What is the approximation of trigonometric function by simple function

Question

for
$f(x)=\sin x$,
$g(x)=\cos x$,
$h(x)=\tan x$,

What is the approximation of each function by using simple function?

Answer 1

Don you want the following:

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots$$

$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots $$

Answer 2

Near to zero it is usual to use the following approximation:

$$\sin x \approx x $$ $$\cos x \approx 1-\displaystyle \frac{x^2}{2}$$ $$\tan x \approx x $$

you can convince yourself, for example, using the relationships that @paul gives.

Answer 3

There is a general construction that can be used here. Consider any nonnegative measurable function $f$. Now define

$$A_{n,m} = \left \{ x : \frac{m-1}{n} \leq f(x) \leq \frac{m}{n} \right \}, \quad m = 1,2,\dots,n^2, \\ A_{n,n^2+1} = \{ x : f(x) \geq n \}.$$

Then the sequence of simple functions

$$s_n(x) = \sum_{m=1}^{n^2} \frac{m-1}{n} 1_{A_{n,m}}(x) + n 1_{A_{n,n^2+1}}(x)$$

converge to $f$.

To generalize this to functions which are not nonnegative, apply this construction to $f^+ = \max \{ f,0 \}$ and $f^- = \max \{ -f,0 \}$, obtaining sequences $s_n^+$ and $s_n^-$ respectively. Then take $s_n = s_n^+ - s_n^-$.

Answer 4

You can establish quite nice approximations using Pade expansions of the functions. For example $$\sin(x)\approx \frac{x-\frac{7 x^3}{60}}{1+\frac{x^2}{20}}$$ $$\cos(x)\approx\frac{1-\frac{5 x^2}{12}}{1+\frac{x^2}{12}}$$ $$\tan(x)\approx\frac{x-\frac{x^3}{15}}{1-\frac{2 x^2}{5}}$$ are quite good.

For sure, if you increase the degrees of numerator and denominator, it will be better but more complex. For example $$\sin(x)\approx\frac{x-\frac{31 x^3}{294}}{1+\frac{3 x^2}{49}+\frac{11 x^4}{5880}}$$