Övningar 4.4

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Versionen från 16 juli 2007 kl. 08.17 (redigera)
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(Ny sida: __NOTOC__ ==&Ouml;vning 4.4:1== <div class="ovning"> För vilka vinklar $\,v\,$, där $\,0 \leq v\leq 2\pi\,$, gäller att <table width="100%" cellspacing="10px"> <tr align="left"> <td clas...)
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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==&Ouml;vning 4.4:1==+'''&Ouml;vning 4.4:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
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">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{v}=\displaystyle \frac{1}{2}$</td> <td class="ntext" width="50%">$\sin{v}=\displaystyle \frac{1}{2}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\cos{v}=\displaystyle \frac{1}{2}$</td> <td class="ntext" width="50%">$\cos{v}=\displaystyle \frac{1}{2}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{v}=1$</td> <td class="ntext" width="50%">$\sin{v}=1$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\tan{v}=1$</td> <td class="ntext" width="50%">$\tan{v}=1$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\cos{v}=2$</td> <td class="ntext" width="50%">$\cos{v}=2$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{v}=-\displaystyle \frac{1}{2}$</td> <td class="ntext" width="50%">$\sin{v}=-\displaystyle \frac{1}{2}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">g)</td>+<td class="ntext">g)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$</td> <td class="ntext" width="50%">$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$</td>
</tr> </tr>
Rad 30: Rad 30:
</div> </div>
-==&Ouml;vning 4.4:2==+'''&Ouml;vning 4.4:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
L&ouml;s ekvationen L&ouml;s ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\sin{x}=\displaystyle \frac{\sqrt{3}}{2}$</td> <td class="ntext" width="33%">$\sin{x}=\displaystyle \frac{\sqrt{3}}{2}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\cos{x}=\displaystyle \frac{1}{2} $</td> <td class="ntext" width="33%">$\cos{x}=\displaystyle \frac{1}{2} $</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\sin{x}=0$</td> <td class="ntext" width="33%">$\sin{x}=0$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} $</td> <td class="ntext" width="33%">$\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} $</td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\sin{5x}=\displaystyle \frac{1}{2}$</td> <td class="ntext" width="33%">$\sin{5x}=\displaystyle \frac{1}{2}$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}$</td> <td class="ntext" width="33%">$\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}$</td>
</tr> </tr>
Rad 54: Rad 54:
</div> </div>
-==&Ouml;vning 4.4:3==+'''&Ouml;vning 4.4:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\cos{x}=\cos{\displaystyle \frac{\pi}{6}}$</td> <td class="ntext" width="50%">$\cos{x}=\cos{\displaystyle \frac{\pi}{6}}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{x}=\sin{\displaystyle \frac{\pi}{5}}$</td> <td class="ntext" width="50%">$\sin{x}=\sin{\displaystyle \frac{\pi}{5}}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
-<td class="ntext" width="50%">$\sin{(x+40^\circ)}=\sin{65^\circ}$</td>+<td class="ntext" width="50%">$\sin{(x+40^\circ)&nbsp;&nbsp;&nbsp;}=\sin{65^\circ}$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{3x}=\sin{15^\circ}$</td> <td class="ntext" width="50%">$\sin{3x}=\sin{15^\circ}$</td>
</tr> </tr>
Rad 74: Rad 74:
</div> </div>
-==&Ouml;vning 4.4:4==+'''&Ouml;vning 4.4:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
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>
-==&Ouml;vning 4.4:5==+'''&Ouml;vning 4.4:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{3x}=\sin{x}$</td> <td class="ntext" width="50%">$\sin{3x}=\sin{x}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\tan{x}=\tan{4x}$</td> <td class="ntext" width="50%">$\tan{x}=\tan{4x}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\cos{5x}=\cos(x+\pi/5)$</td> <td class="ntext" width="50%">$\cos{5x}=\cos(x+\pi/5)$</td>
</tr> </tr>
Rad 97: Rad 97:
</div> </div>
-==&Ouml;vning 4.4:6==+'''&Ouml;vning 4.4:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin x\cdot \cos 3x = 2\sin x$</td> <td class="ntext" width="50%">$\sin x\cdot \cos 3x = 2\sin x$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sqrt{2}\sin{x}\cos{x}=\cos{x}$</td> <td class="ntext" width="50%">$\sqrt{2}\sin{x}\cos{x}=\cos{x}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin 2x = -\sin x$</td> <td class="ntext" width="50%">$\sin 2x = -\sin x$</td>
</tr> </tr>
Rad 115: Rad 115:
</div> </div>
-==&Ouml;vning 4.4:7==+'''&Ouml;vning 4.4:7'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$2\sin^2{x}+\sin{x}=1$</td> <td class="ntext" width="50%">$2\sin^2{x}+\sin{x}=1$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$2\sin^2{x}-3\cos{x}=0$</td> <td class="ntext" width="50%">$2\sin^2{x}-3\cos{x}=0$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\cos{3x}=\sin{4x}$</td> <td class="ntext" width="50%">$\cos{3x}=\sin{4x}$</td>
</tr> </tr>
Rad 133: Rad 133:
</div> </div>
-==&Ouml;vning 4.4:8==+'''&Ouml;vning 4.4:8'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{2x}=\sqrt{2}\cos{x}$</td> <td class="ntext" width="50%">$\sin{2x}=\sqrt{2}\cos{x}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\sin{x}=\sqrt{3}\cos{x}$</td> <td class="ntext" width="50%">$\sin{x}=\sqrt{3}\cos{x}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}$</td> <td class="ntext" width="50%">$\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}$</td>
</tr> </tr>
Rad 150: Rad 150:
</table> </table>
</div> </div>
 +<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
 +<br><br><br><br><br><br><br><br><br><br><br><br>

Nuvarande version

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)    $\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$

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)    $\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}$

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)    $\sin{3x}=\sin{15^\circ}$

Övning 4.4:4

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

Övning 4.4:5

Lös ekvationen

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

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)    $\sin 2x = -\sin x$

Övning 4.4:7

Lös ekvationen

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

Övning 4.4:8

Lös ekvationen

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














































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