The Cosine Rule
Use a² = b² + c² - 2bc cos A when the sine rule does not apply: three sides given, or two sides with the angle between them.
What a learner can do afterwards
- Find the third side from two sides and the included angle
- Rearrange the rule to find an angle from three known sides
- Recognise Pythagoras as the case where the angle is 90 degrees
1 · Read
When the sine rule cannot start, reach for the cosine rule: c squared equals a squared plus b squared minus 2ab cos C. Use it for two setups only: two sides with the angle trapped between them to find the third side, or all three sides to find an angle.
Finding an angle means rearranging first. Make cos of the wanted angle the subject: cos A equals b squared plus c squared minus a squared, all over 2bc. Substitute the three sides, then use inverse cosine to land on the angle.
Two habits save marks. First, check what is known before you pick a rule, and label the triangle before you calculate. Second, remember cos 90 is 0: at a right angle the extra term vanishes and Pythagoras is all that remains, which is why 3, 4, 5 works exactly.
Two sides plus their trapped angle, or three sides rearranged, and Pythagoras appears when the angle is 90.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.