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Cosine Rule

Learn how to use the Cosine Rule, a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2− 2bc\cos A, to find sides or angles in non-right-angled triangles. Let’s get started! 🚀

Cosine Rule - introduction visual

Video Lesson

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Flashcards

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Cosine Rule formula a² = b² +c² -2bc·cos(A) explained with a triangle diagram labelled with sides a, b, c, and angle A.Using the cosine rule to find a side in a triangle with sides 3 cm and 5 cm and an angle of 60 degrees between them.Cosine rule formulas for finding angles in a triangle, with labelled triangle and angle A highlighted in pink.Triangle GEF with sides 4 cm, 3 cm, and 2 cm, applying the cosine rule to find angle F as 47 degrees using a calculator.

🛎️ The Cosine Rule

  • The cosine rule is used to find a missing side or angle in any triangle.
  • It is used when the triangle is not right-angled (otherwise use Pythagoras’ theorem).

🛎️ Example: The Cosine Rule for Finding a Side

  • If you know two sides and the angle between them, you can find the third side.
  • The rule is a2=b2+c2−2bccos⁡(A)a^2 = b^2 + c^2 - 2bc\cos(A), where aa is the unknown side and AA is the angle between bb and cc.

🛎️ Rearranging the Cosine Rule to Find an Angle

  • If you know all three sides, you can find a missing angle.
  • Rearrange to cos⁡(A)=b2+c2−a22bc\cos(A) = \dfrac{b^2 + c^2 - a^2}{2bc}. Once you find cos⁡(A)\cos(A), you can find angle AA.

🛎️ Example: Finding an Angle Using the Cosine Rule

  • Substitute the three side lengths into the rearranged formula.
  • Once you find cos⁡(F)\cos(F), you can find angle FF.
  • Use cos⁡−1\cos^{-1} on your calculator to find the angle in degrees.

Practice Questions

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Interactive Activity

Practice using the Cosine Rule to find a missing side or angle of a non-right-angled triangle

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