Sine and cosine rules
Cambridge IGCSE Mathematics (0580) · Unit 6: Trigonometry · 9 flashcards
Sine and cosine rules is topic 6.2 in the Cambridge IGCSE Mathematics (0580) syllabus , positioned in Unit 6 — Trigonometry , alongside Trigonometric ratios, 3D trigonometry and Trigonometric graphs. In one line: The Sine Rule: a/sin(A) = b/sin(B) = c/sin(C). It's used to find unknown sides or angles in non-right-angled triangles when you know two angles and a side opposite one of them, or two sides and an angle opposite one of them.
This topic is examined across Paper 1 (Core) or Paper 2 (Extended) — non-calculator — and Paper 3 (Core) or Paper 4 (Extended) — calculator. It is a Supplement (Extended-tier) topic, so it appears only on the Extended-tier papers.
The deck below contains 9 flashcards — 3 definitions, 3 key concepts and 1 application card — covering the precise wording mark schemes reward. Use the 3 definition cards to lock down command-word answers (define, state), then move on to the concept and application cards to handle explain, describe and compare questions.
State the Sine Rule. When is it used
The Sine Rule: a/sin(A) = b/sin(B) = c/sin(C). It's used to find unknown sides or angles in non-right-angled triangles when you know two angles and a side opposite one of them, or two sides and an angle opposite one of them.
Questions this Sine and cosine rules deck will help you answer
- › Explain how to determine if the ambiguous case exists when solving a triangle using the Sine Rule.
- › When finding an angle using the Cosine Rule, what must you ensure your calculator is set to?
- › A surveyor needs to find the distance across a lake. From point A, she measures the distance to point B as 500m and to point C as 750m. The angle BAC is 72°. Find the distance BC.
- › Explain how to find the largest angle in a triangle given the lengths of all three sides.
State the Sine Rule. When is it used?
The Sine Rule: a/sin(A) = b/sin(B) = c/sin(C). It's used to find unknown sides or angles in non-right-angled triangles when you know two angles and a side opposite one of them, or two sides and an angle opposite one of them.
State the Cosine Rule for finding a side. When is it used?
The Cosine Rule (for finding a side): a² = b² + c² - 2bc cos(A). Use it when you know two sides and the included angle (the angle between them) of a non-right-angled triangle and want to find the third side.
State the Cosine Rule for finding an angle. When is it used?
The Cosine Rule (for finding an angle): cos(A) = (b² + c² - a²) / 2bc. Use it when you know all three sides of a non-right-angled triangle and want to find an angle.
In triangle ABC, AB = 8cm, BC = 6cm, and angle ABC = 60°. Find the area of the triangle.
Area = (1/2) * ab * sin(C). Area = (1/2) * 8cm * 6cm * sin(60°) = 20.78 cm² (2 d.p.)
Explain how to determine if the ambiguous case exists when solving a triangle using the Sine Rule.
The ambiguous case occurs when given two sides and a non-included angle (SSA). Check if there are two possible triangles by calculating both possible angles using the arcsin. If both angles are less than 180 degrees and add up to less than 180 degrees with the known angle, there are two solutions.
When finding an angle using the Cosine Rule, what must you ensure your calculator is set to?
Ensure your calculator is in degree mode. Otherwise, your angle calculations will be incorrect, as the functions such as sine, cosine, and tangent, operate differently depending on whether the angle provided is in degrees or radians.
In triangle PQR, PQ = 10cm, QR = 7cm, and angle QPR = 30°. Find the possible values of angle PRQ.
Use the Sine Rule: sin(PRQ)/10 = sin(30)/7. sin(PRQ) = (10*sin(30))/7. PRQ = arcsin((10*sin(30))/7) = 45.58°. Another solution: 180° - 45.58° = 134.42° (Ambiguous case).
A surveyor needs to find the distance across a lake. From point A, she measures the distance to point B as 500m and to point C as 750m. The angle BAC is 72°. Find the distance BC.
Use the Cosine Rule: BC² = AB² + AC² - 2(AB)(AC)cos(BAC). BC² = 500² + 750² - 2(500)(750)cos(72°). BC = √(500² + 750² - 2(500)(750)cos(72°)) = 762.2 m (1 d.p.)
Explain how to find the largest angle in a triangle given the lengths of all three sides.
The largest angle is always opposite the longest side. Use the Cosine Rule to find the angle opposite the longest side: cos(A) = (b² + c² - a²) / 2bc, where 'a' is the longest side.
Key Questions: Sine and cosine rules
State the Sine Rule. When is it used?
The Sine Rule: a/sin(A) = b/sin(B) = c/sin(C). It's used to find unknown sides or angles in non-right-angled triangles when you know two angles and a side opposite one of them, or two sides and an angle opposite one of them.
State the Cosine Rule for finding a side. When is it used?
The Cosine Rule (for finding a side): a² = b² + c² - 2bc cos(A). Use it when you know two sides and the included angle (the angle between them) of a non-right-angled triangle and want to find the third side.
State the Cosine Rule for finding an angle. When is it used?
The Cosine Rule (for finding an angle): cos(A) = (b² + c² - a²) / 2bc. Use it when you know all three sides of a non-right-angled triangle and want to find an angle.
More topics in Unit 6 — Trigonometry
Sine and cosine rules sits alongside these Mathematics decks in the same syllabus unit. Each uses the same spaced-repetition system, so progress in one informs the next.
Cambridge syllabus keywords to use in your answers
These are the official Cambridge 0580 terms tagged to this section. Mark schemes credit responses that use the exact term — weave them into your answers verbatim rather than paraphrasing.
Key terms covered in this Sine and cosine rules deck
Every term below is defined in the flashcards above. Use the list as a quick recall test before your exam — if you can't define one of these in your own words, flip back to that card.
Related Mathematics guides
Long-read articles that go beyond the deck — cover the whole subject's common mistakes, high-yield content and revision pacing.
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