The Cosine Rule · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The rule for trapped angles

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. Which formula finds a third side from two sides and the angle between them?

    • a squared equals b squared plus c squared minus 2bc cos A
    • a over sin A equals b over sin B
    • Opposite divided by adjacent

  2. Why is Pythagoras just the cosine rule with a 90 degree angle?

    • Because cos 90 is 1, so the extra term doubles
    • Because cos 90 is 0, so the extra term vanishes
    • Because sin 90 is 0, so the rule flips

  3. Two sides of a triangle are 3 cm and 4 cm, with a 90 degree angle between them. Find the third side, in cm, using the cosine rule.

    Answer: ______________

  4. Two sides of a triangle are 3 cm and 4 cm, and the angle between them is 90 degrees. Find the third side in cm using the cosine rule.

    Answer: ______________

  5. A triangle has sides 3 cm, 4 cm and 5 cm. What is the size of the angle opposite the 5 cm side? Give your answer in degrees.

    Answer: ______________

  6. A triangle has sides 3 cm, 4 cm, and 5 cm. What is the size of the angle opposite the 5 cm side, in degrees?

    Answer: ______________

  7. Two sides are 5 cm and 12 cm with an included angle of 90 degrees. Find the third side in cm.

    Answer: ______________

  8. Two sides are 6 cm and 8 cm with an included angle of 90 degrees. What is the third side?

    • 10
    • 14
    • 7
    • 48

  9. Two sides of a triangle are both 7 cm, and the angle between them is 60 degrees. Find the third side, in cm.

    Answer: ______________

  10. You know all three sides and need one angle. Which rearranged form do you use?

    • cos A equals b squared plus c squared minus a squared, over 2bc
    • a squared equals b squared plus c squared minus 2bc cos A
    • sin A over a equals sin B over b
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Answer key

For grown-ups. Fold this page away before handing over the rest.

The rule for trapped angles W1-mt_yy73gK5CEP-s1

  1. a squared equals b squared plus c squared minus 2bc cos A · That is the cosine rule, built for the trapped-angle setup.
  2. Because cos 90 is 0, so the extra term vanishes · With C at 90 degrees the 2ab cos C term becomes zero.
  3. 5 · c squared equals 9 plus 16 minus zero, which is 25, so c is 5.
  4. 5 · c squared = 9 + 16 - 2 x 3 x 4 x cos 90 = 25, so c = 5.
  5. 90 · 3 squared + 4 squared = 9 + 16 = 25 = 5 squared, so the triangle is right angled and the angle opposite the 5 cm side is 90 degrees.
  6. 90 · 9 plus 16 equals 25, so the triangle is right angled at that corner.
  7. 13 · c squared = 25 + 144 = 169, so c = 13.
  8. 10 · 36 + 64 = 100 with no adjustment, since cos 90 is 0, and the square root of 100 is 10.
  9. 7 · c squared equals 49 plus 49 minus 49, which is 49, so c is 7.
  10. cos A equals b squared plus c squared minus a squared, over 2bc · Making cos A the subject pulls the wanted angle out of three known sides.
Worksheet · LightMySky