The sine rule and cosine rule let us solve triangles that are not right-angled. Label each side with the small letter of the angle opposite it: side a is opposite angle A, and so on.
The sine rule
Use it when you know an angle and its opposite side, plus one more angle or side. To find a missing angle instead, turn the ratios upside down and write sin A / a = sin B / b = sin C / c, then multiply to make the unknown sine the subject.
Key idea
a / sin A = b / sin B = c / sin C
The cosine rule
Use it for two sides and the included angle (to find the third side), or all three sides (to find an angle). Rearranged, cos A = (b2 + c2 − a2)/(2bc), so the cosine of an angle can be found directly from the three side lengths. A negative value tells you the angle is obtuse.
Key idea
a2 = b2 + c2 − 2bc cos A. The area of any triangle is Area = ½ ab sin C.
Worked example
Worked example
In a triangle a = 3 cm, b = 5 cm and the included angle C = 120°. Find side c.
c2 = 32 + 52 − 2(3)(5) cos 120° = 9 + 25 − 30 × (−0.5) = 34 + 15 = 49.
So c = √49 = 7 cm. The area = ½ × 3 × 5 × sin 120° = 7.5 × 0.866 = 6.50 cm2.
Remember
- Sine rule: use with an angle and its opposite side.
- Cosine rule: use for SAS (find a side) or SSS (find an angle).
- Always match each side to the angle opposite it.