The Cosine Rule & the Sine Rule

The Cosine Rule for any triangle is: 

\[ a^2 = b^2+ c^2 - 2bc\cos A\]

For example: find the missing side length: 

Note that we need to rewrite the Cosine Rule to be in terms of \(c\) (instead of \(a\)), which gives:
\begin{eqnarray*}
c^{2}&=& a^{2} + b^{2} - 2ab\cos C \\
&=& 2^{2}+3^{2}-2\times 2\times3\cos 120^{\circ} \\
&=& 19 \\
c &=&\sqrt{19} \approx 4.359 \end{eqnarray*}

The Sine Rule for any triangle is: 

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] 

For example: find the side length \(h\):

Using the Sine Rule:
\begin{eqnarray*}
\frac{h}{\sin32^{\circ}} &=& \frac{12}{\sin25^{\circ}} \\
h &=& \frac{12\sin32^{\circ}}{\sin 25^{\circ}} \\
&\approx& 15.05 
\end{eqnarray*}