Triangles & Circular Functions (AA HL)
Once a triangle isn't right-angled, you need the sine and cosine rules to find missing sides and angles - and the same ideas extend naturally into bearings and angles of elevation. This topic also covers the circular functions \(\sin x\), \(\cos x\) and \(\tan x\) themselves: their periodic graphs, and how amplitude, period and shifts let you model repeating real-world quantities like tides or Ferris wheels.
What the syllabus says
This topic maps onto three points in the official IB Analysis & Approaches syllabus, all shared with SL.
| Code | Syllabus content |
|---|---|
| SL3.2 | The sine rule \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) (excluding the ambiguous case). The cosine rule \(c^2=a^2+b^2-2ab\cos C\). Area of a triangle \(\tfrac12ab\sin C\). |
| SL3.3 | Applications of right and non-right angled trigonometry, including Pythagoras's theorem; contexts may include bearings. Angles of elevation and depression. |
| SL3.7 | The circular functions \(\sin x\), \(\cos x\) and \(\tan x\); amplitude, their periodic nature, and their graphs. Composite functions of the form \(f(x)=a\sin(b(x+c))+d\). |
These are core AA SL syllabus points that are also examinable at AA HL.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the sine rule?
The sine rule relates each side of a triangle to the sine of its opposite angle: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). It's the tool to reach for when you know an angle-side pair plus one more piece of information.
e.g. If \(A=30^\circ,\ a=5,\ B=45^\circ\), then \(b=\dfrac{5\sin45^\circ}{\sin30^\circ}=\dfrac{5(0.7071)}{0.5}\approx7.07\).
What is the cosine rule?
The cosine rule links all three sides of a triangle to one angle: \(c^2=a^2+b^2-2ab\cos C\). Unlike the sine rule, it works even when you only know sides and no angle-side pair.
e.g. If \(a=7,\ b=8,\ C=60^\circ\), then \(c^2=49+64-2(7)(8)(0.5)=113-56=57\), so \(c\approx7.55\).
What is the area formula for a non-right triangle?
When you know two sides and the angle between them, the area is \(\tfrac12ab\sin C\) - no need to find the height first, since \(\sin C\) already accounts for it.
e.g. If \(a=6,\ b=9,\ C=40^\circ\), area \(=\tfrac12(6)(9)\sin40^\circ=27(0.6428)\approx17.4\).
What is a three-figure bearing?
A bearing gives a direction as an angle measured clockwise from north, always written with three digits. A bearing of \(070^\circ\) points slightly east of north; a bearing of \(270^\circ\) points due west.
e.g. A ship sailing on a bearing of \(050^\circ\) is heading \(50^\circ\) clockwise from north, i.e. north-east.
What are amplitude and period?
Amplitude is how far a circular function swings above and below its midline; period is how long it takes to complete one full cycle. For \(f(x)=a\sin(bx)+d\), the amplitude is \(|a|\) and the period is \(\tfrac{2\pi}{b}\).
e.g. \(y=3\sin(2x)\) has amplitude \(3\) and period \(\tfrac{2\pi}{2}=\pi\).
Key formulas
A handful of formulas cover almost every question on this topic. The two tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The sine rule, cosine rule and area formula are printed in the official formula booklet. Amplitude and period are read directly off the equation of a circular function rather than being "formulas" to look up.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) | Sine rule - finds a side or angle | ✓ Yes |
| \(c^2=a^2+b^2-2ab\cos C\) | Cosine rule - finds a side | ✓ Yes |
| \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\) | Cosine rule, rearranged - finds an angle | ✓ Yes |
| \(\text{Area}=\tfrac12ab\sin C\) | Area from two sides and the included angle | ✓ Yes |
| Amplitude \(=|a|\), period \(=\dfrac{2\pi}{b}\) | Reading a circular function's graph from its equation | Not in booklet - read from the equation |
Sine rule vs cosine rule
Both rules relate sides and angles of a triangle, but they suit different combinations of known information.
| Situation | Sine rule | Cosine rule |
|---|---|---|
| What you're given | Two angles and one side, or two sides and a non-included angle | Two sides and the included angle, or all three sides |
| What it finds | A remaining side or angle | The third side, or any angle |
| Watch out for | The ambiguous case (two possible triangles) - not part of the core SL content | No ambiguity - always gives a unique answer |
Solving triangles
Every non-right-angled triangle question comes down to choosing the correct rule for the information you have.
Sine rule
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\]
Set up the ratio using the angle-side pair you know, then cross-multiply to solve for the unknown.
✓ In the formula bookletCosine rule
\[c^2=a^2+b^2-2ab\cos C\]
Label the angle you know (or want) as \(C\), with \(a,b\) the two sides enclosing it.
✓ In the formula bookletArea formula
\[\text{Area}=\tfrac12ab\sin C\]
Works for any triangle where two sides and the angle between them are known - not just right triangles.
✓ In the formula bookletBearings and elevation/depression
These are applications of right and non-right angled trigonometry to real-world direction and height problems.
Bearings convention
Always measured clockwise from north, given as three digits (e.g. \(045^\circ\), not \(45^\circ\)). Sketch a north line at each point mentioned.
Elevation and depression
The angle of elevation is measured upward from the horizontal to an object above; the angle of depression is measured downward to an object below. They're equal for a straight line of sight between two points.
Setting up the triangle
Draw the horizontal and vertical legs first, mark the given angle at the horizontal, then decide whether the sine rule, cosine rule or simple right-angle trigonometry applies.
Circular functions
The graphs of \(\sin x\), \(\cos x\) and \(\tan x\) repeat forever - transformations of the basic graphs are described using amplitude, period and shifts.
Amplitude
For \(f(x)=a\sin(bx+c)+d\), the amplitude is \(|a|\) - how far the graph swings above and below its midline \(y=d\).
Period
The period is \(\dfrac{2\pi}{b}\) (or \(\dfrac{360^\circ}{b}\) in degrees) - the horizontal distance for one complete cycle.
Phase shift and midline
The constant \(c\) shifts the graph horizontally; \(d\) shifts it vertically and sets the midline \(y=d\), which is also the mean value of the function.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A triangle has sides \(a=6\) cm and \(b=10\) cm with included angle \(C=30^\circ\).
(a) Find the exact area.
(b) Find the exact length of the third side, using the cosine rule.
Worked solution
(a) Area: \(\tfrac12 ab\sin C=\tfrac12(6)(10)\sin30^\circ.\) M1
With \(\sin30^\circ=\tfrac12\): \(=30\cdot\tfrac12=15\) cm². A1
(b) \(c^2=36+100-120\cos30^\circ=136-60\sqrt3,\) so \(c\) M1
\(=\sqrt{136-60\sqrt3}\) cm. A1
In triangle \(ABC\), \(a=7\), \(b=8\), \(c=13\).
(a) Show that \(C=120^\circ.\)
(b) Hence find the exact area of the triangle.
Worked solution
(a) Cosine rule: \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}=\dfrac{49+64-169}{2(7)(8)}.\) M1
\(=\dfrac{-56}{112}=-\tfrac12.\) A1
Since \(\cos C=-\tfrac12\), \(C=120^\circ.\) AG
(b) \(\text{Area}=\tfrac12(7)(8)\sin120^\circ=28\cdot\tfrac{\sqrt3}{2}\) M1
\(=14\sqrt3.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Using the sine rule when the cosine rule is needed. If you know two sides and the angle between them, or all three sides, the sine rule can't be set up directly - reach for the cosine rule instead.
- Measuring bearings from the wrong direction. Bearings are always clockwise from north, not from east and not anticlockwise - sketch a north arrow at every point mentioned in the question.
- Rounding too early. Carry full calculator accuracy through cosine rule and area calculations, and only round the final answer - early rounding (especially before a square root) can shift the last significant figure.
- Mixing up degree and radian mode. Triangle problems are usually in degrees, but circular function models are usually in radians - check your GDC's mode before evaluating anything.
Using your GDC
There's no single calculator trick specific to this topic - the habits you need here are the general ones you'll use everywhere.
For triangle problems, put your calculator in degree mode and evaluate the sine rule, cosine rule and area expressions directly on the home screen - there's no need for a special app. For circular function questions, switch to radian mode (the IB assumes radians on exam papers unless stated otherwise) and use the graphing screen: graph the function, then use the maximum/minimum tool to read off key values, or the intersection tool to solve equations like \(d(t)=7.5\) graphically. Always check your calculator's angle mode before you start - a triangle answer in radian mode, or a graph in degree mode, will be wrong even though the method is right.
See the full GDC guide for calculator-specific button sequences on evaluating trig expressions, graphing, and finding intersections.
Ready to practise properly?
Triangles & circular functions questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
When do I use the sine rule instead of the cosine rule?
Use the sine rule when you know two angles and one side, or two sides and a non-included angle. Use the cosine rule when you know two sides and the included angle, or all three sides - the cosine rule copes with situations the sine rule can't handle directly.
What is a three-figure bearing?
A three-figure bearing is an angle measured clockwise from north, always written with three digits, e.g. \(070^\circ\) or \(245^\circ\). It's the standard way exam questions describe direction of travel.
Do circular functions use degrees or radians?
Either can appear in a question, but on examination papers radian measure is assumed unless the question states otherwise - and radians are essential once you differentiate or integrate a circular function later in the course.
Is the ambiguous case of the sine rule examined?
Not at SL. The core sine rule content excludes the ambiguous case (where two different triangles fit the same data), though AA HL students meet it as an extension alongside the compound angle work. See the GDC guide for how to confirm a triangle solution numerically.
Sub-topics
Triangles & Circular Functions broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AA HL syllabus unit, in case you want to keep going.