Unit Overview

Geometry & Trigonometry

IB Mathematics: Analysis and Approaches SL· 5 min read 📊 18-21% of overall exam

1. 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, ending with vectors and their applications to lines in 2D and 3D space.

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

Calculate distances, midpoints, gradients and equations of lines in 2D and 3D space.

★★⏱ 8 min

Right triangle trigonometry and the unit circle

Define trigonometric ratios for right triangles and extend to any angle using the unit circle.

★★⏱ 7 min

Sine rule, cosine rule and area of triangles

Apply core rules to find unknown sides, angles and areas of any triangle, including the ambiguous sine rule case.

★★★⏱ 8 min

Graphs of trigonometric functions

Identify amplitude, period, phase shift and vertical shift, and sketch/interpret sine, cosine and tangent graphs.

★★★⏱ 9 min

Pythagorean and double angle trigonometric identities

Simplify expressions and solve problems using core trigonometric identities.

★★★★⏱ 10 min

Solving trigonometric equations

Solve linear and quadratic trigonometric equations over specified domains.

★★★★⏱ 10 min

2D and 3D vectors, dot product and lines

Perform vector operations, use the dot product to find angles, and write vector equations of lines.

★★★★⏱ 12 min

2. Common Pitfalls

Wrong move:

Forgetting the ambiguous case of the sine rule for SSA triangle problems

Why:

Missing the second valid angle solution leads to incomplete marks on exam questions

Correct move:

Always check if a second obtuse solution $18 0^{\circ} - θ$ is possible by verifying the total angle sum is less than $18 0^{\circ}$

Wrong move:

Mixing up period formulas for transformed trigonometric functions

Why:

Incorrect period calculation leads to wrong graph sketches and equation solutions

Correct move:

Period is $\frac{2 π}{∣ b ∣}$ for sine/cosine, and $\frac{π}{∣ b ∣}$ for tangent, regardless of phase/vertical shift

Wrong move:

Forgetting that vectors require direction when solving line intersection problems

Why:

Assuming any common coordinate means lines intersect, ignoring parallel skew lines in 3D

Correct move:

Always check for consistency of parameters when testing for intersections of 3D lines

3. Quick Reference Cheatsheet

Concept	Key Formula
3D Distance between two points	$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
Sine Rule	$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$
Cosine Rule	$c^{2} = a^{2} + b^{2} - 2 ab cos C$
Area of any triangle	$Area = \frac{1}{2} ab sin C$
Pythagorean Trig Identity	$sin^{2} θ + cos^{2} θ = 1$
Vector Dot Product	$a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Period of transformed sine/cosine	$Period = \frac{2 π}{∣ B ∣}$ for $y = A sin (B x + C) + D$

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 vectors and 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.