Question

In $△ABC$, if $a=100,c=100\sqrt{2}$ and $A={30}^o$ then the angle $B=$

• A. ${105}^o$ or ${15}^o$
• B. ${45}^o$ or ${135}^o$
• C. ${60}^o$ or ${120}^o$
• D. ${30}^o$ or ${60}^o$

Answer 1

The correct option is A ${105}^o$ or ${15}^o$

Using cosine rule in triangle $ABC$,

$\cos A=\frac{b^2+c^2-a^2}{2bc}$
$\frac{\sqrt{3}}{2}=\frac{b^2+20000-10000}{2\times b\times 100\sqrt{2}}$
$b=51.763$ or $b=193.185$

Using sine rule, we have
$\frac{100}{\sin 30}=\frac{51.763}{\sin B}$
$\sin B=0.258$
$\therefore B=14.99={15}^o$

When $b=193.185$,
$\frac{100}{\sin {30}^{\circ }}=\frac{193.185}{\sin B}$
$\sin B=0.9659$
$\therefore B={105}^o$