\(\def \u#1{\,\mathrm{#1}}\) \(\def \us#1{\,\mathrm{\scriptsize #1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Appendix A: Vectors and Trigonometry
11. (Incomplete)

Small Angles

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When the angle $\theta$ is small, then the sine of the angle and the tangent of the angle are approximately equal to each other, AND are approximately equal to $\theta$ in radians. $$\sin\theta \approx \tan\theta \approx \theta \hbox{ (in radians)}$$

This approximation is pretty darn good up to about $\pi/10$ or 18°: at this value, the angle, sine, and tangent are all within 5% of each other:

$$\begin{align*} {\pi\over 10}&=0.314\\ \sin{\pi\over 10}&=0.309\\ \tan{\pi\over10}&=0.325\\ \end{align*}$$ Of course, with a calculator it's just as easy to push the SIN or TAN buttons, but this approximation will be useful in some chapters for deriving approximate formulas.
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The cosine isn't small when $\theta$ is small; to the contrary it is very close to 1. If you want a better approximation, you can write $$\cos\theta \approx 1-\frac12\theta^2$$

This is a better approximation than for the sine, and is within 2% even at $\theta=\pi/4$ (or 45°). We won't need this approximation in this chapter, however.