Assume that angle $x$ is small, using small angle approximation,
$\sin(x)=x$;
$\cos(x)=1-\frac{x^2}{2}$;
and $\tan(x)=x$.
I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?
$\endgroup$ 22 Answers
$\begingroup$Hint: for small $x$, $\frac{1}{\cos{x}}\approx\frac{1}{1-\frac{x^2}{2}}\approx 1+\frac{x^2}{2}$
$\endgroup$ $\begingroup$Fairly simple actually:
$$\tan(x)=\frac{\sin(x)}{\cos(x)}\approx\frac x{1-\frac{x^2}2}=x\left(1+\frac{x^2}2+\frac{x^4}4+\dots\right)\approx x\big(1+\mathcal O(x^2)\big)$$
where we used known approximations and the geometric series.
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