39. Oppsummering: Trigonometri#
Sinus, cosinus og tangens
\[\begin{split}
\begin{align*}
\sin v &= \dfrac{\mathrm{mot}}{\mathrm{hyp}} \\
\\
\cos v &= \dfrac{\mathrm{hos}}{\mathrm{hyp}} \\
\\
\tan v &= \dfrac{\mathrm{mot}}{\mathrm{hos}} = \dfrac{\sin v}{\cos v}
\end{align*}
\end{split}\]
Enhetssirkelen
\[\begin{split}
\begin{align*}
x &= \cos v \\
\\
y &= \sin v
\end{align*}
\end{split}\]
Identiteter
\[\begin{split}
\begin{align*}
\sin (180\degree - v) &= \sin v \\
\\
\cos (180\degree - v) &= -\cos v \\
\\
\sin(90\degree - v) &= \cos v \\
\\
\cos(90\degree - v) &= \sin v
\end{align*}
\end{split}\]
Arealsetningen
\[\begin{split}
\begin{align*}
T &= \dfrac{1}{2} \cdot b \cdot c \cdot \sin A && (\mathrm{hjørne \, A})\\
\\
T &= \dfrac{1}{2} \cdot a \cdot c \cdot \sin B && (\mathrm{hjørne \, B})\\
\\
T &= \dfrac{1}{2} \cdot a \cdot b \cdot \sin C && (\mathrm{hjørne \, C})
\end{align*}
\end{split}\]
Sinussetningen
\[
\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}
\]
Cosinussetningen
\[\begin{split}
\begin{align*}
a^2 &= b^2 + c^2 - 2\cdot b \cdot c \cdot \cos A \\
\\
b^2 &= a^2 + c^2 - 2\cdot a \cdot c \cdot \cos B \\
\\
c^2 &= a^2 + b^2 - 2\cdot a \cdot b \cdot \cos C
\end{align*}
\end{split}\]