IB Maths AI HLTrigonometryPaper 1 & 2Non-right triangles~10 min read
Sine Rule, Cosine Rule & Area of a Triangle
For non-right triangles, three rules cover everything. The sine rule handles opposite-pair information; the cosine rule handles two-sides-plus-included-angle or all three sides; and the area formulaA = ½absinC uses two sides and the included angle. All three are in the formula booklet — the work is choosing the right one.
📘 What you need to know
Standard labelling: angles uppercase (A,B,C), the side opposite each angle is the matching lowercase letter (a,b,c).
Sine rule: asinA = bsinB = csinC. Flip for angles: sinAa = sinBb = sinCc.
Area: A = ½absinC — sides a and b with the angle Cbetween them.
Ambiguous sine rule: when finding an angle, the obtuse option 180°−θ may also work — check whether the angle sum stays under 180°.
All three rules are in the formula booklet (geometry & trigonometry section).
Which rule, and when?
Each rule fits a specific kind of information. Start by listing what you’ve got: sides, angles, and how they’re positioned relative to each other. Opposite pairs (a side and its opposite angle, plus one more piece) → sine rule. Two sides with the angle between them, or all three sides → cosine rule. Two sides with the angle between them, looking for area → area formula. If your data don’t fit one rule directly, find a missing piece first (often using the angle sum of 180°).
Standard labelling: vertices are uppercase A, B, C; opposite sides are lowercase a, b, c. Pick the rule by matching what you know.
The three rules
sine: asinA = bsinB = csinCcosine: c2 = a2+b2−2abcosC · area: A = 12absinC
The ambiguous case (sine rule for an angle)
When you use the sine rule to find an angle, the inverse-sine function only returns the acute answer. The obtuse option 180°−θ has the same sine value and is sometimes also a valid triangle. To check, add the obtuse option to the known angle: if the total is below 180°, both angles work and the question has two solutions; if not, only the acute one is valid. Diagrams in the exam often state “acute” or “obtuse” explicitly — read carefully.
Quick ambiguity test: when finding B via sine rule with known A, compute Bacute first, then check A + (180°−Bacute). If this sum is < 180°, the obtuse option is also a valid triangle and the answer has two parts.
🧭 Recipe — choose & apply
Sketch & label: name vertices A, B, C and opposite sides a, b, c to match.
Pick the rule: opposite pair → sine; two sides + included angle (or three sides) → cosine; area question → ½absinC.
Substitute & rearrange: write the formula, plug in, isolate the unknown.
Inverse for angles: use sin−1 or cos−1. For sine rule, also check the obtuse option.
Multi-step problems: if no rule fits directly, use angle-sum 180° or apply two rules in sequence.
Worked examples
WE 1
Sine rule — find a missing side
A triangular sail PQR has angles P = 48° and R = 35°. The side opposite P (the base p) measures 7 m. Find the length of the side opposite R (side r), to 3 s.f.
In triangle ABC, angle A = 110°, side a = 18 cm (opposite A), and side b = 8 cm. Find angle B, to 1 d.p.
flip sine rule for anglessin B / 8 = sin(110°) / 18sin B = 8 sin(110°) / 18 = 0.4176…inverse sineB = sin⁻¹(0.4176…) = 24.685…°B ≈ 24.7° (1 d.p.)check ambiguityobtuse option: 180 − 24.7 = 155.3°A + B_obtuse = 110 + 155.3 = 265.3° > 180°obtuse impossible — A is already obtuse, so B must be acute. Unique answer.
WE 3
Sine rule — the ambiguous case
In triangle ABC, AB = 12 cm, BC = 9 cm, and angle BAC = 40°. Find both possible values of angle ACB, to 1 d.p.
label: a = BC = 9 (opp A), c = AB = 12 (opp C)sin C / 12 = sin(40°) / 9sin C = 12 sin(40°) / 9 = 0.8571…acute solutionC₁ = sin⁻¹(0.8571…) = 58.99°C ≈ 59.0° (1 d.p.)obtuse optionC₂ = 180 − 59.0 = 121.0°check: 40 + 121.0 = 161.0° < 180° ✓C ≈ 121.0° (1 d.p.)two valid triangles — both answers required.
WE 4
Cosine rule — find a side
A ship sails 22 km from port P to point Q, then turns and sails a further 35 km to point R. The angle at Q between the two legs of the journey is 105°. Find the straight-line distance PR from port to final position, to 3 s.f.
two sides + included angle → cosine rulePR² = 22² + 35² − 2(22)(35) cos(105°) = 484 + 1225 − 1540 × (−0.2588…) = 1709 + 398.6 = 2107.6take the positive rootPR = √2107.6 = 45.91…PR ≈ 45.9 km (3 s.f.)cos(105°) is negative because 105° is obtuse — that adds to PR² rather than subtracting.
WE 5
Cosine rule — largest angle of a garden bed
A triangular flower bed has sides of length 8 m, 11 m, and 15 m. Find the largest angle of the bed, to 1 d.p.
largest ∠ is opposite the longest side (15 m)let a = 8, b = 11, c = 15rearranged cosine rulecos C = (8² + 11² − 15²) / (2·8·11) = (64 + 121 − 225) / 176 = −40 / 176 = −0.2273inverse cosineC = cos⁻¹(−0.2273) = 103.14°C ≈ 103.1° (1 d.p.)negative cosine → obtuse angle, consistent with “largest in a triangle 8,11,15”.
WE 6
Area — using sine rule first
In triangle XYZ, angle X = 38°, angle Y = 72°, and side XY = 18 cm. Find the area of the triangle, to 3 s.f.
find the third angleZ = 180 − 38 − 72 = 70°XY is opposite Z, so z = 18find side y (opp Y) by sine ruley / sin(72°) = 18 / sin(70°)y = 18 sin(72°) / sin(70°) = 18.22 cmarea: sides y, z with included angle XA = ½ · y · z · sin X = ½ · 18.22 · 18 · sin(38°) = 100.95…A ≈ 101 cm² (3 s.f.)use full calculator values for y when computing the area — don’t round mid-step.
💡 Top tips
Relabel to match the rule — the vertex letters in the question may not match A/B/C in the formula. Rewrite to match.
Sine rule for angle: flip it (sin form on top); cosine rule for angle: use the rearranged version.
Always check ambiguity when sine rule gives an angle — does the obtuse option also fit?
Area formula needs the angle BETWEEN the two given sides — not opposite one of them. If your data don’t fit, use sine or cosine rule first.
Keep full calculator accuracy for multi-step problems; round only at the final answer.
⚠ Common mistakes
Forgetting the ambiguous case — when sine rule gives an angle, always check whether the obtuse alternative is also valid.
Using the wrong angle in cosine rule — the angle C in c2 = a2+b2−2abcosC must be the angle opposite side c (and between sides a, b).
Sign error with obtuse cosines — cos of an obtuse angle is negative, so −2abcosC becomes positive and adds to the answer.
Wrong angle in area formula — the included angle must lie between the two given sides, not opposite one.
Calculator in radian mode — AI HL questions use degrees; check before computing.
Next up — Angles of Elevation & Depression. Once you can solve any triangle, the next step is applying that to real-world up/down angles seen from observation points.
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