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Sine rule, cosine rule and 3D problems
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Any triangle can be solved with the sine and cosine rules. In 3D, find a right-angled triangle that contains the angle you want.
Sine and cosine rules
Sine rule: asin A = bsin B = csin C (learn this)
Cosine rule: a2 = b2 + c2 − 2bc cos A (given in the exam)
Area = 12ab sin C (learn this)
Cosine rule: a2 = b2 + c2 − 2bc cos A (given in the exam)
Area = 12ab sin C (learn this)
Which rule?
Two angles and a side, or two sides and an angle opposite one of them: sine rule.
Two sides and the angle between them, or all three sides: cosine rule.
To find an angle from three sides: cos A = (b2 + c2 − a2)2bc
Two sides and the angle between them, or all three sides: cosine rule.
To find an angle from three sides: cos A = (b2 + c2 − a2)2bc
Lines and planes in 3D
The angle between a line and a plane is the angle between the line and its projection (shadow) on the plane.
The angle between two planes is measured between two lines, one in each plane, both perpendicular to the common edge.
Cuboid 4 × 3 × 12: base diagonal 5, space diagonal 13, angle with base = tan−1(125) = 67.4°
The angle between two planes is measured between two lines, one in each plane, both perpendicular to the common edge.
Cuboid 4 × 3 × 12: base diagonal 5, space diagonal 13, angle with base = tan−1(125) = 67.4°
A triangle has sides 5, 6 and 7. Find its largest angle.
- The largest angle is opposite 7
- cos C = (25 + 36 − 49)60 = 0.2
- C = cos−1 0.2
Answer: 78.5°
Sine rule for opposite pairs, cosine rule for two sides and the included angle or three sides. Area = 12ab sin C. In 3D, find a right-angled triangle.
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