6.2

Sine and cosine rules

9 flashcards to master Sine and cosine rules

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Definition Flip

State the Sine Rule. When is it used?

Answer Flip

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.

Definition Flip

State the Cosine Rule for finding a side. When is it used?

Answer Flip

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.

Definition Flip

State the Cosine Rule for finding an angle. When is it used?

Answer Flip

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.

Key Concept Flip

In triangle ABC, AB = 8cm, BC = 6cm, and angle ABC = 60°. Find the area of the triangle.

Answer Flip

Area = (1/2) * ab * sin(C). Area = (1/2) * 8cm * 6cm * sin(60°) = 20.78 cm² (2 d.p.)

Key Concept Flip

Explain how to determine if the ambiguous case exists when solving a triangle using the Sine Rule.

Answer Flip

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.

Key Concept Flip

When finding an angle using the Cosine Rule, what must you ensure your calculator is set to?

Answer Flip

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.

Key Concept Flip

In triangle PQR, PQ = 10cm, QR = 7cm, and angle QPR = 30°. Find the possible values of angle PRQ.

Answer Flip

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).

Key Concept Flip

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.

Answer Flip

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.)

Key Concept Flip

Explain how to find the largest angle in a triangle given the lengths of all three sides.

Answer Flip

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.

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6.1 Trigonometric ratios 6.3 3D trigonometry

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.

About Sine and cosine rules (6.2)

These 9 flashcards cover everything you need to know about Sine and cosine rules for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.

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