Small Angle Approximation in Trigonometry

Our previous blog post used the method of proving by contradiction to prove for the inequality $\sin 10^{\circ}>\dfrac{1}{6}$ (Prove that $\sin 10^{\circ}>\dfrac{1}{6}$).

Alternatively, we could also use the "Small-angle Approximation" to prove that $\sin 10^{\circ}>\dfrac{1}{6}$, provided we know fully well that as the angle approaches zero (or the angle is less than $0.2 \,\,\text{radians}$, it is clear that the gap between the approximation and the original function quickly vanishes.
 (Graph taken from Small-angle_approximation (Wikipedia))