Unit Overview

Geometry & Trigonometry

IB Mathematics: Analysis and Approaches Higher Level· 5 min read 📊 18-20% of total IB exam

1. 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

Learn radian angle measurement and define trigonometric functions using the unit circle.

★⏱ 8 min

Right triangle trigonometry

Define sine, cosine, and tangent for right triangles and solve for unknown sides and angles.

★★⏱ 8 min

Non-right triangle trigonometry

Apply the sine rule, cosine rule, and area formula to any non-right triangle.

★★⏱ 10 min

Trigonometric equations

Solve linear and quadratic trigonometric equations over specified domains.

★★★⏱ 12 min

Compound and double angle identities

Use addition and double angle formulas to simplify expressions and solve equations.

★★★⏱ 15 min

2D coordinate geometry

Explore lines, circles, and geometric properties on the 2D coordinate plane.

★★⏱ 10 min

3D coordinate geometry

Extend coordinate concepts to 3-dimensional coordinate systems.

★★★⏱ 10 min

Vector fundamentals and dot product

Learn vector notation, operations, and the dot product for angles and magnitudes.

★★★⏱ 12 min

Cross product of vectors (HL only)

Calculate cross products and use them to find areas and normals to planes.

★★★★⏱ 10 min

Vector equations of lines and planes (HL only)

Write and interpret vector forms for lines and planes in 3D space.

★★★★⏱ 15 min

Intersections and angles between lines and planes (HL only)

Calculate intersection points/lines and angles between lines and planes.

★★★★⏱ 15 min

Distance from point to plane (HL only)

Use vector formulas to find the shortest distance from a point to a plane.

★★★★★⏱ 10 min

Congruence and similarity

Prove geometric properties using congruence and similarity criteria for triangles.

★★⏱ 8 min

2. Common Pitfalls

Wrong move:

Forgetting all solutions when solving trigonometric equations over large domains

Why:

Trigonometric functions are periodic, so multiple solutions usually exist

Correct move:

Always check your solution interval and list all valid angles

Wrong move:

Mixing up degrees and radians in calculator calculations

Why:

Most IB trigonometry questions require radians, but calculators default to degrees

Correct move:

Confirm your calculator mode before starting any calculation

Wrong move:

Omitting magnitude terms when using the dot product to find angles

Why:

The formula relies on normalized vectors to get the correct cosine value

Correct move:

Always divide by the product of the vector magnitudes

3. Quick Reference Cheatsheet

Concept	Key Formula
Sine Rule	$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$
Cosine Rule	$c^{2} = a^{2} + b^{2} - 2 ab cos C$
Compound Sine Identity	$sin (A + B) = sin A cos B + cos A sin B$
Double Angle Cosine	$cos 2 θ = 2 cos^{2} θ - 1 = 1 - 2 sin^{2} θ$
Dot Product	$a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Cross Product Magnitude	$∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$ (area of parallelogram)
Vector Plane Equation	$r \cdot n = d$ , where $n$ is the unit normal
Distance Point to Plane	$\frac{∣ a x _{0} + b y _{0} + c z _{0} + d ∣}{a ^{2} + b ^{2} + c ^{2}}$

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.