4.4 Övningar

Sommarmatte 1

(Skillnad mellan versioner)

Nuvarande version

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1 Övning 4.4:1
2 Övning 4.4:2
3 Övning 4.4:3
4 Övning 4.4:4
5 Övning 4.4:5
6 Övning 4.4:6
7 Övning 4.4:7
8 Övning 4.4:8

[redigera] Övning 4.4:1

För vilka vinklar $\,v\,$, där $\,0 \leq v\leq 2\pi\,$, gäller att

a)	$\sin{v}=\displaystyle \frac{1}{2}$	b)	$\cos{v}=\displaystyle \frac{1}{2}$
c)	$\sin{v}=1$	d)	$\tan{v}=1$
e)	$\cos{v}=2$	f)	$\sin{v}=-\displaystyle \frac{1}{2}$
g)	$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$

Facit

Facit till alla delfrågor

a)	$\displaystyle v=\frac{\pi}{6}\,$, $\,\displaystyle v=\frac{5\pi}{6}$	b)	$\displaystyle v=\frac{\pi}{3}\,$, $\,\displaystyle v=\frac{5\pi}{3}$
c)	$\displaystyle v=\frac{\pi}{2}$	d)	$\displaystyle v=\frac{\pi}{4}\,$, $\,\displaystyle v=\frac{5\pi}{4}$
e)	lösning saknas	f)	$\displaystyle v=\frac{11\pi}{6}\,$, $\,\displaystyle v=\frac{7\pi}{6}$
g)	$\displaystyle v=\frac{5\pi}{6}\,$, $\,\displaystyle v=\frac{11\pi}{6}$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

Lösning d)

Lösning till delfråga d)

Lösning e)

Lösning till delfråga e)

Lösning f)

Lösning till delfråga f)

Lösning g)

Lösning till delfråga g)

[redigera] Övning 4.4:2

Lös ekvationen

a)	$\sin{x}=\displaystyle \frac{\sqrt{3}}{2}$	b)	$\cos{x}=\displaystyle \frac{1}{2} $	c)	$\sin{x}=0$
d)	$\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} $	e)	$\sin{5x}=\displaystyle \frac{1}{2}$	f)	$\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}$

Facit

Facit till alla delfrågor

a)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{3}+2n\pi\cr x&=\displaystyle\frac{2\pi}{3}+2n\pi } \right.$	b)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{3}+2n\pi\cr x&=\displaystyle\frac{5\pi}{3}+2n\pi } \right.$	c)	$x=n\pi$
d)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{20}+\displaystyle\frac{2n\pi}{5}\cr x&=\displaystyle\frac{3\pi}{20}+\displaystyle\frac{2n\pi}{5} } \right.$	e)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{30}+\displaystyle\frac{2n\pi}{5}\cr x&=\displaystyle\frac{\pi}{6}+\displaystyle\frac{2n\pi}{5}} \right.$	f)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{4}+\displaystyle\frac{2n\pi}{3}\cr x&=\displaystyle\frac{5\pi}{12}+\displaystyle\frac{2n\pi}{3}} \right.$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

Lösning d)

Lösning till delfråga d)

Lösning e)

Lösning till delfråga e)

Lösning f)

Lösning till delfråga f)

[redigera] Övning 4.4:3

Lös ekvationen

a)	$\cos{x}=\cos{\displaystyle \frac{\pi}{6}}$	b)	$\sin{x}=\sin{\displaystyle \frac{\pi}{5}}$
c)	$\sin{(x+40^\circ)}=\sin{65^\circ}$	d)	$\sin{3x}=\sin{15^\circ}$

Facit

Facit till alla delfrågor

a)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{6}+2n\pi\cr x&=\displaystyle\frac{11\pi}{6}+2n\pi }\right.$	b)	$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{5}+2n\pi\cr x&=\displaystyle\frac{4\pi}{5}+2n\pi }\right.$
c)	$\left\{\eqalign{ x&=25^\circ + n\cdot 360^\circ\cr x&=75^\circ + n\cdot 360^\circ }\right.$	d)	$\left\{\eqalign{ x&=5^\circ + n \cdot 120^\circ \cr x&= 55^\circ + n \cdot 120^\circ }\right.$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

Lösning d)

Lösning till delfråga d)

[redigera] Övning 4.4:4

Bestäm de vinklar $\,v\,$ i intervallet $\,0^\circ \leq v \leq 360^\circ\,$ som uppfyller $\ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,$.

Facit

$v_1=50^\circ$, $\ \ v_2=120^\circ$, $\ \ v_3=230^\circ\ \ $ och $\ \ v_4=300^\circ$

Lösning

[redigera] Övning 4.4:5

Lös ekvationen

a)	$\sin{3x}=\sin{x}$	b)	$\tan{x}=\tan{4x}$
c)	$\cos{5x}=\cos(x+\pi/5)$

Facit

Facit till alla delfrågor

a)	$\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{\pi}{4}+\displaystyle \frac{n\pi}{2} }\right.$	b)	$x=\displaystyle \frac{n\pi}{3}$
c)	$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3} }\right.$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

[redigera] Övning 4.4:6

Lös ekvationen

a)	$\sin x\cdot \cos 3x = 2\sin x$	b)	$\sqrt{2}\sin{x}\cos{x}=\cos{x}$
c)	$\sin 2x = -\sin x$

Facit

Facit till alla delfrågor

a)	$x=n\pi$	b)	$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{4}+2n\pi\cr x&=\displaystyle \frac{\pi}{2}+n\pi\cr x&=\displaystyle \frac{3\pi}{4}+2n\pi}\right.$
c)	$\left\{\eqalign{ x&=\displaystyle \frac{2n\pi}{3}\cr x&=\displaystyle \pi + 2n\pi\cr }\right.$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

[redigera] Övning 4.4:7

Lös ekvationen

a)	$2\sin^2{x}+\sin{x}=1$	b)	$2\sin^2{x}-3\cos{x}=0$
c)	$\cos{3x}=\sin{4x}$

Facit

Facit till alla delfrågor

a)	$\left\{ \matrix{ x=\displaystyle \frac{\pi}{6}+2n\pi\cr x=\displaystyle \frac{5\pi}{6}+2n\pi\cr x=\displaystyle \frac{3\pi}{2}+2n\pi }\right.$	b)	$x=\pm \displaystyle \frac{\pi}{3} + 2n\pi $
c)	$\left\{ \matrix{ x=\displaystyle \frac{\pi}{2}+2n\pi\cr x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7} }\right.$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

[redigera] Övning 4.4:8

Lös ekvationen

a)	$\sin{2x}=\sqrt{2}\cos{x}$	b)	$\sin{x}=\sqrt{3}\cos{x}$
c)	$\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}$

Facit

Facit till alla delfrågor

a)	$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{4}+2n\pi\cr x&=\displaystyle \frac{\pi}{2}+n\pi\cr x&=\displaystyle \frac{3\pi}{4}+2n\pi }\right.$	b)	$x=\displaystyle \frac{\pi}{3}+n\pi$
c)	$\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{3\pi}{4}+n\pi }\right.$

Lösning a)

Lösning till delfråga a)

Lösning b)

Lösning till delfråga b)

Lösning c)

Lösning till delfråga c)

Den här artikeln är hämtad från http://wiki.math.se/wikis/sf0600_0701/index.php/4.4_%C3%96vningar

4.4 Övningar

Sommarmatte 1

Nuvarande version

Innehåll

[redigera] Övning 4.4:1

[redigera] Övning 4.4:2

[redigera] Övning 4.4:3

[redigera] Övning 4.4:4

[redigera] Övning 4.4:5

[redigera] Övning 4.4:6

[redigera] Övning 4.4:7

[redigera] Övning 4.4:8

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-==Övning 4.4:1==
+==&Ouml;vning 4.4:1==
 <div class="ovning">
-För vilka vinklar $v$, där $0 \leq v\leq 2\pi$, gäller att
+För vilka vinklar $\,v\,$, där $\,0 \leq v\leq 2\pi\,$, gäller att
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\sin{v}=\displaystyle \frac{1}{2}$</td>
+<td class="ntext" width="50%">$\sin{v}=\displaystyle \frac{1}{2}$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$\cos{v}=\displaystyle \frac{1}{2}$</td>
+<td class="ntext" width="50%">$\cos{v}=\displaystyle \frac{1}{2}$</td>
-<td class="ntext">$\textrm{c) }$</td>
-<td class="ntext">$\sin{v}=1$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{d) }$</td>
+<td class="ntext">c)</td>
-<td class="ntext">$\tan{v}=1$</td>
+<td class="ntext" width="50%">$\sin{v}=1$</td>
-<td class="ntext">$\textrm{e) }$</td>
+<td class="ntext">d)</td>
-<td class="ntext">$\cos{v}=2$</td>
+<td class="ntext" width="50%">$\tan{v}=1$</td>
-<td class="ntext">$\textrm{f) }$</td>
-<td class="ntext">$\sin{v}=-\displaystyle \frac{1}{2}$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{g) }$</td>
+<td class="ntext">e)</td>
-<td class="ntext">$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$</td>
+<td class="ntext" width="50%">$\cos{v}=2$</td>
-<td class="ntext"></td>
+<td class="ntext">f)</td>
-<td class="ntext"></td>
+<td class="ntext" width="50%">$\sin{v}=-\displaystyle \frac{1}{2}$</td>
-<td class="ntext"></td>
+</tr>
-<td class="ntext"></td>
+<tr align="left">
+<td class="ntext">g)</td>
+<td class="ntext" width="50%">$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-Facit till alla delfrågorna<br \>
+Facit till alla delfrågor<br \>
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\displaystyle v=\frac{\pi}{6}, \displaystyle v=\frac{5\pi}{6}$</td>
+<td class="ntext" width="50%">$\displaystyle v=\frac{\pi}{6}\,$, $\,\displaystyle v=\frac{5\pi}{6}$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$\displaystyle v=\frac{\pi}{3}, \displaystyle v=\frac{5\pi}{3}$</td>
+<td class="ntext" width="50%">$\displaystyle v=\frac{\pi}{3}\,$, $\,\displaystyle v=\frac{5\pi}{3}$</td>
-<td class="ntext">$\textrm{c) }$</td>
-<td class="ntext">$\displaystyle v=\frac{\pi}{2}$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{d) }$</td>
+<td class="ntext">c)</td>
-<td class="ntext">$\displaystyle v=\frac{\pi}{4},\displaystyle v=\frac{5\pi}{4}$</td>
+<td class="ntext" width="50%">$\displaystyle v=\frac{\pi}{2}$</td>
-<td class="ntext">$\textrm{e) }$</td>
+<td class="ntext">d)</td>
-<td class="ntext">$\textrm{lösning saknas}$</td>
+<td class="ntext" width="50%">$\displaystyle v=\frac{\pi}{4}\,$, $\,\displaystyle v=\frac{5\pi}{4}$</td>
-<td class="ntext">$\textrm{f) }$</td>
-<td class="ntext">$\displaystyle v=\frac{11\pi}{6}, \displaystyle v=\frac{7\pi}{6}$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{g) }$</td>
+<td class="ntext">e)</td>
-<td class="ntext">$\displaystyle v=\frac{5\pi}{6}, \displaystyle v=\frac{11\pi}{6}$</td>
+<td class="ntext" width="50%">lösning saknas</td>
-<td class="ntext"></td>
+<td class="ntext">f)</td>
-<td class="ntext"></td>
+<td class="ntext" width="50%">$\displaystyle v=\frac{11\pi}{6}\,$, $\,\displaystyle v=\frac{7\pi}{6}$</td>
-<td class="ntext"></td>
+</tr>
-<td class="ntext"></td>
+<tr align="left">
+<td class="ntext">g)</td>
+<td class="ntext" width="50%">$\displaystyle v=\frac{5\pi}{6}\,$, $\,\displaystyle v=\frac{11\pi}{6}$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning a&nbsp;</div>
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga a<br />[[Bild:4_4_1a.gif]]
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1a.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning b&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga b<br />[[Bild:4_4_1b.gif]]
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1b.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning c&nbsp;</div>
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga c<br />[[Bild:4_4_1c.gif]]
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1c.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning d&nbsp;</div>
+<div class=NavHead>L&ouml;sning d)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga d<br />[[Bild:4_4_1d.gif]]
+Lösning till delfråga d)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1d.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning e&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning e)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga e<br />[[Bild:4_4_1e.gif]]
+Lösning till delfråga e)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1e.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning f&nbsp;</div>
+<div class=NavHead>L&ouml;sning f)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga f<br />[[Bild:4_4_1f.gif]]
+Lösning till delfråga f)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1f.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning g&nbsp;</div>
+<div class=NavHead>L&ouml;sning g)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga g<br />[[Bild:4_4_1g.gif]]
+Lösning till delfråga g)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_1g-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_1g-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
-==Övning 4.4:2==
+==&Ouml;vning 4.4:2==
 <div class="ovning">
 L&ouml;s ekvationen
 <table width="100%" cellspacing="10px">
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\sin{x}=\displaystyle \frac{\sqrt{3}}{2}$</td>
+<td class="ntext" width="33%">$\sin{x}=\displaystyle \frac{\sqrt{3}}{2}$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$\cos{x}=\displaystyle \frac{1}{2} $</td>
+<td class="ntext" width="33%">$\cos{x}=\displaystyle \frac{1}{2} $</td>
-<td class="ntext">$\textrm{c) }$</td>
+<td class="ntext">c)</td>
-<td class="ntext">$\sin{x}=0$</td>
+<td class="ntext" width="33%">$\sin{x}=0$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{d) }$</td>
+<td class="ntext">d)</td>
-<td class="ntext">$\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} $</td>
+<td class="ntext" width="33%">$\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} $</td>
-<td class="ntext">$\textrm{e) }$</td>
+<td class="ntext">e)</td>
-<td class="ntext">$\sin{5x}=\displaystyle \frac{1}{2}$</td>
+<td class="ntext" width="33%">$\sin{5x}=\displaystyle \frac{1}{2}$</td>
-<td class="ntext">$\textrm{f) }$</td>
+<td class="ntext">f)</td>
-<td class="ntext">$\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}$</td>
+<td class="ntext" width="33%">$\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-Facit till alla delfrågorna<br \>
+Facit till alla delfrågor<br \>
 <table width="100%" cellspacing="10px">
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">
+<td class="ntext" width="33%">
-$\left\{  \matrix{
+$\left\{\eqalign{
-x=\displaystyle\frac{\pi}{3}+2n\pi\cr
+x&=\displaystyle\frac{\pi}{3}+2n\pi\cr
-x=\displaystyle\frac{2\pi}{3}+2n\pi  } \right.$
+x&=\displaystyle\frac{2\pi}{3}+2n\pi  } \right.$
 </td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">
+<td class="ntext" width="33%">
-$\left\{  \matrix{
+$\left\{\eqalign{
-x=\displaystyle\frac{\pi}{3}+2n\pi\cr
+x&=\displaystyle\frac{\pi}{3}+2n\pi\cr
-x=\displaystyle\frac{5\pi}{3}+2n\pi  } \right.$
+x&=\displaystyle\frac{5\pi}{3}+2n\pi  } \right.$
 </td>
-<td class="ntext">$\textrm{c) }$</td>
+<td class="ntext">c)</td>
-<td class="ntext">
+<td class="ntext" width="33%">
 $x=n\pi$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{d) }$</td>
+<td class="ntext">d)</td>
-<td class="ntext">
+<td class="ntext" width="33%">
-$\left\{  \matrix{
+$\left\{\eqalign{
-x=\displaystyle\frac{\pi}{20}+\displaystyle\frac{2n\pi}{5}\cr
+x&=\displaystyle\frac{\pi}{20}+\displaystyle\frac{2n\pi}{5}\cr
-x=\displaystyle\frac{3\pi}{20}+\displaystyle\frac{2n\pi}{5}  } \right.$
+x&=\displaystyle\frac{3\pi}{20}+\displaystyle\frac{2n\pi}{5}  } \right.$
 </td>
-<td class="ntext">$\textrm{e) }$</td>
+<td class="ntext">e)</td>
-<td class="ntext">
+<td class="ntext" width="33%">
-$\left\{  \matrix{
+$\left\{\eqalign{
-x=\displaystyle\frac{\pi}{30}+\displaystyle\frac{2}{5}n\pi\cr
+x&=\displaystyle\frac{\pi}{30}+\displaystyle\frac{2n\pi}{5}\cr
-x=\displaystyle\frac{\pi}{6}+\displaystyle\frac{2}{5}n\pi  } \right.$
+x&=\displaystyle\frac{\pi}{6}+\displaystyle\frac{2n\pi}{5}} \right.$
+</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="33%">
+$\left\{\eqalign{
+x&=\displaystyle\frac{\pi}{4}+\displaystyle\frac{2n\pi}{3}\cr
+x&=\displaystyle\frac{5\pi}{12}+\displaystyle\frac{2n\pi}{3}} \right.$
 </td>
-<td class="ntext">$\textrm{f) }$</td>
-<td class="ntext">$\left\{  \matrix{x=\displaystyle\frac{\pi}{4}+\displaystyle\frac{2}{3}n\pi\cr
-x=\displaystyle\frac{5\pi}{12}+\displaystyle\frac{2}{3}n\pi  } \right.$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning a&nbsp;</div>
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga a<br />[[Bild:4_4_2a.gif]]
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_2a-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_2a-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning b&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga b<br />[[Bild:4_4_2b.gif]]
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_2b.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning c&nbsp;</div>
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga c<br />[[Bild:4_4_2c.gif]]
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_2c.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning d&nbsp;</div>
+<div class=NavHead>L&ouml;sning d)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga d<br />[[Bild:4_4_2d.gif]]
+Lösning till delfråga d)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_2d-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_2d-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning e&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning e)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga e<br />[[Bild:4_4_2e.gif]]
+Lösning till delfråga e)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_2e.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning f&nbsp;</div>
+<div class=NavHead>L&ouml;sning f)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga f<br />[[Bild:4_4_2f.gif]]
+Lösning till delfråga f)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_2f.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
-==Övning 4.4:3==
+==&Ouml;vning 4.4:3==
 <div class="ovning">
 Lös ekvationen
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\cos{x}=\cos{\left( \displaystyle \frac{\pi}{6} \right)}$</td>
+<td class="ntext" width="50%">$\cos{x}=\cos{\displaystyle \frac{\pi}{6}}$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$\sin{x}=\sin{\left( \displaystyle \frac{\pi}{5} \right)}$</td>
+<td class="ntext" width="50%">$\sin{x}=\sin{\displaystyle \frac{\pi}{5}}$</td>
-<td class="ntext">$\textrm{c) }$</td>
-<td class="ntext">$\sin{(x+40^\circ)}=\sin{65^\circ}$</td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{d) }$</td>
+<td class="ntext">c)</td>
-<td class="ntext">$\sin{3x}=\sin{15^\circ}$</td>
+<td class="ntext" width="50%">$\sin{(x+40^\circ)}=\sin{65^\circ}$</td>
-<td class="ntext"></td>
+<td class="ntext">d)</td>
-<td class="ntext"></td>
+<td class="ntext" width="50%">$\sin{3x}=\sin{15^\circ}$</td>
-<td class="ntext"></td>
-<td class="ntext"></td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-Facit till alla delfrågorna<br \>
+Facit till alla delfrågor<br \>
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">
+<td class="ntext" width="50%">
-$\left\{ \matrix{
+$\left\{\eqalign{
-x=\displaystyle\frac{\pi}{6}+2n\pi\cr
+x&=\displaystyle\frac{\pi}{6}+2n\pi\cr
-x=\displaystyle\frac{11\pi}{6}+2n\pi
+x&=\displaystyle\frac{11\pi}{6}+2n\pi
 }\right.$
 </td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">
+<td class="ntext" width="50%">
-$\left\{ \matrix{
+$\left\{\eqalign{
-x=\displaystyle\frac{\pi}{5}+2n\pi\cr
+x&=\displaystyle\frac{\pi}{5}+2n\pi\cr
-x=\displaystyle\frac{4\pi}{5}+2n\pi
+x&=\displaystyle\frac{4\pi}{5}+2n\pi
-}\right.$
-</td>
-<td class="ntext">$\textrm{c) }$</td>
-<td class="ntext">
-$\left\{ \matrix{
-x=25^\circ + n\cdot 360^\circ\cr
-x=75^\circ + n\cdot 360^\circ
 }\right.$
 </td>
 </tr>
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext">$\textrm{d) }$</td>
+<td class="ntext">c)</td>
-<td class="ntext">
+<td class="ntext" width="50%">
-$\left\{ \matrix{
+$\left\{\eqalign{
-x=5^\circ + n \cdot 120^\circ \cr
+x&=25^\circ + n\cdot 360^\circ\cr
-x= 55^\circ + n \cdot 120^\circ
+x&=75^\circ + n\cdot 360^\circ
+}\right.$
+</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">
+$\left\{\eqalign{
+x&=5^\circ + n \cdot 120^\circ \cr
+x&= 55^\circ + n \cdot 120^\circ
 }\right.$
 </td>
-<td class="ntext"></td>
-<td class="ntext"></td>
-<td class="ntext"></td>
-<td class="ntext"></td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning a&nbsp;</div>
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga a<br />[[Bild:4_4_3a.gif]]
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_3a.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning b&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga b<br />[[Bild:4_4_3b.gif]]
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_3b.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning c&nbsp;</div>
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga c<br />[[Bild:4_4_3c.gif]]
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_3c.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning d&nbsp;</div>
+<div class=NavHead>L&ouml;sning d)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga d<br />[[Bild:4_4_3d.gif]]
+Lösning till delfråga d)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_3d.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
-==Övning 4.4:4==
+==&Ouml;vning 4.4:4==
 <div class="ovning">
-Bestäm de vinklar $v$ i intervallet $0^\circ \leq v \leq 360^\circ$ som uppfyller $\cos{\left(2v+10^\circ\right)}=\cos{110^\circ}$.
+Bestäm de vinklar $\,v\,$ i intervallet $\,0^\circ \leq v \leq 360^\circ\,$ som uppfyller $\ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,$.
 </div>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-$v=50^\circ, v=120^\circ, v=230^\circ$ och $v=300^\circ$
+<table width="100%" cellpadding="10px">
+<tr>
+<td align="left">
+$v_1=50^\circ$, $\ \ v_2=120^\circ$, $\ \ v_3=230^\circ\ \ $ och $\ \ v_4=300^\circ$
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavHead>L&ouml;sning</div>
 <div class=NavContent>
-[[Bild:4_4_4.gif]]
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_4-1(3).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_4-2(3).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_4-3(3).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
-==Övning 4.4:5==
+==&Ouml;vning 4.4:5==
 <div class="ovning">
 Lös ekvationen
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\sin{3x}=\sin{x}$</td>
+<td class="ntext" width="50%">$\sin{3x}=\sin{x}$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$\tan{x}=\tan{4x}$</td>
+<td class="ntext" width="50%">$\tan{x}=\tan{4x}$</td>
-<td class="ntext">$\textrm{c) }$</td>
+</tr>
-<td class="ntext">$\cos{5x}=\cos(x+\pi/5)$</td>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\cos{5x}=\cos(x+\pi/5)$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-Facit till alla delfrågorna<br \>
+Facit till alla delfrågor<br \>
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">
+<td class="ntext" width="50%">
-$\left\{ \matrix{
+$\left\{\eqalign{
-x=n\pi\cr
+x&=n\pi\cr
-x=\displaystyle \frac{\pi}{4}+\displaystyle \frac{1}{2}n\pi
+x&=\displaystyle \frac{\pi}{4}+\displaystyle \frac{n\pi}{2}
 }\right.$
 </td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$x=\displaystyle \frac{1}{3}n\pi$</td>
+<td class="ntext" width="50%">$x=\displaystyle \frac{n\pi}{3}$</td>
-<td class="ntext">$\textrm{c) }$</td>
+</tr>
-<td class="ntext">
+<tr align="left">
-$\left\{\matrix{
+<td class="ntext">c)</td>
-x=\displaystyle \frac{\pi}{20}+\displaystyle \frac{1}{2}n\pi\cr
+<td class="ntext" width="50%">
-x=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{1}{3}n\pi
+$\left\{\eqalign{
+x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr
+x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3}
 }\right.$
 </td>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning a&nbsp;</div>
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga a<br />[[Bild:4_4_5a.gif]]
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_5a-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_5a-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning b&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga b<br />[[Bild:4_4_5b.gif]]
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_5b-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_5b-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning c&nbsp;</div>
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga c<br />[[Bild:4_4_5c.gif]]
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_5c-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_5c-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
-==Övning 4.4:6==
+==&Ouml;vning 4.4:6==
 <div class="ovning">
 Lös ekvationen
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\sin x\cdot \cos 3x = 2\sin x$</td>
+<td class="ntext" width="50%">$\sin x\cdot \cos 3x = 2\sin x$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$\sqrt{2}\sin{x}\cos{x}=\cos{x}$</td>
+<td class="ntext" width="50%">$\sqrt{2}\sin{x}\cos{x}=\cos{x}$</td>
-<td class="ntext">$\textrm{c) }$</td>
+</tr>
-<td class="ntext"></td>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\sin 2x = -\sin x$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-Facit till alla delfrågorna<br \>
+Facit till alla delfrågor<br \>
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">
+<td class="ntext" width="50%">
 $x=n\pi$
 </td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">
+<td class="ntext" width="50%">
-$\left\{ \matrix{x=\displaystyle \frac{\pi}{4}+2n\pi\cr
+$\left\{\eqalign{
-x=\displaystyle \frac{\pi}{2}+n\pi\cr
+x&=\displaystyle \frac{\pi}{4}+2n\pi\cr
-x=\displaystyle \frac{3\pi}{4}+2n\pi}\right.$
+x&=\displaystyle \frac{\pi}{2}+n\pi\cr
+x&=\displaystyle \frac{3\pi}{4}+2n\pi}\right.$
 </td>
-<td class="ntext">$\textrm{c) }$</td>
+</tr>
-<td class="ntext">
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">
+$\left\{\eqalign{
+x&=\displaystyle \frac{2n\pi}{3}\cr
+x&=\displaystyle \pi + 2n\pi\cr
+}\right.$
 </td>
 </tr>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning a&nbsp;</div>
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga a<br />[[Bild:4_4_6a.gif]]
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_6a.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning b&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga b<br />[[Bild:4_4_6b.gif]]
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_6b-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_6b-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning c&nbsp;</div>
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga c<br />[[Bild:4_4_6c.gif]]
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_6c.gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
-==Övning 4.4:7==
+==&Ouml;vning 4.4:7==
 <div class="ovning">
 Lös ekvationen
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$2\sin^2{x}+\sin{x}=1$</td>
+<td class="ntext" width="50%">$2\sin^2{x}+\sin{x}=1$</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$2\sin^2{x}-3\cos{x}=0$</td>
+<td class="ntext" width="50%">$2\sin^2{x}-3\cos{x}=0$</td>
-<td class="ntext">$\textrm{c) }$</td>
+</tr>
-<td class="ntext">$\cos{3x}=\sin{4x}$</td>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\cos{3x}=\sin{4x}$</td>
 </tr>
 <tr><td height="5px"/></tr>
 <div class=NavHead>Facit&nbsp;</div>
 <div class=NavContent>
-Facit till alla delfrågorna<br \>
+Facit till alla delfrågor<br \>
 <table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext">$\textrm{a) }$</td>
+<td class="ntext">a)</td>
-<td class="ntext">$\left\{
+<td class="ntext" width="50%">
-\matrix{
+$\left\{ \matrix{
 x=\displaystyle \frac{\pi}{6}+2n\pi\cr
 x=\displaystyle \frac{5\pi}{6}+2n\pi\cr
 x=\displaystyle \frac{3\pi}{2}+2n\pi
-}
+}\right.$
-\right.$</td>
+</td>
-<td class="ntext">$\textrm{b) }$</td>
+<td class="ntext">b)</td>
-<td class="ntext">$x=\pm \displaystyle \frac{\pi}{3} + 2n\pi $</td>
+<td class="ntext" width="50%">$x=\pm \displaystyle \frac{\pi}{3} + 2n\pi $</td>
-<td class="ntext">$\textrm{c) }$</td>
+</tr>
-<td class="ntext">
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">
 $\left\{ \matrix{
 x=\displaystyle \frac{\pi}{2}+2n\pi\cr
-x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2}{7}n\pi
+x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7}
 }\right.$
 </td>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning a&nbsp;</div>
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga a<br />[[Bild:4_4_7a.gif]]
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_7a-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_7a-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning b&nbsp;</DIV>
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
 <div class=NavContent>
-Lösning till delfråga b<br />[[Bild:4_4_7b.gif]]
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_7b-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_7b-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>
 <div class=NavFrame style="CLEAR: both">
-<div class=NavHead>L&ouml;sning c&nbsp;</div>
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
 <div class=NavContent>
-Lösning till delfråga c<br />[[Bild:4_4_7c.gif]]
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_7c-1(3).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_7c-2(3).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_7c-3(3).gif]]
+</td>
+</tr>
+</table>
+</div>
+</div>
+==&Ouml;vning 4.4:8==
+<div class="ovning">
+Lös ekvationen
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$\sin{2x}=\sqrt{2}\cos{x}$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\sin{x}=\sqrt{3}\cos{x}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}$</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
+</div>
+<div class=NavFrame style="CLEAR: both">
+<div class=NavHead>Facit&nbsp;</div>
+<div class=NavContent>
+Facit till alla delfrågor<br \>
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">
+$\left\{\eqalign{
+x&=\displaystyle \frac{\pi}{4}+2n\pi\cr
+x&=\displaystyle \frac{\pi}{2}+n\pi\cr
+x&=\displaystyle \frac{3\pi}{4}+2n\pi
+}\right.$
+</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$x=\displaystyle \frac{\pi}{3}+n\pi$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">
+$\left\{\eqalign{
+x&=n\pi\cr
+x&=\displaystyle \frac{3\pi}{4}+n\pi
+}\right.$
+</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
+</div>
+</div>
+<div class=NavFrame style="CLEAR: both">
+<div class=NavHead>L&ouml;sning a)&nbsp;</div>
+<div class=NavContent>
+Lösning till delfråga a)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_8a-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_8a-2(2).gif]]
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class=NavFrame style="CLEAR: both">
+<div class=NavHead>L&ouml;sning b)&nbsp;</DIV>
+<div class=NavContent>
+Lösning till delfråga b)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_8b.gif]]
+</td>
+</tr>
+</table>
+</div>
+</div>
+<div class=NavFrame style="CLEAR: both">
+<div class=NavHead>L&ouml;sning c)&nbsp;</div>
+<div class=NavContent>
+Lösning till delfråga c)
+<table width="100%">
+<tr>
+<td align="center">
+[[Bild:4_4_8c-1(2).gif]]
+</td>
+</tr>
+<tr>
+<td align="center">
+[[Bild:4_4_8c-2(2).gif]]
+</td>
+</tr>
+</table>
 </div>
 </div>