4.4 Övningar
Sommarmatte 1
Versionen från 30 april 2007 kl. 16.57 (redigera) Ossiang (Diskussion | bidrag) (→Övning 4.4:7) ← Gå till föregående ändring |
Nuvarande version (31 maj 2007 kl. 07.05) (redigera) (ogör) KTH.SE:u1rp004j (Diskussion | bidrag) |
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(9 mellanliggande versioner visas inte.) | |||
Rad 1: | Rad 1: | ||
- | ==Övning 4.4:1== | + | |
+ | ==Övning 4.4:1== | ||
<div class="ovning"> | <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"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <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> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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> | + | </tr> |
- | <td class="ntext">\sin{v}=-\displaystyle \frac{1}{2}</td> | + | <tr align="left"> |
+ | <td class="ntext">e)</td> | ||
+ | <td class="ntext" width="50%">\cos{v}=2</td> | ||
+ | <td class="ntext">f)</td> | ||
+ | <td class="ntext" width="50%">\sin{v}=-\displaystyle \frac{1}{2}</td> | ||
</tr> | </tr> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <tr align="left"> | ||
- | <td class="ntext">$\textrm{g) }$</td> | + | <td class="ntext">g)</td> |
- | <td class="ntext">\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}</td> | + | <td class="ntext" width="50%">\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}</td> |
- | <td class="ntext"></td> | + | |
- | <td class="ntext"></td> | + | |
- | <td class="ntext"></td> | + | |
- | <td class="ntext"></td> | + | |
</tr> | </tr> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 36: | Rad 33: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <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> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 70: | Rad 63: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning d </div> | + | <div class=NavHead>Lösning d) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning e </DIV> | + | <div class=NavHead>Lösning e) </DIV> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning f </div> | + | <div class=NavHead>Lösning f) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning g </div> | + | <div class=NavHead>Lösning g) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga g<br />[[Bild:4_4_1g-1(2).gif]]<br />[[Bild:4_4_1g-2(2).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> | ||
</div> | </div> | ||
- | ==Övning 4.4:2== | + | ==Övning 4.4:2== |
<div class="ovning"> | <div class="ovning"> | ||
Lös ekvationen | Lös ekvationen | ||
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 147: | Rad 192: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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> | ||
- | <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> | ||
- | <td class="ntext">$\textrm{c) }$</td> | + | <td class="ntext">c)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="33%"> |
x=n\pi</td> | x=n\pi</td> | ||
</tr> | </tr> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <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">e)</td> | ||
+ | <td class="ntext" width="33%"> | ||
+ | $\left\{\eqalign{ | ||
+ | x&=\displaystyle\frac{\pi}{30}+\displaystyle\frac{2n\pi}{5}\cr | ||
+ | x&=\displaystyle\frac{\pi}{6}+\displaystyle\frac{2n\pi}{5}} \right.$ | ||
</td> | </td> | ||
- | <td class="ntext">$\textrm{e) }$</td> | + | <td class="ntext">f)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="33%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=\displaystyle\frac{\pi}{30}+\displaystyle\frac{2}{5}n\pi\cr | + | x&=\displaystyle\frac{\pi}{4}+\displaystyle\frac{2n\pi}{3}\cr |
- | x=\displaystyle\frac{\pi}{6}+\displaystyle\frac{2}{5}n\pi } \right.$ | + | x&=\displaystyle\frac{5\pi}{12}+\displaystyle\frac{2n\pi}{3}} \right.$ |
</td> | </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> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 191: | Rad 237: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga a<br />[[Bild:4_4_2a-1(2).gif]]<br />[[Bild:4_4_2a-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning d </div> | + | <div class=NavHead>Lösning d) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga d<br />[[Bild:4_4_2d-1(2).gif]]<br />[[Bild:4_4_2d-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning e </DIV> | + | <div class=NavHead>Lösning e) </DIV> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning f </div> | + | <div class=NavHead>Lösning f) </div> |
<div class=NavContent> | <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> | ||
</div> | </div> | ||
- | ==Övning 4.4:3== | + | ==Övning 4.4:3== |
<div class="ovning"> | <div class="ovning"> | ||
Lös ekvationen | Lös ekvationen | ||
Rad 238: | Rad 336: | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">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">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> |
</tr> | </tr> | ||
- | <tr align="left"><td height="5px"\></tr> | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext">\sin{(x+40^\circ)}=\sin{65^\circ}</td> | + | <td class="ntext" width="50%">\sin{(x+40^\circ)}=\sin{65^\circ}</td> |
<td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
- | <td class="ntext">\sin{3x}=\sin{15^\circ}</td> | + | <td class="ntext" width="50%">\sin{3x}=\sin{15^\circ}</td> |
</tr> | </tr> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 256: | Rad 353: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <tr align="left"> | ||
- | <td class="ntext">$\textrm{a) }$</td> | + | <td class="ntext">a)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=\displaystyle\frac{\pi}{6}+2n\pi\cr | + | x&=\displaystyle\frac{\pi}{6}+2n\pi\cr |
- | x=\displaystyle\frac{11\pi}{6}+2n\pi | + | x&=\displaystyle\frac{11\pi}{6}+2n\pi |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
- | <td class="ntext">$\textrm{b) }$</td> | + | <td class="ntext">b)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=\displaystyle\frac{\pi}{5}+2n\pi\cr | + | x&=\displaystyle\frac{\pi}{5}+2n\pi\cr |
- | x=\displaystyle\frac{4\pi}{5}+2n\pi | + | x&=\displaystyle\frac{4\pi}{5}+2n\pi |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
</tr> | </tr> | ||
- | <tr><td height="5px"/></tr> | ||
<tr align="left"> | <tr align="left"> | ||
- | <td class="ntext">$\textrm{c) }$</td> | + | <td class="ntext">c)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=25^\circ + n\cdot 360^\circ\cr | + | x&=25^\circ + n\cdot 360^\circ\cr |
- | x=75^\circ + n\cdot 360^\circ | + | x&=75^\circ + n\cdot 360^\circ |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
- | <td class="ntext">$\textrm{d) }$</td> | + | <td class="ntext">d)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=5^\circ + n \cdot 120^\circ \cr | + | x&=5^\circ + n \cdot 120^\circ \cr |
- | x= 55^\circ + n \cdot 120^\circ | + | x&= 55^\circ + n \cdot 120^\circ |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
Rad 297: | Rad 393: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning d </div> | + | <div class=NavHead>Lösning d) </div> |
<div class=NavContent> | <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> | ||
</div> | </div> | ||
- | ==Övning 4.4:4== | + | ==Övning 4.4:4== |
<div class="ovning"> | <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> | ||
Rad 332: | Rad 456: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | v_1=50^\circ, v_2=120^\circ, v_3=230^\circ och v_4=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> | </div> | ||
Rad 339: | Rad 469: | ||
<div class=NavHead>Lösning</div> | <div class=NavHead>Lösning</div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | [[Bild:4_4_4-1(3).gif]]<br\>[[Bild:4_4_4-2(3).gif]]<br\>[[Bild:4_4_4-3(3).gif]]<br\> | + | <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> | ||
</div> | </div> | ||
- | ==Övning 4.4:5== | + | ==Övning 4.4:5== |
<div class="ovning"> | <div class="ovning"> | ||
Lös ekvationen | Lös ekvationen | ||
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <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> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 362: | Rad 510: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <tr align="left"> | ||
- | <td class="ntext">$\textrm{a) }$</td> | + | <td class="ntext">a)</td> |
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=n\pi\cr | + | x&=n\pi\cr |
- | x=\displaystyle \frac{\pi}{4}+\displaystyle \frac{1}{2}n\pi | + | x&=\displaystyle \frac{\pi}{4}+\displaystyle \frac{n\pi}{2} |
}\right.$ | }\right.$ | ||
</td> | </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 | + | $\left\{\eqalign{ |
+ | x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr | ||
+ | x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3} | ||
}\right.$ | }\right.$ | ||
</td> | </td> | ||
Rad 388: | Rad 538: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga a<br />[[Bild:4_4_5a-1(2).gif]]<br />[[Bild:4_4_5a-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga b<br />[[Bild:4_4_5b-1(2).gif]]<br />[[Bild:4_4_5b-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga c<br />[[Bild:4_4_5c-1(2).gif]]<br />[[Bild:4_4_5c-2(2).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> | ||
</div> | </div> | ||
- | ==Övning 4.4:6== | + | ==Övning 4.4:6== |
<div class="ovning"> | <div class="ovning"> | ||
Lös ekvationen | Lös ekvationen | ||
Rad 414: | Rad 600: | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">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">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> |
+ | </tr> | ||
+ | <tr align="left"> | ||
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext">\sin 2x = -\sin x</td> | + | <td class="ntext" width="50%">\sin 2x = -\sin x</td> |
</tr> | </tr> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 427: | Rad 615: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
x=n\pi | x=n\pi | ||
</td> | </td> | ||
- | <td class="ntext">$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> | ||
+ | </tr> | ||
+ | <tr align="left"> | ||
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=\displaystyle \frac{2n\pi}{3}\cr | + | x&=\displaystyle \frac{2n\pi}{3}\cr |
- | x=\displaystyle \pi + 2n\pi\ | + | x&=\displaystyle \pi + 2n\pi\cr |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
Rad 454: | Rad 645: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga b<br />[[Bild:4_4_6b-1(2).gif]]<br />[[Bild:4_4_6b-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <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> | ||
</div> | </div> | ||
Rad 480: | Rad 697: | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">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">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> |
+ | </tr> | ||
+ | <tr align="left"> | ||
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext">\cos{3x}=\sin{4x}</td> | + | <td class="ntext" width="50%">\cos{3x}=\sin{4x}</td> |
</tr> | </tr> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 493: | Rad 712: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
$\left\{ \matrix{ | $\left\{ \matrix{ | ||
x=\displaystyle \frac{\pi}{6}+2n\pi\cr | x=\displaystyle \frac{\pi}{6}+2n\pi\cr | ||
Rad 505: | Rad 724: | ||
</td> | </td> | ||
<td class="ntext">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> |
+ | </tr> | ||
+ | <tr align="left"> | ||
<td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
$\left\{ \matrix{ | $\left\{ \matrix{ | ||
x=\displaystyle \frac{\pi}{2}+2n\pi\cr | x=\displaystyle \frac{\pi}{2}+2n\pi\cr | ||
- | x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2}{7}n\pi | + | x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7} |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
Rad 520: | Rad 741: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga a<br />[[Bild:4_4_7a-1(2).gif]]<br />[[Bild:4_4_7a-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga b<br />[[Bild:4_4_7b-1(2).gif]]<br />[[Bild:4_4_7b-2(2).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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga c<br />[[Bild:4_4_7c-1(3).gif]]<br />[[Bild:4_4_7c-2(3).gif]]<br />[[Bild:4_4_7c-3(3).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> | ||
</div> | </div> | ||
Rad 546: | Rad 808: | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
- | <td class="ntext">\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)</td> | ||
- | <td class="ntext">\sin{x}=\sqrt{3}\cos{x}</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">c)</td> | ||
- | <td class="ntext">\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> | ||
<tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
Rad 559: | Rad 823: | ||
<div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
<div class=NavContent> | <div class=NavContent> | ||
- | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
<table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
<tr align="left"> | <tr align="left"> | ||
<td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=\displaystyle \frac{\pi}{4}+2n\pi\cr | + | x&=\displaystyle \frac{\pi}{4}+2n\pi\cr |
- | x=\displaystyle \frac{\pi}{2}+n\pi\cr | + | x&=\displaystyle \frac{\pi}{2}+n\pi\cr |
- | x=\displaystyle \frac{3\pi}{4}+2n\pi | + | x&=\displaystyle \frac{3\pi}{4}+2n\pi |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
<td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
- | <td class="ntext">x=\displaystyle \frac{\pi}{3}+n\pi</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">c)</td> | ||
- | <td class="ntext"> | + | <td class="ntext" width="50%"> |
- | $\left\{ \matrix{ | + | $\left\{\eqalign{ |
- | x=n\pi\cr | + | x&=n\pi\cr |
- | x=\displaystyle \frac{3\pi}{4}+n\pi | + | x&=\displaystyle \frac{3\pi}{4}+n\pi |
}\right.$ | }\right.$ | ||
</td> | </td> | ||
Rad 586: | Rad 852: | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga a<br />[[Bild:4_4_8a-1(2).gif]][[Bild:4_4_8a-2(2).gif]] | + | 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> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning b </DIV> | + | <div class=NavHead>Lösning b) </DIV> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga b<br />[[Bild:4_4_8b.gif]] | + | Lösning till delfråga b) |
+ | <table width="100%"> | ||
+ | <tr> | ||
+ | <td align="center"> | ||
+ | [[Bild:4_4_8b.gif]] | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
</div> | </div> | ||
</div> | </div> | ||
<div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
- | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
<div class=NavContent> | <div class=NavContent> | ||
- | Lösning till delfråga c<br />[[Bild:4_4_8c-1(2).gif]][[Bild:4_4_8c-2(2).gif]] | + | 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> | ||
</div> | </div> |
Nuvarande version
Innehåll[göm] |
[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 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} | ||
[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 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. |
[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 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. |
[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}\,.
v_1=50^\circ, \ \ v_2=120^\circ, \ \ v_3=230^\circ\ \ och \ \ v_4=300^\circ |
[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 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. |
||
[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 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. |
||
[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 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. |
||
[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 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. |
||