$\cos \left(\gamma \right)=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(\backslash gamma\backslash right)=\backslash frac12\backslash sqrt2$

Der Kosinus ist im ersten sowie im vierten Quadranten positiv.

1. Winkel

$\cos \left(\gamma \right)=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(\backslash gamma\backslash right)=\backslash frac12\backslash sqrt2$

${\gamma }_1={45}^{\circ }\backslash gamma\_1=45^\backslash circ$

2. Winkel

$\cos ({360}^{\circ }-{\gamma }_1)=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(360^\backslash circ-\backslash gamma\_1\backslash right)=\backslash frac12\backslash sqrt2$

$\cos ({360}^{\circ }-{45}^{\circ })=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(360^\backslash circ-45^\backslash circ\backslash right)=\backslash frac12\backslash sqrt2$

${\gamma }_2={315}^{\circ }\backslash gamma\_2=315^\backslash circ$

3. Winkel

$\cos ({360}^{\circ }+{\gamma }_1)=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(360^\backslash circ+\backslash gamma\_1\backslash right)=\backslash frac12\backslash sqrt2$

$\cos ({360}^{\circ }+{45}^{\circ })=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(360^\backslash circ+45^\backslash circ\backslash right)=\backslash frac12\backslash sqrt2$

${\gamma }_3={405}^{\circ }\backslash gamma\_3=405^\backslash circ$

4. Winkel

$\cos ({720}^{\circ }-{\gamma }_1)=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(720^\backslash circ-\backslash gamma\_1\backslash right)=\backslash frac12\backslash sqrt2$

$\cos ({720}^{\circ }-{45}^{\circ })=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(720^\backslash circ-45^\backslash circ\backslash right)=\backslash frac12\backslash sqrt2$

${\gamma }_3={675}^{\circ }\backslash gamma\_3=675^\backslash circ$

5. Winkel

$\cos (0^{\circ }-{\gamma }_1)=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(0^\backslash circ-\backslash gamma\_1\backslash right)=\backslash frac12\backslash sqrt2$

$\cos (0^{\circ }-{45}^{\circ })=\frac{1}{2}\sqrt{2}\backslash cos\backslash left(0^\backslash circ-45^\backslash circ\backslash right)=\backslash frac12\backslash sqrt2$

${\gamma }_4=-{45}^{\circ }\backslash gamma\_4=-45^\backslash circ$

$\sin \left(\frac{x}{2}\right)=1\backslash sin\backslash left(\backslash frac\; x2\backslash right)=1$

Sinus ist im ersten und zweiten Quadranten positiv.

$y\in [-\pi ;3\pi ]y\backslash in\backslash left[-\backslash mathrm\backslash pi;\backslash ;3\backslash mathrm\backslash pi\backslash right]$

1.Winkel im Bogenmaß

${\displaystyle \frac{x}{2}}=y\backslash dfrac\; x2=y$

$\sin \left(y\right)=1\backslash sin\backslash left(y\backslash right)=1$

$y_1={\displaystyle \frac{1}{2}}\pi y\_1=\backslash dfrac12\backslash mathrm\backslash pi$

$x_1=\pi x\_1=\backslash mathrm\backslash pi$

2. Winkel im Bogenmaß

$\sin (2\pi +y_1)=1\backslash sin\backslash left(2\backslash mathrm\backslash pi+y\_1\backslash right)=1$

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

${\gamma }_2=\mathrm{2,5}\pi \backslash gamma\_2=2\{,\}5\backslash mathrm\backslash pi$

$x=5\pi x=5\backslash mathrm\backslash pi$

3. Winkel im Bogenmaß

$\sin (0-y_1)=1\backslash sin\backslash left(0-y\_1\backslash right)=1$

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

$y_3=-\frac{1}{2}\pi y\_3=-\backslash frac12\backslash mathrm\backslash pi$

$x=-\pi x=-\backslash mathrm\backslash pi$

$\sin \left(x\right)=-2\backslash sin\backslash left(x\backslash right)=-2$

Geht nicht, da gilt: $-1\le \sin \left(x\right)\le 1-1\backslash leq\backslash sin\backslash left(x\backslash right)\backslash leq1$