$\Delta ABC$ में जिसका कोण $B$ समकोण है, $AB =5 \,cm$ और $\angle ACB =30^{\circ}($ देखिए आकृति $)$ भुजाओं $BC$ और $AC$ की लंबाइयाँ ज्ञात करें।

To find the length of the side $BC ,$ we will choose the trigonometric ratio involving $BC$ and the given side $AB$. since $BC$ is the side adjacent to angle $C$ and $AB$ is the side opposite to angle $C ,$ therefore

$\frac{ AB }{ BC }=\tan C$

$\frac{5}{ BC }=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$

which gives $BC =5 \sqrt{3} \,cm$

To find the length of the side $AC ,$ we consider

$\sin 30^{\circ}=\frac{ AB }{ AC }$

$\frac{1}{2}=\frac{5}{ AC }$

$AC =10 \,cm$

Note that alternatively we could have used Pythagoras theorem to determine the third side in the example above,

$AC =\sqrt{ AB ^{2}+ BC ^{2}}=\sqrt{5^{2}+(5 \sqrt{3})^{2}} cm =10 \,cm$