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Higher tier essential: apply the sine rule, cosine rule and ½absinC to non-right-angled triangles. Includes the ambiguous case and rule-selection strategy.
Advanced trigonometry (Topic 4.7, Higher tier only) covers three formulae for non-right-angled triangles: the sine rule (a/sinA = b/sinB = c/sinC), the cosine rule (a² = b² + c² − 2bc·cosA), and the area formula (Area = ½ab·sinC). Students must choose the correct rule based on the given information, handle the ambiguous case of the sine rule, and combine rules in multi-step problems. This topic typically accounts for 4–6 marks on each Higher tier paper.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The sine rule connects sides and angles in any triangle (not just right-angled ones). It states:
a / sin A = b / sin B = c / sin C
Here, lowercase letters represent sides and uppercase letters represent the angle opposite that side. The rule can also be flipped to sin A/a = sin B/b = sin C/c, which is more convenient when you are finding an angle rather than a side.
The cosine rule is an extension of Pythagoras' theorem for non-right-angled triangles. It states:
a² = b² + c² − 2bc·cos A
Notice that if A = 90°, then cos A = 0 and the formula reduces to a² = b² + c², which is Pythagoras' theorem. The “minus 2bc·cos A” term adjusts for the angle not being a right angle.
The standard area formula ½ × base × height requires a perpendicular height, which is not always given. The trigonometric area formula works with any two sides and the included angle:
Area = ½ab·sin C
Here, a and b are any two sides and C is the angle between them (the included angle). This formula is given on the IGCSE formula sheet, but you must remember when to apply it.
Use this decision flowchart every time you face a non-right-angled triangle problem:
Practising this flowchart until it becomes automatic is the single most effective revision strategy for this topic.
When you use the sine rule to find an angle, you calculate sin B and then take the inverse sine. However, the inverse sine function on your calculator only gives an acute angle (between 0° and 90°). There is always a second possible value: 180° minus the calculator answer.
For example, if sin B = 0.866, your calculator gives B = 60°. But B could also be 120° (since sin 120° = 0.866 as well). You must check whether 120° creates a valid triangle by adding it to the other known angle — if the sum exceeds 180°, that solution is impossible and only the acute angle works.
At IGCSE, the question often specifies whether the triangle is acute or obtuse. If it does not, present both solutions or state which one is geometrically valid.
In triangle ABC, angle A = 42°, angle B = 73° and side a = 8 cm. Find the length of side b.
Step 1 — Identify the rule: We have angle A and side a (a complete pair), plus angle B. We need side b. This is an angle-side pair situation → sine rule.
Step 2 — Set up the sine rule: a / sin A = b / sin B
8 / sin 42° = b / sin 73°
Step 3 — Solve for b:
b = (8 × sin 73°) / sin 42°
b = (8 × 0.9563) / 0.6691
b = 7.6504 / 0.6691
b = 11.4 cm (3 s.f.)
Always check you have a matching angle-side pair before applying the sine rule. Here, angle A = 42° is opposite side a = 8 cm.
In triangle PQR, p = 11 cm, q = 7 cm and r = 9 cm. Find angle P.
Step 1 — Identify the rule: We have all three sides (SSS) and need an angle → cosine rule (rearranged).
Step 2 — Rearrange for cos P:
cos P = (q² + r² − p²) / (2qr)
cos P = (49 + 81 − 121) / (2 × 7 × 9)
cos P = 9 / 126
cos P = 0.07143
Step 3 — Find the angle:
P = cos−1(0.07143) = 85.9° (1 d.p.)
When rearranging the cosine rule, the side opposite the angle you want goes in the “minus” position (here p² is subtracted). Keep at least 4 significant figures in intermediate steps to avoid rounding errors.
In triangle XYZ, x = 14 cm, y = 10 cm and angle Z = 52°. Find the area of the triangle, then use the sine rule to find angle X.
Part (a) — Find the area:
Area = ½ × x × y × sin Z
Area = ½ × 14 × 10 × sin 52°
Area = 70 × 0.7880
Area = 55.2 cm² (3 s.f.)
Part (b) — Find side z first (cosine rule):
z² = x² + y² − 2xy·cos Z
z² = 196 + 100 − 2(14)(10)cos 52°
z² = 296 − 280 × 0.6157
z² = 296 − 172.4 = 123.6
z = 11.12 cm
Part (c) — Find angle X (sine rule):
sin X / x = sin Z / z
sin X / 14 = sin 52° / 11.12
sin X = (14 × 0.7880) / 11.12 = 0.9920
X = sin−1(0.9920) = 82.7° (1 d.p.)
Check: X + Y + Z = 82.7 + Y + 52 = 180 → Y = 45.3°. All angles are positive and sum to 180°, so the solution is valid.
This Grade 9 question combines three techniques: the area formula, the cosine rule and the sine rule. Use the ANS button on your calculator between steps to avoid premature rounding.
Choosing the wrong rule: Students apply the sine rule when they have SAS (no angle-side pair), or the cosine rule when they have an angle-side pair. Always check: do I have a matching angle and its opposite side? If yes, sine rule. If no, cosine rule.
Using the wrong angle in the area formula: The angle in ½ab·sinC must be the included angle (between the two sides). Using any other angle gives the wrong answer.
Forgetting the ambiguous case: When using the sine rule to find an angle, always check whether 180° minus your answer is also a valid solution. If the question does not specify acute or obtuse, present both possibilities.
Rounding too early: The cosine rule involves squaring, subtracting, and then taking a square root. Rounding intermediate values introduces significant error. Keep at least 4 significant figures or use the ANS button throughout.
Label sides and angles consistently: Use lowercase for sides and uppercase for the opposite angle. This prevents substitution errors in both the sine and cosine rules.
Write the formula before substituting: The mark scheme awards a method mark for writing a/sinA = b/sinB or a² = b² + c² − 2bc·cosA before putting numbers in. Never skip this step.
Check your answer makes sense: A side cannot be longer than the sum of the other two sides (triangle inequality). An angle must be between 0° and 180°. If your answer breaks these rules, re-check your working.
Use the formula sheet wisely: The sine rule, cosine rule and area formula are all given on the IGCSE formula sheet. Focus your revision on knowing when to use each formula rather than memorising the formulae themselves.
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