--- title: "The Cosine Rule" description: "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." canonical: https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP source: https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP ## Ready when they can - 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 ## Lesson: The rule for trapped angles 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. **Tip.** 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. **Recap.** Two sides plus their trapped angle, or three sides rearranged, and Pythagoras appears when the angle is 90. ## Practice 15 questions on this page, each with its working shown. ## Needs first - [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho) ## Opens up - [Choosing a Triangle Method](https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn) - [Triangle Area with Half ab sin C](https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b) - [Bearing Problems with the Sine and Cosine Rules](https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP)