MA0001/Exercise 3/tasks/4.tex

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2020-09-20 15:09:03 +02:00
\begin{deloppgaver}
\delo
\begin{align*}
cos(x) &= -1 \\
x &= acos(-1) \\
x &= \pm \pi + 2n\pi
\end{align*}
\begin{minipage}{0.42\textwidth}
\begin{graphbox}
\input{figures/4a.tex}
\end{graphbox}
\end{minipage}
Ettersom forskjellen mellom $\pi$ og $-\pi$ er $2\pi$ som er et element i $2n\pi$, kan vi slå sammen svarene og si at
\[ x = \pi + 2n\pi \]
\delo
\[cos(2x)=1-2sin^2(x)\]
Vi substituerer $cos(2x)=cos^2(x)-sin^2(x)$ og $cos^2(x)+sin^2(x)=1$
\begin{align*}
cos^2(x)-sin^2(x) &= cos^2(x)+sin^2(x) - 2sin^2(x) \\
cos^2(x) &= cos^2(x)+2sin^2(x) - 2sin^2(x) \\
cos^2(x) &= cos^2(x)
\end{align*}
\delo
\[cos(\frac{\pi}{8})\]
Vi bruker
\[ sin(2a) = 2 sin(a)cos(a) \]
Hvor $a = \frac{\pi}{8}$
\begin{align*}
sin\left(\frac{\pi}{4}\right) &= 2sin\left(\frac{\pi}{8}\right)cos\left(\frac{\pi}{8}\right) \\[1em]
cos\left(\frac{\pi}{8}\right) &= \frac{sin\left(\frac{\pi}{4}\right)}{2sin\left(\frac{\pi}{8}\right)} \\[1em]
&= \frac{\frac{1}{\sqrt{2}}}{2sin\left(\frac{\pi}{8}\right)}
\end{align*}
\end{deloppgaver}