/ib-math-aa-sl-u3-radians-arc-length-and-sectors/) — Convert between degrees and radians, then use radian measure to find arc length and the area of a sector.
- [Sine rule, cosine rule and area of triangles](https://www.owlsprep.com/study/ib-math-aa-sl-u3-sine-rule-cosine-rule-and/) — Apply core rules to find unknown sides, angles and areas of any triangle, including the ambiguous sine rule case.
- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-sl-u3-graphs-of-trigonometric-functions/) — Identify amplitude, period, phase shift and vertical shift, and sketch/interpret sine, cosine and tangent graphs.
- [Pythagorean and double angle trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-sl-u3-pythagorean-and-double-angle-trigonometric/) — Simplify expressions and solve problems using core trigonometric identities.
- [Solving trigonometric equations](https://www.owlsprep.com/study/ib-math-aa-sl-u3-solving-trigonometric-equations/) — Solve linear and quadratic trigonometric equations over specified domains.

## Common pitfalls

- **Wrong:** Forgetting the ambiguous case of the sine rule for SSA triangle problems
  - Why it fails: Missing the second valid angle solution leads to incomplete marks on exam questions
  - Correct: Always check if a second obtuse solution $180^\circ - \theta$ is possible by verifying the total angle sum is less than $180^\circ$
- **Wrong:** Mixing up period formulas for transformed trigonometric functions
  - Why it fails: Incorrect period calculation leads to wrong graph sketches and equation solutions
  - Correct: Period is $\frac{2\pi}{|b|}$ for sine/cosine, and $\frac{\pi}{|b|}$ for tangent, regardless of phase/vertical shift

## Cheatsheet

| Concept | Key Formula |
| --- | --- |
| 3D Distance between two points | $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ |
| Sine Rule | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ |
| Cosine Rule | $c^2 = a^2 + b^2 - 2ab\cos C$ |
| Area of any triangle | $\text{Area} = \frac{1}{2}ab\sin C$ |
| Pythagorean Trig Identity | $\sin^2 \theta + \cos^2 \theta = 1$ |
| Period of transformed sine/cosine | $\text{Period} = \frac{2\pi}{\|B\|}$ for $y = A\sin(Bx + C) + D$ |

## What's next

Begin your study of this unit with the first sub-topic on 2D and 3D coordinate geometry, which forms a foundation for all later topics in this unit, including trigonometric spatial problems. Once you complete all sub-topics in this unit, you can move on to the next core unit on statistics and probability for IB AA SL.

- [2D and 3D coordinate geometry](https://www.owlsprep.com/study/ib-math-aa-sl-u3-2d-and-3d-coordinate-geometry/)
- [Statistics & Probability](https://www.owlsprep.com/study/ib-math-aa-sl-u4-overview/)
- [Right triangle trigonometry and the unit circle](https://www.owlsprep.com/study/ib-math-aa-sl-u3-right-triangle-trigonometry-and-the/)

---

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-sl-u3-overview/