Proof of the small angle approximation sinθ≈θ using the geometry and motion of a simple pendulum

Figures & data

Figure 1. Pendulum OB of length $l$ and bob mass $m$, swinging through an angle $\theta$, along an arc AB of length $s=l\theta$. The weight of the bob is $mg$, such that the restoring force $F$ along the arc (i.e. at right angles to the pendulum) is $F=mg\text{sin}(\theta )$.

Table 1. Increasing accuracy of the small angle approximation $\text{sin}(\theta )/\theta \approx 1$ as $x/l$ gets smaller, and $\sqrt{1-{(x/l)}^2}=\text{cos}(\theta )$ approaches 1 (cf. inequalities (11) and (18)).

$x/l$	$\sqrt{1-{(x/l)}^2}$	$\le$	$\text{sin}(\theta )/\theta$	$\le$	1/$\sqrt{1-{(x/l)}^2}$
0.01000	0.99995	$\le$	0.99998	$\le$	1.00005
0.02000	0.99980	$\le$	0.99993	$\le$	1.00020
0.02500	0.99969	$\le$	0.99990	$\le$	1.00031
0.05000	0.99875	$\le$	0.99958	$\le$	1.00125
0.10000	0.99499	$\le$	0.99833	$\le$	1.00504
0.20000	0.97980	$\le$	0.99335	$\le$	1.02062
0.25000	0.96825	$\le$	0.98962	$\le$	1.03280
0.50000	0.86603	$\le$	0.95885	$\le$	1.15470