Berechne $\sin \left(\frac{5\pi }{2}\right)\backslash sin\; \backslash left(\; \backslash frac\{5\backslash pi\}\{2\}\; \backslash right)$. Die Sinusfunktion ist periodisch mit Periode $2\pi 2\backslash pi$.

$\sin (\cos \left(\frac{\pi }{2}\right))+\tan (1-\sin \left(\frac{5\pi }{2}\right))\backslash sin\backslash left(\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\backslash right)+\backslash tan\backslash left(1-\backslash sin\backslash left(\backslash frac\{5\backslash pi\}\{2\}\backslash right)\backslash right)$

$=\sin (\cos \left(\frac{\pi }{2}\right))+\tan (1-\sin (\frac{\pi }{2}+2\pi ))=\backslash sin\backslash left(\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\backslash right)\; +\; \backslash tan\backslash left(1\; -\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}+2\backslash pi\backslash right)\backslash right)$

$=\sin (\cos \left(\frac{\pi }{2}\right))+\tan (1-\sin \left(\frac{\pi }{2}\right))=\backslash sin\backslash left(\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\backslash right)+\backslash tan\backslash left(1-\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\backslash right)$

Berechne $\cos \left(\frac{\pi }{2}\right)\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)$ und $\sin \left(\frac{\pi }{2}\right)\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)$.Die Werte von $\sin \left(\frac{\pi }{2}\right)\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)$ und $\cos \left(\frac{\pi }{2}\right)\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)$ kannst du dir leicht merken. Deswegen brauchst du dafür keinen Taschenrechner.Als Hilfe kannst du dir den Graphen zu den Funktionen anschauen.

Daraus kannst du ablesen:

• $\cos \left(\frac{\pi }{2}\right)=0\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 0$

• $\sin \left(\frac{\pi }{2}\right)=1\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 1$

Diese Werte setzt du in den Term ein.

$\mathrm{.}\mathrm{.}\mathrm{.}=\sin (\cos \left(\frac{\pi }{2}\right))+\tan (1-\sin \left(\frac{\pi }{2}\right))...=\backslash \; \backslash sin\backslash left(\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\backslash right)+\backslash tan\backslash left(1-\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\backslash right)$

$=\sin \left(0\right)+\tan (1-1)=\backslash sin\backslash left(0\backslash right)\; +\; \backslash tan\backslash left(1\; -\; 1\backslash right)$

$=\sin \left(0\right)+\tan \left(0\right)=\backslash sin\backslash left(0\backslash right)+\backslash tan\backslash left(0\backslash right)$

Benutze nun die Definition der Tangensfunktion: $\tan (x)=\frac{\sin (x)}{\cos (x)}\backslash tan(x)=\backslash frac\{\backslash sin(x)\}\{\backslash cos(x)\}$.

$\mathrm{.}\mathrm{.}\mathrm{.}=\sin \left(0\right)+\tan \left(0\right)...=\backslash sin\backslash left(0\backslash right)+\backslash tan\backslash left(0\backslash right)$

$=\sin (0)+\frac{\sin (0)}{\cos (0)}=\; \backslash sin(0)\; +\; \backslash frac\{\backslash sin(0)\}\{\backslash cos(0)\}$

Setze die Funktionswerte ein. Zur Hilfe kannst du wieder den Graphen betrachten.

$\mathrm{.}\mathrm{.}\mathrm{.}=\sin (0)+\frac{\sin (0)}{\cos (0)}=0+\frac{0}{1}=0...=\backslash sin(0)+\backslash frac\{\backslash sin(0)\}\{\backslash cos(0)\}=0+\backslash frac\{0\}\{1\}=0$

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