Differentiation rules

When angles $\theta$ or $h$ are measured in radians and close to zero, trigonometric functions can be approximated using basic polynomials. Specifically, as $h\to 0$, the ratio $(\sin h)/h$ approaches $1$, which gives the small-angle approximation $\sin h\approx h$. Using the fundamental identity ${\cos }^{2}h=1-{\sin }^{2}h$, we find ${\cos }^{2}h\approx 1-h^2$. Taking the square root via binomial expansion yields $\cos h\approx 1-\frac{1}{2}{h}^2$. These small-angle approximations allow us to evaluate trigonometric limits near zero by replacing sine and cosine terms with simple algebraic expressions in $h$.