Question

In a ∆ABC, if $\angle C=\frac{\mathrm{\pi }}{2},\angle A=\frac{\mathrm{\pi }}{6},$ c = 20, then a = ____________.

Answer 1

In ∆ABC

∠C = $\frac{\mathrm{\pi }}{2}$, ∠ A = $\frac{\mathrm{\pi }}{6}$
⇒∠B = $\frac{\mathrm{\pi }}{3}$ (By angle sum property A + B + C = $\mathrm{\pi }$)
using sina formula,

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

⇒ $\frac{a}{\sin {\displaystyle \frac{\mathrm{\pi }}{6}}}=\frac{20}{s\mathrm{in}{\displaystyle \frac{\mathrm{\pi }}{2}}}$ ($\because$c = 20 given)
⇒ a = sin$\frac{\mathrm{\pi }}{6}\times 20$ ($\because$ sin π/2 = 1)
$\Rightarrow a=\frac{1}{2}\times 20$
i.e a = 10