IB Mathematics: AA HL topic guide

Geometry and Trigonometry

Geometry and Trigonometry is a core part of IB Mathematics: AA HL. This guide connects the syllabus ideas behind Introduction to Geometry and Trigonometry, 3D Geometry and Mensuration, The Sine Rule and Cosine Rule, Radian Measure and the Unit Circle, Trigonometric Identities and Graphs and 6 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.

What you will learn

Geometry and Trigonometry syllabus outline

The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.

3.0

Introduction to Geometry and Trigonometry

What You Will Learn in Geometry and Trigonometry

3.1

3D Geometry and Mensuration

Distance & Midpoint in 3D · Volume, Surface Area & Angles in 3D Solids

3.2

The Sine Rule and Cosine Rule

The Sine Rule · The Cosine Rule & Area of a Triangle

3.3

Radian Measure and the Unit Circle

Radians, Arc Length & Sector Area · The Unit Circle & Exact Values

3.4

Trigonometric Identities and Graphs

The Pythagorean Identity · Graphs of Trig Functions & Transformations

3.5

Solving Trigonometric Equations

Solving Trig Equations Graphically & Analytically · Trigonometric Equations with Multiple Angles and Domains

3.6

Further Trigonometric Identities

Compound & Double Angle Identities · Reciprocal Trig Functions

3.7

Inverse Trigonometric Functions

arcsin, arccos and arctan · Domain Restrictions and Composite Inverse Trig

3.8

Vectors: Fundamentals

Vector Representation & Operations · Position Vectors

3.9

Vectors: The Scalar Product and Lines

The Scalar (Dot) Product · Vector Equation of a Line

3.10

Vectors: The Vector Product and Planes

The Vector (Cross) Product · Vector Equations of Planes & Intersections

Free worked preview

Compound & Double Angle Identities

This complete preview comes from the Further Trigonometric Identities unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.

Compound Angle Identities

The sine of a sum is not the sum of the sines — sin⁡(A+B)\sin(A+B) expands in a more subtle way that combines both sine and cosine of the individual angles. These compound and double-angle identities (HL only) are the essential toolkit for proving more complex trig identities, solving equations like sin⁡(2x)=cos⁡x\sin(2x)=\cos x, and for the integration of sin⁡2x\sin^2 x and cos⁡2x\cos^2 x in calculus. By the end you will be able to state and apply the compound-angle formulas for sine, cosine, and tangent, derive the double-angle formulas from them, and use them to simplify expressions and solve equations.
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B) = \cos A\cos B \mp \sin A\sin B
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A\pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

Double Angle Identities

Setting A = B = θ in the compound identities gives:

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1-2\sin^2\theta

The three forms of cos(2θ) are all equivalent (using the Pythagorean identity to substitute) — pick whichever suits the problem, e.g. the last form is useful when only sinθ is known.

Common Errorsin⁡(2θ)\sin(2\theta) does not equal 2sin⁡θ2\sin\theta — this is one of the most common algebra slips in trigonometry. The double angle identity requires both sinθ and cosθ.
Worked Example Given sin⁡θ=35\sin\theta=\tfrac35, θ acute, find sin⁡(2θ)\sin(2\theta) and cos⁡(2θ)\cos(2\theta).
1From the Pythagorean identity, θ acute so cosine positivecos⁡θ=45\cos\theta = \tfrac45
2Double angle formula for sinesin⁡(2θ)=2(35)(45)=2425\sin(2\theta) = 2\left(\tfrac35\right)\left(\tfrac45\right) = \tfrac{24}{25}
3Double angle formula for cosine (using the sin-only form)cos⁡(2θ)=1−2(35)2=725\cos(2\theta) = 1-2\left(\tfrac35\right)^2 = \tfrac{7}{25}
Practice QuestionProve that cos⁡(A+B)+cos⁡(A−B)=2cos⁡Acos⁡B\cos(A+B)+\cos(A-B) = 2\cos A\cos B using the compound angle identities.

Use the formulas

Keep the formula booklet beside you

Revision becomes faster when you practise locating the right equation and understand what every variable means. The full reference is public and searchable on any device.

Open IB Math AA HL Formula Booklet →

Reviewed by the Study to Learn editorial team · Updated 2026-07-24