Unit 3 · Topic 3: Geometry and Trigonometry
The Complete IB Mathematics: Analysis and Approaches Guide · IB Mathematics: Analysis and ApproachesUnit 3 · Topic 3: Geometry and Trigonometry
- The Complete IB Mathematics: Analysis and Approaches Guide
- IB Mathematics: Analysis and Approaches
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. Trigonometric identity work transforms one side using established relations; checking a few angles is evidence, not proof. Trigonometric equation work needs a declared interval so periodic families can be reduced to the requested solutions.
- How IB Mathematics: Analysis and Approaches assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Analysis and Approaches; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a labeled triangle or three-dimensional coordinate diagram, a unit-circle or periodic graph, a vector equation for a line or plane
- Key skills Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles, Sine Rule, Cosine Rule, Area of a Triangle, and Applications, Radian Measure, Arcs and Sectors, the Unit Circle, and Exact Values, Pythagorean and Double Angle Identities and Trigonometric Equations, Circular Functions, Their Graphs, Transformations, and Modeling, Reciprocal and Inverse Trigonometric Functions and Compound Angle Identities (HL), Vectors: Algebra, Scalar Product, and Vector Product (HL), Vector Equations of Lines and Planes, Intersections, and Angles (HL)
- How to study for Unit 3 This page turns a labeled triangle or three-dimensional coordinate diagram, a unit-circle or periodic graph, a vector equation for a line or plane into one route: Write the known sides, angles, vectors, interval, or period on a diagram before solving..
- The organizing decision Draw and label the geometry, state the interval, and then choose the relation that preserves every possible case.
What IB Mathematics: Analysis and Approaches Unit 3 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles
three-dimensional geometry; distance; angle02Sine Rule, Cosine Rule, Area of a Triangle, and Applications
sine rule; cosine rule; ambiguous case03Radian Measure, Arcs and Sectors, the Unit Circle, and Exact Values
radian measure; unit circle; exact value04Pythagorean and Double Angle Identities and Trigonometric Equations
double-angle identity; solution interval; trigonometric equation05Circular Functions, Their Graphs, Transformations, and Modeling
midline; amplitude; angular frequency06Reciprocal and Inverse Trigonometric Functions and Compound Angle Identities (HL)
inverse trigonometric function; compound angle; reciprocal function07Vectors: Algebra, Scalar Product, and Vector Product (HL)
dot product; cross product; normal vector08Vector Equations of Lines and Planes, Intersections, and Angles (HL)
vector equation; line parameter; intersectionInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Identity, measure, and spatial reasoning
Connect the published share to the unit model
Trigonometric identity work transforms one side using established relations; checking a few angles is evidence, not proof. Trigonometric equation work needs a declared interval so periodic families can be reduced to the requested solutions.
Triangle solution begins with a labeled diagram and a decision among sine rule, cosine rule, and area relations. The ambiguous sine case requires checking whether zero, one, or two triangles meet the data.
In three dimensions, vectors encode direction, lines, planes, angles, and distances. A dot product establishes perpendicularity, a cross product supplies a normal vector, and a parameter must be interpreted in the original geometric setting.
The decision that organizes this unit
Define the system and choose the route before calculating
Draw and label the geometry, state the interval, and then choose the relation that preserves every possible case.
Write the known sides, angles, vectors, interval, or period on a diagram before solving.
Mechanism route and repair branches
Relationships to preserve
- A trigonometric identity is proved by transforming expressions, while an equation is solved on a stated interval.
- The sine rule can produce a supplementary angle that must pass the triangle angle-sum test.
- Two lines intersect only when one consistent parameter pair makes all coordinates agree.
Representations to read
- a labeled triangle or three-dimensional coordinate diagram
- a unit-circle or periodic graph
- a vector equation for a line or plane
Branches to reject
- using selected angle checks as proof of an identity
- keeping only the first inverse-sine angle
- matching one coordinate with inconsistent parameter values
| Key concept | Why it's hard | What scores |
|---|---|---|
| Identity or equation | A correct-looking graph does not prove an identity | Names the identity and gives every solution in the interval |
| Triangle data | The sine rule may admit a second angle | Labels sides, tests feasibility, and states units |
| Vectors in space | Coordinates alone do not explain incidence | Shows vector relations and interprets the parameter |
How IB Mathematics: Analysis and Approaches assesses Topic 3: Geometry and Trigonometry
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles | three-dimensional geometry; distance; angle | using selected angle checks as proof of an identity |
| Sine Rule, Cosine Rule, Area of a Triangle, and Applications | sine rule; cosine rule; ambiguous case | keeping only the first inverse-sine angle |
| Radian Measure, Arcs and Sectors, the Unit Circle, and Exact Values | radian measure; unit circle; exact value | matching one coordinate with inconsistent parameter values |
| Pythagorean and Double Angle Identities and Trigonometric Equations | double-angle identity; solution interval; trigonometric equation | using selected angle checks as proof of an identity |
| Circular Functions, Their Graphs, Transformations, and Modeling | midline; amplitude; angular frequency | keeping only the first inverse-sine angle |
| Reciprocal and Inverse Trigonometric Functions and Compound Angle Identities (HL) | inverse trigonometric function; compound angle; reciprocal function | matching one coordinate with inconsistent parameter values |
| Vectors: Algebra, Scalar Product, and Vector Product (HL) | dot product; cross product; normal vector | using selected angle checks as proof of an identity |
| Vector Equations of Lines and Planes, Intersections, and Angles (HL) | vector equation; line parameter; intersection | keeping only the first inverse-sine angle |
Resolve the ambiguous sine case
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Keep both valid sine-rule triangles
In triangle ABC, side a is opposite angle A and side b is opposite angle B.
Given a=8, b=11 and A=35°, find all possible values of B and the corresponding value of c in each case.
Show your working.
Give angles and lengths to three significant figures.
A. Use sin B=11sin35°/8, test B≈52.1° and 127.9°, then obtain c≈13.9 and 4.1 respectively.
✅ Correct! 😊 Correct. Both supplementary angles leave a positive third angle, and applying the sine rule separately pairs each side length with its triangle.
B. Use B=sin⁻¹(11sin35°/8)≈52.1° and report only c≈13.9.
❌ Not quite 😭 Inverse sine gives one principal value, but the supplementary angle also has the same sine. Omitting it loses a valid triangle.
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C. Use B≈52.1° and 127.9°, but pair them with c≈4.1 and 13.9 respectively.
❌ Not quite 😭 The angle branches are found, but the side values are swapped. Each value of c must be calculated using the matching remaining angle C.
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D. Use B≈52.1° and 307.9° because sine is positive in two quadrants.
❌ Not quite 😭 An interior triangle angle must lie between 0° and 180°. The relevant second solution is the supplement 180°−52.1°, not a reflex angle.
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Full explanation
- Step 1The sine rule gives sin B/11=sin35°/8, so sin B≈0.7887.
- Step 2The candidates are B≈52.1° and 180°−52.1°=127.9°; both satisfy A+B<180°.
- Step 3The corresponding third angles are approximately 92.9° and 17.1°.
- Step 4Applying the sine rule in each triangle gives c≈13.9 and c≈4.1 respectively.
Practice 2: Verify a three-dimensional line intersection
Two lines are given in vector form.
L₁ has equation r=(0,1,2)+t(1,2,−1), and L₂ has equation r=(3,7,−1)+s(−2,1,1). Determine whether the lines intersect and, if so, find the intersection point.
Show that all three coordinates are consistent.
A. t=3 gives (3,7,−1) on L₁, while s=0 gives the same point on L₂, so the lines intersect there.
✅ Correct! 😊 Correct. One consistent parameter value on each line produces agreement in all three coordinates at the same point.
B. The x-coordinates can both equal 3, so the lines intersect at some point with x=3.
❌ Not quite 😭 Matching one coordinate is insufficient in three dimensions. The y- and z-coordinates must agree for the same parameter pair.
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C. The direction vectors are not scalar multiples, so the lines must intersect.
❌ Not quite 😭 Nonparallel lines in three dimensions may be skew. Their position equations must still produce a common point.
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D. Use t=3 for the x- and y-coordinates but t=−3 for the z-coordinate, giving (3,7,−1).
❌ Not quite 😭 A single point on a vector line requires one shared parameter value across every coordinate. Mixing t-values does not describe any point on L₁.
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Full explanation
- Step 1Equating the y-coordinates with s=0 gives 1+2t=7, so t=3.
- Step 2Substitute t=3 into L₁ to obtain (3,7,−1).
- Step 3Substitute s=0 into L₂ to obtain its fixed point (3,7,−1).
- Step 4All coordinates agree for the same parameter pair, so the intersection is (3,7,−1).
Key terms for Unit 3: Topic 3: Geometry and Trigonometry
Models, uses, and boundaries
- Pythagorean identity
- Sine squared plus cosine squared equals 1. Use it to transform trigonometric expressions without relying on sampled angles.
- Double-angle identity
- One minus cosine of twice the angle equals twice sine squared of the angle. Use it to prove the identity and then solve on the stated cycle.
- Sine rule
- A side divided by the sine of its opposite angle equals another side divided by the sine of its opposite angle. Use it while checking both possible angles in an ambiguous case.
- Vector line
- A line position equals a fixed point plus a parameter times a direction vector. Use it to compare coordinates with one consistent parameter value.
- Periodic height
- Height equals 10 minus 8 times cosine of pi times time divided by 20. Use it for a ride with midline 10, amplitude 8, period 40 seconds, and a minimum start.
IB Mathematics: Analysis and Approaches Unit 3 FAQ
How much of IB Mathematics: Analysis and Approaches does Unit 3 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 3: Geometry and Trigonometry problem?
Write the known sides, angles, vectors, interval, or period on a diagram before solving.
Which relationships should I preserve?
A trigonometric identity is proved by transforming expressions, while an equation is solved on a stated interval. The sine rule can produce a supplementary angle that must pass the triangle angle-sum test. Two lines intersect only when one consistent parameter pair makes all coordinates agree.
Which representations should I practice?
Practice moving among a labeled triangle or three-dimensional coordinate diagram, a unit-circle or periodic graph, a vector equation for a line or plane.
What error should I check before submitting an answer?
Check for using selected angle checks as proof of an identity; keeping only the first inverse-sine angle; matching one coordinate with inconsistent parameter values.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Prove an identity and solve on one cycle
- Resolve the ambiguous sine case
- Intersect two lines in three dimensions
- Generalize a periodic height model
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Analysis and Approaches unit guides
IB Mathematics: Analysis and Approaches Exam Guide & Review
The whole exam and its official unit sequence.01Topic 1: Number and Algebra
International Baccalaureate does not publish a per-unit weighting for this exam.02Topic 2: Functions
International Baccalaureate does not publish a per-unit weighting for this exam.04Topic 4: Statistics and Probability
International Baccalaureate does not publish a per-unit weighting for this exam.05Topic 5: Calculus
International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Analysis and Approaches Unit 3
Start with the organizing decision
Before solving, restate the decision in operational terms: Draw and label the geometry, state the interval, and then choose the relation that preserves every possible case.. Your first written move should be to Write the known sides, angles, vectors, interval, or period on a diagram before solving..
Practice the same idea in several representations
Rotate through a labeled triangle or three-dimensional coordinate diagram, a unit-circle or periodic graph, a vector equation for a line or plane. Use each representation to practice Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles, Sine Rule, Cosine Rule, Area of a Triangle, and Applications, Radian Measure, Arcs and Sectors, the Unit Circle, and Exact Values, Pythagorean and Double Angle Identities and Trigonometric Equations, Circular Functions, Their Graphs, Transformations, and Modeling, Reciprocal and Inverse Trigonometric Functions and Compound Angle Identities (HL), Vectors: Algebra, Scalar Product, and Vector Product (HL), Vector Equations of Lines and Planes, Intersections, and Angles (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using selected angle checks as proof of an identity; keeping only the first inverse-sine angle; matching one coordinate with inconsistent parameter values. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Analysis and Approaches to see every unit and plan your revision.