به انگلیسی

The rule for trapped angles

Mathematics · Geometry · ages 15-16

Name ______________________ Date ____________

Which formula finds a third side from two sides and the angle between them?
- a over sin A equals b over sin B, which is the sine rule
- a squared equals b squared plus c squared minus 2bc cos A
- Opposite divided by adjacent, which gives tangent
Why is Pythagoras just the cosine rule with a 90 degree angle?
- Because cos 90 is 0, so the extra term vanishes
- Because cos 90 is 1, so the extra term doubles the result
- Because sin 90 is 0, so the rule flips
Two sides of a triangle are 5 cm and 7 cm, with a 60 degree angle between them. Find the third side, in cm, using the cosine rule. Round your answer to 1 decimal place.
Answer: ______________
Two sides of a triangle are 5 cm and 7 cm, with a 60 degree angle between them. Find the third side in cm using the cosine rule. Round your answer to 1 decimal place.
Answer: ______________
A triangle has sides 7 cm, 8 cm, and 10 cm. Find the angle opposite the 8 cm side, in degrees. Round your answer to 1 decimal place.
Answer: ______________
A triangle has sides 5 cm, 7 cm, and 8 cm. Find the angle opposite the 7 cm side, in degrees.
Answer: ______________
Two sides are 9 cm and 11 cm with an included angle of 60 degrees. Find the third side in cm. Round your answer to 1 decimal place.
Answer: ______________
Two sides are 7 cm and 8 cm with an included angle of 120 degrees. What is the third side?
- 13 cm
- 15.5 cm
- 3.6 cm
Two sides of a triangle are both 7 cm, and the angle between them is 60 degrees. Find the third side, in cm.
Answer: ______________
You know all three sides and need one angle. Which rearranged form do you use?
- a squared equals b squared plus c squared minus 2bc cos A, the cosine rule
- sin A over a equals sin B over b, the sine rule
- cos A equals b squared plus c squared minus a squared, over 2bc

Answer key

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

The rule for trapped angles W1-mt_yy73gK5CEP-s1

a squared equals b squared plus c squared minus 2bc cos A · That is the cosine rule, built for the trapped-angle setup.
Because cos 90 is 0, so the extra term vanishes · With C at 90 degrees the 2ab cos C term becomes zero.
6.2 · c squared equals 25 plus 49 minus 2 times 5 times 7 times cos 60, which is 74 minus 35, so 39, and the root is about 6.2.
6.2 · c squared equals 25 plus 49 minus 35, which is 39, so the root is about 6.2.
52.6 · Cos A equals 49 plus 100 minus 64, all over 140, which is about 0.607, so A is about 52.6 degrees.
60 · Cos A equals 25 plus 64 minus 49, all over 80, which is 0.5, so A is 60 degrees.
10.1 · c squared equals 81 plus 121 minus 99, which is 103, so the root is about 10.1.
13 cm · 49 plus 64 is 113, and cos 120 is minus 0.5, so the cosine term adds 56, giving 169, whose root is 13.
7 · c squared equals 49 plus 49 minus 49, which is 49, so c is 7.
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.