From the standard expansion series, when $x$ is small, ${{x}^{3}}\approx 0$. Thus, $\sin x\approx x$ $\cos x\approx 1-\frac{{{x}^{2}}}{2}$ $\tan x\approx x$

Higher-Order Small-Angle Approximations

Higher-order approximations

Order of approximation

Function	Terms of order $x^3$ and above neglected	Terms of order $x^4$ and above neglected
$\sin x$	$x$	$x-\dfrac{x^3}{3!}$
$\cos x$	$1-\dfrac{x^2}{2!}$	$1-\dfrac{x^2}{2!}$

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Function	Terms of order \(x^3\) and above neglected	Terms of order \(x^4\) and above neglected
\(\sin x\)	\(x\)	\(x-\dfrac{x^3}{3!}\)
\(\cos x\)	\(1-\dfrac{x^2}{2!}\)	\(1-\dfrac{x^2}{2!}\)