Cosine Rule — Finding a Side

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The Cosine Rule — Finding a Side

When the Sine Rule won’t work

If you have two sides and the angle between them (and no opposite pair), use the Cosine Rule: $a^2 = b^2 + c^2 - 2bc\cos A$. Side $a$ faces the known angle $A$.

Worked example — two sides and the angle between

In a triangle $b = 5$, $c = 6$ and the angle between them $A = 60^\circ$. Find $a$.

$a^2 = 5^2 + 6^2 - 2(5)(6)\cos 60$

$a^2 = 61 - 60(0.5) = 31 \Rightarrow a = \sqrt{31} = 5.6$

$a = 5.6$

Worked example — another

In a triangle $b = 7$, $c = 8$ and $A = 50^\circ$. Find $a$.

$a^2 = 7^2 + 8^2 - 2(7)(8)\cos 50 = 113 - 112\cos 50 = 41$

$a = \sqrt{41} = 6.4$

$a = 6.4$

Worked example — with named vertices

In $\triangle ABC$, $|AB| = 5.6$, $|AC| = 4.3$ and $|\angle BAC| = 57^\circ$. Find $|BC|$.

$|BC|^2 = 4.3^2 + 5.6^2 - 2(5.6)(4.3)\cos 57 = 23.62$

$|BC| = 4.9$

You try

In a triangle $b = 5$, $c = 8$ and $A = 23^\circ$. Find $a$.

$a^2 = 5^2 + 8^2 - 2(5)(8)\cos 23$.

$a^2 = 89 - 80\cos 23 = 15 \Rightarrow a = 3.91 = 3.9$

$a = 3.9$

You try

In a triangle $b = 5$, $c = 7$ and $A = 30^\circ$. Find $a$.

$a^2 = 5^2 + 7^2 - 2(5)(7)\cos 30$.

$a^2 = 74 - 70\cos 30 = 13.3 \Rightarrow a = 3.65 = 3.7$

$a = 3.7$

That’s Class 8.

The Cosine Rule to find a side. Class 9: the Cosine Rule to find an angle.