They are called second order because they include all terms of the Taylor series up to and including the term of order two, so that any discrepancy is of third order. Note that the second order terms are zero for sine and tan.Mr Davis 97 said:From the Wikipedia article https://en.wikipedia.org/wiki/Small-angle_approximation, it says that they are "second-order approximations." What makes all three second order? Shouldn't sin and tan be first-order and cos be second-order?
So for sin and tan are the second and first-order approximations the same? And for cos, are the zeroth and first-order approximations the same?andrewkirk said:They are called second order because they include all terms of the Taylor series up to and including the term of order two, so that any discrepancy is of third order. Note that the second order terms are zero for sine and tan.
YesMr Davis 97 said:So for sin and tan are the second and first-order approximations the same? And for cos, are the zeroth and first-order approximations the same?
The small angle expansions for sin, cos, and tan are mathematical formulas used to approximate the values of these trigonometric functions for small angles. This is often done to simplify calculations and make them more manageable.
The expansions for sin, cos, and tan are derived using the Taylor series, which is a way of expressing a function as an infinite sum of polynomials. By truncating the series at a certain point, we can get an approximation of the function for specific values.
Small angles are significant because they allow us to use simpler versions of the trigonometric functions without sacrificing much accuracy. This is especially useful in fields such as engineering and physics where precise calculations are necessary.
Small angle expansions are commonly used in fields such as mechanics, optics, and electronics to calculate angles and make approximations without the need for complex calculations. They are also used in astronomy and navigation to determine the position of celestial bodies.
While small angle expansions are useful for approximating values for small angles, they become less accurate as the angle increases. This is because the higher-order terms in the Taylor series become more significant. Additionally, these expansions only work for angles measured in radians, not degrees.