Uppgift 1105

Trigonometriska ettan

$sin^2u+cos^2u=1$

Additionssatserna

$sin(u+v)=sin(u)cos(v)+cos(u)sin(v)$

$sin(u-v)=sin(u)cos(v)-cos(u)sin(v)$

$cos(u+v)=cos(u)cos(v)-sin(u)sin(v)$

$cos(u-v)=cos(u)cos(v)+sin(u)sin(v)$

$tan(u+v)=\frac{tan(u)+tan(v)}{1-tan(u)tan(v)}$

$tan(u-v)=\frac{tan(u)-tan(v)}{1+tan(u)tan(v)}$

Formler  för dubbla vinkeln

$sin(2u)=2sin(u)cos(u)$

$cos(2u)=cos^2u-sin^2u=2cos^2(u)-1=1-2sin^2(u)$

$tan(2u)=\frac{2{\hspace{2mm}}tan(u)}{1-tan^2(u)}$

Formler för halva vinkeln

$sin^2\frac{u}{2}=\frac{1-cos(u)}{2}$

$cos^2\frac{u}{2}=\frac{1+cos(u)}{2}$

Produktformlerna

$2cos(u)cos(v)=cos(u-v)+cos(u+v)$

$2sin(u)sin(v)=cos(u-v)-cos(u+v)$

$2sin(u)cos(v)=sin(u-v)+sin(u+v)$

Uttrycket $a \hspace{1mm} sin(x)+b\hspace{1mm}cos(x)$

$a \hspace{1mm} sin(x)+b\hspace{1mm}cos(x)=\sqrt{a^2+b^2}\hspace{1mm}sin(x+v)$

$a \hspace{1mm} sin(x)-b\hspace{1mm}cos(x)=\sqrt{a^2+b^2}\hspace{1mm}sin(x-v)$

Då $a>0, b>0$ $tan(v)=\frac{b}{a}$ och $v\in (0^{\circ},90^{\circ})$