 # Assume that angle x is small, using small angle approximation, sin(x)=x; cos(x)=1-(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? Lilliana Livingston 2022-07-15 Answered
Justify the small angle approximation for tangent
Assume that angle x is small, using small angle approximation,
$\mathrm{sin}\left(x\right)=x$
$\mathrm{cos}\left(x\right)=1-\frac{{x}^{2}}{2}$;
and $\mathrm{tan}\left(x\right)=x$.
I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?
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Explanation:
For small x, $\frac{1}{\mathrm{cos}x}\approx \frac{1}{1-\frac{{x}^{2}}{2}}\approx 1+\frac{{x}^{2}}{2}$.

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Step 1
Fairly simple actually: $\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}\approx \frac{x}{1-\frac{{x}^{2}}{2}}=x\left(1+\frac{{x}^{2}}{2}+\frac{{x}^{4}}{4}+\dots \right)\approx x\left(1+\mathcal{O}\left({x}^{2}\right)\right)$
Step 2
Where we used known approximations and the geometric series.

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