Wichtige Additionstheoreme kennen

Einführung

Kennt man den Sinus- und den Kosinuswert zweier Winkel $\alpha$ und $\beta$ , so kann man die Sinus- und Kosinuswerte von $\left(\alpha+\beta\right)$ berechnen. Allerdings gilt im Allgemeinen $\sin{\left(\alpha+\beta\right)}\neq\sin{\left(\alpha\right)}+\sin(\beta)$ und $\cos{\left(\alpha+\beta\right)}\neq\cos{\left(\alpha\right)}+\cos{\left(\beta\right)}$ . Deshalb brauchen wir zur Berechnung dieser Summen andere Formeln.

Einfache Additionstheoreme

Sinus	$sin{\left(\alpha+\beta\right)}=sin{\left(\alpha\right)}\cdot c o s{\left(\beta\right)}+sin{\left(\beta\right)}\cdot c o s{\left(\alpha\right)}$
	$sin{\left(\alpha-\beta\right)}=sin{\left(\alpha\right)}\cdot c o s{\left(\beta\right)}-sin{\left(\beta\right)}\cdot c o s{\left(\alpha\right)}$
Kosinus	$cos{\left(\alpha+\beta\right)}=cos{\left(\alpha\right)}\cdot c o s{\left(\beta\right)}-sin{\left(\alpha\right)}\cdot s i n{\left(\beta\right)}$
	$cos{\left(\alpha-\beta\right)}=cos{\left(\alpha\right)}\cdot c o s{\left(\beta\right)}+sin(\alpha)\cdot sin(\beta)$
Tangens	$tan{\left(\alpha+\beta\right)}=\frac{tan{\left(\alpha\right)+tan{\left(\beta\right)}}}{1-tan{\left(\alpha\right)}\cdot t a n{\left(\beta\right)}}$
	$tan{\left(\alpha-\beta\right)}=\frac{tan{\left(\alpha\right)-tan{(\beta)}}}{1+tan(\alpha)\cdot t a n(\beta)}$

Doppelwinkelformel

Sinus und Kosinus	$sin{\left(2\alpha\right)}=2sin{\left(\alpha\right)}cos(\alpha)$
Sinus und Kosinus	$cos{\left(2\alpha\right)}={cos}^2{\left(\alpha\right)}-{sin}^2{(\alpha)}$
Tangens	$tan{\left(2\alpha\right)}=\frac{2\cdot t a n{(\alpha)}}{1-{tan}^2{(\alpha)}}$

Trigonometrischer Pythagoras

${cos}^2{\left(\alpha\right)=1-{sin}^2{(\alpha)}}$

Weitere Formeln

sin{\left(\alpha\right)}+sin{\left(\beta\right)}=2sin{\left(\frac{\alpha+\beta}{2}\right)}cos{\left(\frac{\alpha-\beta}{2}\right)}

sin{\left(\alpha\right)}-sin{\left(\beta\right)}=2cos{\left(\frac{\alpha+\beta}{2}\right)}sin{\left(\frac{\alpha-\beta}{2}\right)}

cos{\left(\alpha\right)}+cos{\left(\beta\right)}=2cos{\left(\frac{\alpha+\beta}{2}\right)}cos{\left(\frac{\alpha-\beta}{2}\right)}

cos{\left(\alpha\right)}-cos{\left(\beta\right)}=-2sin{\left(\frac{\alpha+\beta}{2}\right)}sin{\left(\frac{\alpha-\beta}{2}\right)}