A pendulum is oscillating in a vertical plane. The acceleration at the extreme position and lowest position are equal. Then the angular amplitude of the pendulum is

acceleration at the extreme position gsin$\theta$  and lowest position $\frac{{\mathrm{v}}^2}{\mathrm{l}}$ . 
$\mathrm{gsin\theta }=\frac{{\mathrm{v}}^2}{\mathrm{l}}$ 
From conservation of energy  $\frac{1}{2}{\mathrm{mv}}^2=\mathrm{mgl}\left(1-\mathrm{cos\theta }\right)$

$\frac{{\mathrm{mv}}^2}{\mathrm{l}}=2\mathrm{mg}\left(1-\mathrm{cos\theta }\right)$

$\mathrm{gsin\theta }=2\mathrm{g}\left(1-\mathrm{cos\theta }\right)$

$\mathrm{sin\theta }=4{\sin }^2\left(\frac{\mathrm{\theta }}{2}\right)$

$2\sin \frac{\theta }{2}\cos \frac{\theta }{2}=4{\sin }^2\frac{\theta }{2}$

$\tan \frac{\mathrm{\theta }}{2}=\frac{1}{2}$

$\frac{\mathrm{\theta }}{2}={\tan }^{-1}\left(\frac{1}{2}\right)$

$\mathrm{\theta }=2{\tan }^{-1}\frac{1}{2}$