Central angle intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .) Let be the minor arc of the circle between points and , and let be the radius of the circle. If the central angle is subtended by , then If the central angle is not subtended by the minor arc , then is a reflex angle and If a tangent at and a tangent at intersect at the exterior point , then denoting the center as , the angles (convex) and are supplementary (sum to 180°). A
Pendulum (mathematics) spring from its equilibrium position assumes the small angle approximation. The Lagrangian is then which leads to the following set of coupled differential equations: Adding and subtracting these two equations in turn, and applying the small angle approximation, gives two harmonic oscillator equations in the variables formula_61 and formula_62: with the corresponding solutions where and formula_66, formula_67, formula_47, formula_69 are constants of integration. Expressing the solutions in terms of formula_53 and formula_54 alone: If the bobs are not given an initial push, then the condition formula_73 requires formula_74, which gives (after some rearranging): Pendulum (mathematics) The mathematics of pendulums are