Triangle Trigonometry (AI SL)
Triangle trigonometry is the toolkit for finding unknown sides and angles once a triangle stops being right-angled. This topic covers the sine rule, the cosine rule and the ½ab sinC area formula, then applies them to real contexts like bearings, surveying, and angles of elevation and depression. Sketching a clearly labelled diagram before you start is often half the battle.
What the syllabus says
This topic maps onto three points in the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL3.2 | Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles. The sine rule \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). The cosine rule \(c^2=a^2+b^2-2ab\cos C\). Area of a triangle as \(\tfrac12 ab\sin C\). This section does not include the ambiguous case of the sine rule. |
| SL3.3 | Applications of right and non-right angled trigonometry, including Pythagoras' theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements. Contexts may include the use of bearings. |
Radians are not required anywhere in this topic at SL - all angles are worked in degrees.
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 links each side of a triangle to the sine of its opposite angle: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). Use it whenever you know an angle-side pair (opposite each other) plus one more piece of information.
e.g. If \(A=40^\circ\), \(B=65^\circ\), \(a=10\), then \(b=\dfrac{10\sin65^\circ}{\sin40^\circ}\approx14.1\).
What is the cosine rule?
The cosine rule finds a side or angle when the sine rule can't be used directly - typically when you know two sides and the angle between them (SAS), or all three sides (SSS). Its form is \(c^2=a^2+b^2-2ab\cos C\).
e.g. With \(b=12\), \(c=9\), \(A=35^\circ\): \(a^2=12^2+9^2-2(12)(9)\cos35^\circ\approx48.06\), so \(a\approx6.93\).
What is the area formula for a triangle?
When you know two sides and the angle between them, the area is \(\tfrac12 ab\sin C\) - no height needed. It uses exactly the same information as one form of the cosine rule, so the two often appear in the same question.
e.g. With sides 15 and 11 and included angle \(64^\circ\): \(A=\tfrac12(15)(11)\sin64^\circ\approx74.2\).
What is a bearing?
A bearing is a direction measured clockwise from north, always written with three digits (e.g. \(060^\circ\), not \(60^\circ\)). The bearing back the way you came is found by adding or subtracting \(180^\circ\).
e.g. A bearing of \(200^\circ\) has a reverse bearing of \(200^\circ-180^\circ=020^\circ\).
What is the angle of elevation?
The angle of elevation is the angle measured upward from the horizontal to a point above you; the angle of depression is the same idea measured downward. They form a right triangle with the horizontal and vertical distances involved.
e.g. Standing 50 m from a tower with elevation \(30^\circ\), the tower's height is \(50\tan30^\circ\approx28.9\) m.
Key formulas
Four formulas cover almost every question on this topic. The two tables below summarise 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 all on the official formula booklet; Pythagoras' theorem is assumed prior knowledge and isn't listed separately.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) | Sine rule - find a side or angle | ✓ Yes |
| \(c^2=a^2+b^2-2ab\cos C\) | Cosine rule - find a side | ✓ Yes |
| \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\) | Cosine rule rearranged - find an angle | ✓ Yes |
| \(\text{Area}=\tfrac12 ab\sin C\) | Area from two sides and the included angle | ✓ Yes |
| \(a^2+b^2=c^2\) | Pythagoras' theorem (right triangles only) | Not in booklet - prior knowledge |
Sine rule vs cosine rule
Choosing the right rule comes down to what information the question already gives you.
| Feature | Sine rule | Cosine rule |
|---|---|---|
| Use when you know | An angle-side pair, plus one more angle or side | Two sides and the included angle (SAS), or all three sides (SSS) |
| Finds | A missing side or a missing angle | A missing side (SAS) or a missing angle (SSS) |
| SL restriction | Ambiguous (two-solution) case not tested | No restriction at SL |
| Typical trigger phrase | "Two angles given" or "angle opposite a known side" | "Two sides and the angle between them" |
Solving non-right-angled triangles
Most exam questions on this topic involve a triangle with no right angle, so the three formulas below do almost all the work.
Sine rule for a side
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}\]
Rearrange to make the unknown side the subject once you know its opposite angle.
✓ In the formula bookletCosine rule for a side
\[c^2=a^2+b^2-2ab\cos C\]
Needs two sides and the angle trapped between them.
✓ In the formula bookletCosine rule for an angle
\[\cos C=\dfrac{a^2+b^2-c^2}{2ab}\]
Use when all three sides are known and an angle is missing.
✓ In the formula bookletApplications in context
Bearings, elevation/depression and surveying problems dress the same triangle skills up in a real-world setting.
Bearings
Directions measured clockwise from north, given as three digits. Two bearings from a common point define an angle you can feed into the cosine rule.
Convention, not a formulaElevation and depression
Right-triangle problems where the angle is measured from the horizontal. Alternate angles make the elevation from the ground equal the depression from above.
Uses SOH CAH TOALabelled diagrams
Exam questions increasingly give only a written description. Sketching and labelling the triangle first turns a wordy problem into a standard sine/cosine rule question.
Technique, not a formulaWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A right-angled triangle has legs \(6\) cm and \(8\) cm. Find the hypotenuse.
Worked solution
\(\sqrt{6^2+8^2}\) M1
\(=10\) cm. A1 🖩 On the GDC (evaluate, in degree mode).\(\texttt{sqrt(6^2+8^2)} \Rightarrow 10\) A right-angled triangle needs no trig here - just Pythagoras on the home screen.
Casio fx-CG50 / fx-CG100: ensure SHIFT → SET UP → Angle: Deg, then evaluate in Run-Matrix.
TI-Nspire CX: ensure the Angle setting is Degree, then evaluate in a Calculator page.
In triangle \(ABC\), \(A=35^\circ,\ b=12\) cm\(,\ c=9\) cm.
(a) Find side \(a\) (3 s.f.).
(b) Find the area of the triangle.
Worked solution
(a) Two sides \(b=12,\ c=9\) with included angle \(A=35^\circ\):
\(a^2 = 12^2 + 9^2 - 2(12)(9)\cos 35^\circ = 144 + 81 - 216(0.8192\ldots)\) M1 \(= 48.05\ldots\) A1 \(a = \sqrt{48.05\ldots} = 6.931\ldots \approx 6.93\) (3 s.f.). A1
(b) Use the two given sides and their included angle: \(A = \tfrac12 bc\sin A\): M1
\(A = \tfrac12(12)(9)\sin 35^\circ = 54(0.5736\ldots) = 30.97\ldots\) A1 \(A \approx 31.0\) units² (3 s.f.). 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.
- Reaching for 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 - you need the cosine rule instead.
- Rounding too early. Carrying a rounded intermediate value (like a rounded side) into a second part of the question compounds the error - keep the unrounded value in your calculator and only round the final answer.
- Measuring a bearing anticlockwise, or forgetting the three-digit convention. Bearings are always clockwise from north and written with three digits, e.g. \(060^\circ\) not \(60^\circ\).
- Leaving the calculator in radian mode. This entire topic is worked in degrees at SL - a radian-mode slip gives a wildly wrong trig value that's easy to miss if you don't sanity-check the answer.
Using your GDC
There isn't a single dedicated calculator routine for triangle trigonometry - you're mostly evaluating the sine rule, cosine rule and area formula directly on the home screen, so accuracy comes down to careful bracketing rather than a special menu.
Once you've set up the equation on paper, type the whole right-hand side into your calculator in one go rather than computing pieces separately and rounding between steps - this avoids rounding errors creeping into a multi-part question. Make sure your calculator's angle mode is set to degrees before you start (check the mode/settings screen), since every triangle trigonometry question at SL is degree-based. If a part asks you to find a missing angle from the cosine rule, use the inverse cosine key on the rearranged formula rather than trying to solve for the angle by trial and error.
See the full GDC guide for model-specific button sequences across other topics.
Ready to practise properly?
Triangle trigonometry 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 and when do I use the cosine rule?
Use the sine rule when you know a matching angle-side pair plus one more angle or side (AAS/ASA/SSA). Use the cosine rule when you know two sides and the included angle (SAS), or all three sides (SSS) and need an angle.
Do I need radians for triangle trigonometry?
No. At SL, all triangle trigonometry (sine rule, cosine rule, area, bearings, elevation and depression) is done in degrees - radians are not required for this part of the syllabus.
What exactly is a bearing?
A bearing is a direction measured clockwise from north, always given as three digits (e.g. 060 degrees, not 60 degrees). It's the standard way exam questions describe direction in navigation and surveying contexts.
Is the ambiguous case of the sine rule tested at SL?
No. The SL syllabus explicitly excludes the ambiguous (two-solution) case of the sine rule - that only appears at AI HL. At SL you can assume a single valid triangle.
Sub-topics
Triangle Trigonometry broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI SL syllabus unit, in case you want to keep going.