# Small angle expansions for sin, cos, and tan

• I
• Mr Davis 97
In summary, the small-angle approximation includes all terms up to and including the second-order term in the Taylor series, making any discrepancies third order. The second-order terms are zero for sine and tangent, while for cosine, the first and zeroth-order approximations are the same. This is represented mathematically by the equation sin x = x + O(x^3).
Mr Davis 97
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?

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?
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.

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.
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?

Mr 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?
Yes

It's better to write something like
$$\sin x=x+\mathcal{O}(x^3).$$
Then it's clear that the next term in the expansion is at order ##x^3##.

## What are small angle expansions for sin, cos, and tan?

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.

## How are these expansions derived?

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.

## What is the significance of small angles in these expansions?

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.

## What are some common applications of small angle expansions for sin, cos, and tan?

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.

## Are there any limitations to using small angle expansions?

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.

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