Regel

Sinusvärdet för en vinkel som ökat med $90^\circ$

Ett cosinusvärde kan omvandlas till ett sinusvärde.

$\cos(v)=\sin(v+90^\circ)$

För att visa detta kan additionsformeln för sinus användas på högerledet.

$\sin\left(v+90^\circ\right)$

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

$\sin(v)\cos(90^\circ)+\cos(v)\sin(90^\circ)$

${\textstyle \ifnumequal{90}{0}{\cos\left(0^\circ\right)=1}{}\ifnumequal{90}{30}{\cos\left(30^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{45}{\cos\left(45^\circ\right)=\dfrac{1}{\sqrt{2}}}{}\ifnumequal{90}{60}{\cos\left(60^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{90}{90}{\cos\left(90^\circ\right)=0}{}\ifnumequal{90}{120}{\cos\left(120^\circ\right)=\text{-} \dfrac{1}{2}}{}\ifnumequal{90}{135}{\cos\left(135^\circ\right)=\text{-} \dfrac{1}{\sqrt{2}}}{}\ifnumequal{90}{150}{\cos\left(150^\circ\right)=\text{-} \dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{180}{\cos\left(180^\circ\right)=\text{-} 1}{}\ifnumequal{90}{210}{\cos\left(210^\circ\right)=\text{-} \dfrac{\sqrt 3}2}{}\ifnumequal{90}{225}{\cos\left(225^\circ\right)=\text{-} \dfrac 1 {\sqrt 2}}{}\ifnumequal{90}{240}{\cos\left(240^\circ\right)=\text{-} \dfrac {1}2}{}\ifnumequal{90}{270}{\cos\left(270^\circ\right)=0}{}\ifnumequal{90}{300}{\cos\left(300^\circ\right)=\dfrac{1}2}{}\ifnumequal{90}{315}{\cos\left(315^\circ\right)=\dfrac 1 {\sqrt 2}}{}\ifnumequal{90}{330}{\cos\left(330^\circ\right)=\dfrac{\sqrt 3}2}{}\ifnumequal{90}{360}{\cos\left(360^\circ\right)=1}{}}$, ${\textstyle \ifnumequal{90}{0}{\sin\left(0^\circ\right)=0}{}\ifnumequal{90}{30}{\sin\left(30^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{90}{45}{\sin\left(45^\circ\right)=\dfrac{1}{\sqrt{2}}}{}\ifnumequal{90}{60}{\sin\left(60^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{90}{\sin\left(90^\circ\right)=1}{}\ifnumequal{90}{120}{\sin\left(120^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{135}{\sin\left(135^\circ\right)=\dfrac{1}{\sqrt{2}}}{}\ifnumequal{90}{150}{\sin\left(150^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{90}{180}{\sin\left(180^\circ\right)=0}{}\ifnumequal{90}{210}{\sin\left(210^\circ\right)=\text{-} \dfrac 1 2}{}\ifnumequal{90}{225}{\sin\left(225^\circ\right)=\text{-} \dfrac 1 {\sqrt 2}}{}\ifnumequal{90}{240}{\sin\left(240^\circ\right)=\text{-} \dfrac {\sqrt 3}2}{}\ifnumequal{90}{270}{\sin\left(270^\circ\right)=\text{-}1}{}\ifnumequal{90}{300}{\sin\left(300^\circ\right)=\text{-}\dfrac {\sqrt 3}2}{}\ifnumequal{90}{315}{\sin\left(315^\circ\right)=\text{-} \dfrac 1 {\sqrt 2}}{}\ifnumequal{90}{330}{\sin\left(330^\circ\right)=\text{-} \dfrac 1 2}{}\ifnumequal{90}{360}{\sin\left(360^\circ\right)=0}{}}$

$\sin(v)\cdot0+\cos(v)\cdot 1$

Multiplicera faktorer

$\cos(v)$

Alltså är $\cos(v) = \sin\left(v+90^\circ\right).$

Q.E.D.