PlanetMath: comparison of $\sin þeta$ and $þeta$ near $þeta = 0$

Proof. Let

, and $% latex2html id marker 357 $ Q=(\cos \theta, \sin \theta)$$ . Note that the circle

passes through

and

and that the shortest arc along this circle from

has length $% latex2html id marker 369 $ \theta$$ . Note also that the line segments $\overline{OP}$ and $\overline{OQ}$ are radii of the circle

and therefore must each have length 1.

$\begin{pspicture} % latex2html id marker 68 (-1,1)(-2,2) \psdots(0,0)(2,0)(1.2,1... ....3}{0}{55.5} \rput[l](0.3,0.2){$\theta$} \psarc(0,0){2}{0}{55.5} \end{pspicture}$

Draw the line segment $\overline{PQ}$ . Since this does not correspond to the arc, its length $% latex2html id marker 379 $ \left\vert \overline{PQ} \right\vert<\theta$$ .

$\begin{pspicture} % latex2html id marker 84 (-1,1)(-2,2) \psdots(0,0)(2,0)(1.2,1... ....3}{0}{55.5} \rput[l](0.3,0.2){$\theta$} \psarc(0,0){2}{0}{55.5} \end{pspicture}$

Drop the perpendicular from to $\overline{OP}$ . Let be the point of intersection. Note that $% latex2html id marker 387 $ \left\vert \overline{OR} \right\vert=\cos \theta$$ and $% latex2html id marker 389 $ \left\vert \overline{QR} \right\vert=\sin \theta$$ .

$\begin{pspicture} % latex2html id marker 107 (-1,1)(-2,2) \psdots(0,0)(2,0)(1.2,... ...0,0){2}{0}{55.5} \psline(1.2,0)(1.2,1.6) \rput[b](1.2,-0.5){$R$} \end{pspicture}$

Since $% latex2html id marker 391 $ \displaystyle 0< \theta <\frac{\pi}{2}$$ , $% latex2html id marker 393 $ 0< \sin \theta <1$$ and $% latex2html id marker 395 $ 0< \cos \theta <1$$ . Thus, $R \neq Q$ and lies strictly in between and . Therefore, $\left\vert \overline{PR} \right\vert>0$ .

By the Pythagorean theorem, $\left\vert \overline{QR} \right\vert^2+\left\vert \overline{PR} \right\vert^2=\left\vert \overline{PQ} \right\vert^2$ . Thus, $\left\vert \overline{QR} \right\vert^2< \left\vert \overline{PQ} \right\vert^2$ . Therefore, $% latex2html id marker 411 $ \sin \theta = \left\vert \overline{QR} \right\vert< \left\vert \overline{PQ} \right\vert < \theta$$ . $\qedsymbol$

The analogous result for $% latex2html id marker 413 $ \theta$$ slightly below 0 is: