In $\triangle$ $ABC,$ right-angled at $B$, $AB =5\, cm$ and $\angle ACB =30^{\circ}$ (see $Fig.$). Determine the lengths of the sides $BC$ and $AC .$
In $\triangle$ $ABC,$ right-angled at $B$, $AB =5\, cm$ and $\angle ACB =30^{\circ}$ (see $Fig.$). Determine the lengths of the sides $BC$ and $AC .$
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,
$AC =\sqrt{ AB ^{2}+ BC ^{2}}=\sqrt{5^{2}+(5 \sqrt{3})^{2}} cm =10 \,cm$
Similar Questions
State whether the following are true or false. Justify your answer.
The value of $\sin \theta$ increases as $\theta$ increases.
State whether the following are true or false. Justify your answer.
The value of $\sin \theta$ increases as $\theta$ increases.
In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:
$(i)$ $\sin A, \cos A$
$(ii)$ $\sin C, \cos C$
In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:
$(i)$ $\sin A, \cos A$
$(ii)$ $\sin C, \cos C$
Given $\tan A=\frac{4}{3},$ find the other trigonometric ratios of the $\angle A$
Given $\tan A=\frac{4}{3},$ find the other trigonometric ratios of the $\angle A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\sqrt{\frac{1+\sin A }{1-\sin A }}=\sec A +\tan A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\sqrt{\frac{1+\sin A }{1-\sin A }}=\sec A +\tan A$