IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~9 min read

Sine Rule, Cosine Rule & Area of a Triangle

When a triangle isn’t right-angled, SOH CAH TOA stops working. The sine rule handles triangles where you have an angle paired with the side opposite it. The cosine rule handles two-sides-and-the-included-angle (find a side) or three-sides (find an angle). And the area formula ½ab sin C works for any triangle, not just right-angled ones. Together they extend your trig toolkit to every triangle the IB will throw at you.

📘 What you need to know

Labelling convention: angles are uppercase (A, B, C); sides are lowercase, named after the opposite angle (a, b, c).
Sine rule: a/sin A = b/sin B = c/sin C. Use when you have an angle paired with its opposite side.
Cosine rule: c² = a² + b² − 2ab cos C. Use for SAS (find third side) or SSS (find any angle).
Area formula: A = ½ab sin C — the angle must be the one between the two given sides.
All three formulas are in the formula booklet under Geometry & Trigonometry.
Ambiguous case of the sine rule: when finding an angle, two solutions may exist (acute and obtuse). Check whether the obtuse alternative is geometrically valid.
Angles in any triangle sum to 180° — useful when one rule alone doesn’t give the answer.
Rule of thumb: for non-right triangles, ask “do I have an angle-opposite-side pair?” — if yes, sine rule; if no, cosine rule.

Labelling non-right-angled triangles

The standard convention: angles get uppercase letters, and each side takes the lowercase version of the letter of the opposite angle. So angle A is opposite side a, angle B is opposite side b, and so on.

Why it matters: every formula in this note pairs an angle with the side opposite it. Mis-pairing one angle and side wrecks the calculation entirely. Always relabel a given triangle to match your chosen rule before substituting.

Sine rule

Sine rule asin A = bsin B = csin C

The same equation can be flipped to put sines on top — useful when finding an angle:

Sine rule (rearranged) sin Aa = sin Bb = sin Cc

Find a side

b = a sin B / sin A

use sides on top — clean rearrangement

Find an angle

B = sin⁻¹(b sin A / a)

use sines on top — easier inverse

The ambiguous case

When you use the sine rule to find an angle from a side opposite to it, there can be two valid answers — an acute one and an obtuse one. The obtuse alternative is 180° minus the acute. Whether it’s geometrically possible depends on whether the three angles can still sum to less than 180° once you include the original given angle.

Quick check: if (acute answer) + (given angle) > 180°, the obtuse case is impossible — only the acute answer works. If their sum is less than 180°, both are valid and you should give both. Your sketch usually clarifies which the question wants.

Cosine rule

When the sine rule won’t work — typically because you don’t have an angle paired with its opposite side — reach for the cosine rule.

Cosine rule — find a side c² = a² + b² − 2ab cos C

Use this when you know two sides and the angle between them (SAS), and want the third side.

Cosine rule — find an angle cos C = (a² + b² − c²) / (2ab)

Use this when you know all three sides (SSS) and want any angle. The angle C is opposite the side c.

Reduces to Pythagoras: when C = 90°, cos C = 0, and the cosine rule collapses to c² = a² + b². The cosine rule is just Pythagoras with a correction term for non-right angles.

Area of a triangle

Area formula A = 12 ab sin C

The angle C must be the one formed between sides a and b — the included angle. If your given setup doesn’t match, use the sine or cosine rule first to find the missing piece.

Which rule to use?

You have	You want	Rule to use
One side, two angles	Another side	Sine rule
Two sides, an angle opposite one of them	The angle opposite the other side	Sine rule (check ambiguous case)
Two sides and the included angle	The third side	Cosine rule
All three sides	Any angle	Cosine rule
Two sides and the included angle	The area	Area formula
The area, two sides	The included angle	Area formula (rearranged)
A right angle	Anything	SOH CAH TOA / Pythagoras

If none of the rules seems to fit, remember the angles in a triangle sum to 180°. Often a missing third angle unlocks the sine rule. And harder problems may chain multiple rules: cosine rule first to find a side, then area formula, for instance.

🧭 Recipe — solving a non-right-triangle problem

Sketch the triangle with all given information labelled. Mark uppercase for angles, lowercase for the sides opposite them.
Identify what you have and what you want. Use the table above to pick a rule.
Substitute carefully. The angle in any rule must pair correctly with its opposite side.
For sine rule angles: check the ambiguous case. Compute 180° minus your answer and see if it can still fit with the given angle.
For cosine rule: take the positive square root for sides. For angles, use cos⁻¹ — which always returns a value between 0° and 180°, so no ambiguity.
For area: confirm the angle is between the two given sides. If not, find another angle first.
Round to 3 s.f. unless told otherwise. Keep exact values during intermediate steps.

Worked examples

WE 1

Sine rule — find a missing side

In triangle PQR, P = 40°, Q = 65°, and side r (opposite R) = 14 cm. Find the length of side q, correct to 3 s.f.

Step 1: Find the third angle R = 180° − 40° − 65° = 75° Step 2: Apply the sine rule with sides on top q/sin Q = r/sin R q/sin 65° = 14/sin 75° Step 3: Solve for q q = 14 × sin 65° / sin 75° q = 14 × 0.9063…/0.9659… q = 13.137… q ≈ 13.1 cm (3 s.f.) two angles given automatically gives the third — then sine rule pairs angle 65° with the unknown side

WE 2

Sine rule — find a missing angle

In triangle ABC, A = 50°, side a = 12 cm, and side b = 10 cm. Find angle B, correct to 3 s.f.

Step 1: Apply the sine rule with sines on top sin B/b = sin A/a sin B/10 = sin 50°/12 Step 2: Solve for sin B sin B = 10 × sin 50°/12 sin B = 10 × 0.766…/12 = 0.6383… Step 3: Take inverse sine and check ambiguous case B = sin⁻¹(0.6383…) = 39.68…° obtuse alternative: 180° − 39.68° = 140.32° check: 50° + 140.32° = 190.32° > 180° → not possible B ≈ 39.7° (unique answer) side a (opposite given angle) is bigger than side b, so the obtuse alternative fails the angle-sum check — only the acute solution works

WE 3

Ambiguous case of the sine rule

In triangle ABC, A = 30°, side a = 5 cm, and side b = 8 cm. Find both possible values of angle B, correct to 3 s.f.

Step 1: Apply sine rule sin B/8 = sin 30°/5 sin B = 8 × 0.5/5 = 0.8 Step 2: Find the acute solution B₁ = sin⁻¹(0.8) = 53.130…° Step 3: Find the obtuse alternative B₂ = 180° − 53.13° = 126.87° Step 4: Check both are valid B₁: 30° + 53.13° = 83.13° < 180° ✓ B₂: 30° + 126.87° = 156.87° < 180° ✓ B ≈ 53.1° or B ≈ 126.9° both answers are geometrically possible — two distinct triangles satisfy the given data; quote both unless context rules one out

WE 4

Cosine rule — find a missing side (SAS)

In triangle ABC, a = 5 cm, c = 7 cm, and the included angle B = 100°. Find the length of side b, correct to 3 s.f.

Step 1: Apply cosine rule with B and its opposite side b b² = a² + c² − 2ac cos B b² = 5² + 7² − 2(5)(7) cos 100° Step 2: Compute b² = 25 + 49 − 70 × (−0.1736…) b² = 74 + 12.155… = 86.155… Step 3: Take the positive square root b = √86.155… = 9.282… b ≈ 9.28 cm (3 s.f.) cos of an obtuse angle is negative — that flipped sign added to the right-hand side, making b larger than either of the two given sides as expected

WE 5

Cosine rule — find an angle (SSS)

In triangle PQR, PQ = 6 cm, QR = 9 cm, and PR = 11 cm. Find the size of angle Q, correct to 3 s.f.

Step 1: Identify side opposite Q angle Q is opposite PR — call this side q = 11 other two sides: p = QR = 9; r = PQ = 6 Step 2: Apply cos formula cos Q = (p² + r² − q²)/(2pr) cos Q = (81 + 36 − 121)/(2 × 9 × 6) cos Q = −4/108 = −0.0370… Step 3: Apply inverse cosine Q = cos⁻¹(−0.0370…) = 92.12…° Q ≈ 92.1° (3 s.f.) cos⁻¹ always returns a unique value in (0°, 180°), so no ambiguous case for the cosine rule — the sign of cos Q tells you whether the angle is acute or obtuse

WE 6

Combined — cosine rule and area formula

In triangle ABC, AB = 9 cm, BC = 12 cm, and the angle at B is 70°. Find (a) the length of AC, and (b) the area of the triangle. Give both answers correct to 3 s.f.

(a) Cosine rule for AC (= side b, opposite B) b² = AB² + BC² − 2(AB)(BC) cos B b² = 9² + 12² − 2(9)(12) cos 70° b² = 81 + 144 − 216 × 0.342… b² = 225 − 73.872… = 151.13… b = √151.13… = 12.294… (a) AC ≈ 12.3 cm (3 s.f.) (b) Area = ½ × AB × BC × sin B A = ½ × 9 × 12 × sin 70° A = 54 × 0.9397… A = 50.74… (b) area ≈ 50.7 cm² (3 s.f.) area used the two sides given (AB and BC) and the angle between them (B = 70°) — a perfect fit for the area formula directly, no need for additional rules

💡 Top tips

Sketch first, label second. Match each side to the angle opposite it before substituting into any rule.
Sine rule for finding a side: keep sides on top. Sine rule for finding an angle: flip so sines are on top — easier inverse step.
Always check the ambiguous case when finding an angle via sine rule. Compute 180° minus your answer and check if it can still fit with the given angle.
Cosine rule has no ambiguous case — cos⁻¹ returns a unique value in (0°, 180°). Use it when in doubt.
For the area formula, the angle must be the one between the two given sides. If not, find a missing piece first.
If no rule seems to fit, find the missing third angle (180° − sum of the other two). It often unlocks the sine rule.
Mind the units: if one side is in cm and another in m, convert first. Mixed units in the same equation give garbage.

⚠ Common mistakes

Mismatching angles and opposite sides. The sine rule and cosine rule both rely on each angle being paired with its opposite side, not an adjacent one.
Forgetting the ambiguous case on the sine rule. Always check whether the obtuse alternative is possible.
Using SOH CAH TOA on a non-right triangle. Only works when one angle is exactly 90°.
Using the area formula with a non-included angle. The angle in ½ab sin C must be between the two sides — not opposite.
Sign error with cos of an obtuse angle. cos 100° is negative, so −2ab cos C becomes positive — adding to the squared sides, making the result larger.
Forgetting to take the square root at the end of the cosine rule when finding a side.
Calculator in radian mode when the angle was given in degrees (or vice versa). Check the mode before each calculation.
Premature rounding. Carry exact values (or many decimals) until the very last step.

With both notes done, you can solve any triangle problem the IB throws at you — right-angled or not. The next note, Angles of Elevation & Depression, applies this trig toolkit to a specific real-world setup: looking up at something (elevation) or down at something (depression). The geometry stays familiar, but you now read the angles from a horizontal sight line rather than from the triangle’s vertex.

Need help with the Sine, Cosine & Area rules?

Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.

Book Free Session →