Geometric For a small angle, and are almost the same length, and therefore is nearly 1. The segment (in red to the right) is the difference between the lengths of the hypotenuse, , and the adjacent side, , and has length \textstyle H - \sqrt{H^2 - O^2}, which for small angles is approximately equal to \textstyle O^2\!/2H \approx \tfrac12 \theta^2H. As a second-order approximation, \cos{\theta} \approx 1 - \frac{\theta^2}{2}. The opposite leg, , is approximately equal to the length of the blue arc, . The arc has length , and by definition and , and for a small angle, and , which leads to: \sin \theta = \frac{O}{H}\approx\frac{O}{A} = \tan \theta = \frac{O}{A} \approx \frac{s}{A} = \frac{A\theta}{A} = \theta. Or, more concisely, \sin \theta \approx \tan \theta \approx \theta.
Calculus Using the
squeeze theorem, \begin{align} \sin \theta &= \theta - \frac16\theta^3 + \frac1{120}\theta^5 - \cdots, \\[6mu] \cos \theta &= 1 - \frac1{2}{\theta^2} + \frac1{24}\theta^4 - \cdots, \\[6mu] \tan \theta &= \theta + \frac{1}{3}\theta^3 + \frac{2}{15}\theta^5 + \cdots. \end{align} where is the angle in radians. For very small angles, higher powers of become extremely small, for instance if , then , just one ten-thousandth of . Thus for many purposes it suffices to drop the cubic and higher terms and approximate the sine and tangent of a small angle using the radian measure of the angle, , and drop the quadratic term and approximate the cosine as . If additional precision is needed the quadratic and cubic terms can also be included, , , and . ==Error of the approximations==