Cosine rule calculator with working

• Missing side: $b = 5$, $c = 7$, $A = 60°$. $a^2 = 25 + 49 - 2 \times 5 \times 7 \times \cos 60° = 74 - 35 = 39$, so $a = \sqrt{39} \approx 6.24$.
• Missing angle: $a = 7$, $b = 5$, $c = 8$. $\cos A = \dfrac{25 + 64 - 49}{80} = 0.5$, so $A = 60°$.

Work out $2bc\cos A$ as one product and subtract it from $b^2 + c^2$. A common error is calculating $(b^2 + c^2 - 2bc) \times \cos A$, which gives a completely different number. Type it carefully into your calculator, or use brackets.

If $\cos A$ comes out negative, the angle is obtuse (more than 90°): $\cos^{-1}$ handles that automatically. Unlike the sine rule, the cosine rule never gives an ambiguous answer, so it's the safer choice for finding the largest angle of a triangle.

Once you have two sides and the angle between them, the area is $\tfrac{1}{2}ab\sin C$. For the first example, the area is $\tfrac{1}{2} \times 5 \times 7 \times \sin 60° \approx 15.16$ square units.

• Using an angle that isn't between the two known sides.
• Forgetting to square-root at the end when finding a side.
• Calculator in radians.

Practise with our cosine rule questions, and use the sine rule calculator when you have a side-angle pair.

• Label the triangle so the angle you know (or want) is $A$.
• Write $a^2 = \dots$ with the numbers in, then the value of $a^2$, then $a$.
• The cosine rule is Higher tier at GCSE and Extended tier at IGCSE.

Two paths leave a gate at an angle of 110°. One is 300 m long and the other 450 m. How far apart are their ends?

$a^2 = 300^2 + 450^2 - 2 \times 300 \times 450 \times \cos 110° = 90000 + 202500 + 92345 = 384845$, so $a \approx 620$ m.

Because the angle is obtuse, $\cos 110°$ is negative, so the last term is added. That's why the distance is longer than you might expect: an obtuse angle pushes the ends further apart.

Choosing the rule: if you can see a side with its opposite angle, use the sine rule; if the known angle is between the two known sides, or you know all three sides, use the cosine rule.