Versionen från 25 juni 2007 kl. 07.52 (redigera) KTH.SE:u1xsetv1 (Diskussion | bidrag) ← Gå till föregående ändring |
Nuvarande version (30 juni 2007 kl. 15.48) (redigera) (ogör) KTH.SE:u1m1gion (Diskussion | bidrag) |
(43 mellanliggande versioner visas inte.) |
Rad 89: |
Rad 89: |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Facit </div> | + | |
- | <div class=NavContent> | + | |
- | Facit till alla delfrågor<br \> | + | |
| <table width="100%" cellspacing="10px"> | | <table width="100%" cellspacing="10px"> |
| <tr align="left"> | | <tr align="left"> |
Rad 105: |
Rad 102: |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | ==Övning 4.1:4== |
| + | <div class="ovning"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="100%">Bestäm avståndet mellan punkterna (1,1) och (5,4).</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">Bestäm avståndet mellan punkterna (-2,5) och (3,-1).</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="100%">Hitta den punkt på x-axeln som ligger lika långt från punkterna (3,3) och (5,1).</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning a) </div> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr align="left"> |
- | Lösning till delfråga a) | + | <td class="ntext">a)</td> |
- | <table width="100%"> | + | <td class="ntext" width="100%">$5 \textrm{ l.e.}$</td> |
- | <tr> | + | |
- | <td align="center"> | + | |
- | [[Bild:4_1_3a.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">\sqrt{61} \textrm{ l.e.}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="100%">(2,0)</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.1:5== |
- | <div class=NavHead>Lösning b) </DIV> | + | <div class="ovning"> |
- | <div class=NavContent> | + | <table width="100%" cellspacing="10px"> |
- | Lösning till delfråga b) | + | <tr align="left" valign="top"> |
- | <table width="100%"> | + | <td class="ntext">a)</td> |
- | <tr> | + | <td class="ntext" width="100%">Bestäm ekvationen för en cirkel med medelpunkt i (1,2) och radie 2.</td> |
- | <td align="center"> | + | |
- | [[Bild:4_1_3b.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">Bestäm ekvationen för den cirkel som har medelpunkt i (2,-1) och innehåller punkten (-1,1).</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning c) </div> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr align="left"> |
- | Lösning till delfråga c) | + | <td class="ntext">a)</td> |
- | <table width="100%"> | + | <td class="ntext" width="100%">(x-1)^2+(y-2)^2=4</td> |
- | <tr> | + | |
- | <td align="center"> | + | |
- | [[Bild:4_1_3c.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">(x-2)^2+(y+1)^2=13</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:4== | + | ==Övning 4.1:6== |
| <div class="ovning"> | | <div class="ovning"> |
| + | Skissera följande cirklar |
| <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" width="100%">Bestäm avståndet mellan punkterna (1,1) och (5,4).</td> | + | <td class="ntext" width="50%">x^2+y^2=9</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">$(x-1)^2+(y-2)^2=3$</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">(3x-1)^2+(3y+7)^2=10</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">En cirkel med radie 3 och medelpunkt i origo.</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="100%">Bestäm avståndet mellan punkterna (-2,5) och (3,-1).</td> | + | <td class="ntext" width="50%">En cirkel med radie \sqrt 3 och medelpunkt i punkten (1, 2).</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| <td class="ntext">c)</td> | | <td class="ntext">c)</td> |
- | <td class="ntext" width="100%">Hitta den punkt på x-axeln som ligger lika långt från punkterna (3,3) och (5,1).</td> | + | <td class="ntext" width="50%">En cirkel med radie $\frac{1}{3}\sqrt 10$ och medelpunkt i punkten (1/3, -7/3).</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
Rad 168: |
Rad 203: |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.1:7== |
- | <div class=NavHead>Facit </div> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Skissera följande cirklar |
- | 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" width="100%">$5 \textrm{ l.e.}$</td> | + | <td class="ntext" width="50%">$x^2+2x+y^2-2y=1$</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">$x^2+y^2+4y=0$</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">x^2-2x+y^2+6y=-3</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">x^2-2x+y^2+2y=-2</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">En cirkel med medelpunkt (-1, 1) och radie \sqrt 3.</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="100%">$\sqrt{61} \textrm{ l.e.}$</td> | + | <td class="ntext" width="50%">En cirkel med medelpunkt (0, -2) och radie 2. </td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| <td class="ntext">c)</td> | | <td class="ntext">c)</td> |
- | <td class="ntext" width="100%">$(2,0)$</td> | + | <td class="ntext" width="50%">En cirkel med medelpunkt (1, -3) och radie $\sqrt 7$.</td> |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">Endast punkten (1, -1). </td> |
| + | </tr> |
| + | |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | ==Övning 4.1:8== |
| + | <div class="ovning"> |
| + | Hur många varv snurrar ett hjul med radie 50 cm när det rullar 10m? |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning a) </div> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr><td height="5px"/></tr> |
- | Lösning till delfråga a) | + | <tr align="left"> |
- | <table width="100%"> | + | <td class="ntext" width="100%"> |
- | <tr> | + | $\displaystyle \frac{10}{\pi}\textrm{ varv }\approx 3,2 \textrm{ varv} $ </td> |
- | <td align="center"> | + | |
- | [[Bild:4_1_4a-1(2).gif]] | + | |
- | [[Bild:4_1_4a-2(2).gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | ==Övning 4.1:9== |
| + | <div class="ovning"> |
| + | På en klocka är sekundvisaren 8 cm lång. Hur stor area sveper den över på 10 sekunder? |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning b) </DIV> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr><td height="5px"/></tr> |
- | Lösning till delfråga b) | + | <tr align="left"> |
- | <table width="100%"> | + | <td class="ntext" width="100%">\displaystyle \frac{32\pi}{3} \textrm{ cm}^2 \approx 33,5 \textrm{ cm}^2</td> |
- | <tr> | + | |
- | <td align="center"> | + | |
- | [[Bild:4_1_4b.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | + | |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.1:10== |
- | <div class=NavHead>Lösning c) </div> | + | <div class="ovning"> |
- | <div class=NavContent> | + | En 5,4 m lång tvättlina hänger mellan två vertikala träd på 4,8 m avstånd från varandra. Linans ena ände är fäst 0,6 m högre än den andra änden, och 1,2 m från trädet där linan har sin lägre infästning hänger en kavaj på en galge. Bestäm hur mycket under den nedre infästningspunkten som galgen hänger (dvs. avståndet \,x\, i figuren). |
- | Lösning till delfråga c) | + | |
- | <table width="100%"> | + | <table width="100%" cellspacing="10px"> |
| + | <tr><td height="5px"/></tr> |
| <tr> | | <tr> |
| <td align="center"> | | <td align="center"> |
- | [[Bild:4_1_4c-1(2).gif]] | + | [[Bild:Uppg_4_1_10.gif]] |
- | </td> | + | |
- | </tr> | + | |
- | <tr> | + | |
- | <td align="center"> | + | |
- | [[Bild:4_1_4c-2(2).gif]] | + | |
| </td> | | </td> |
| + | </tr><tr><td height="5px"/></tr> |
| + | </table> |
| + | |
| + | |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr><td height="5px"/></tr> |
| + | <tr align="left"> |
| + | <td class="ntext" width="100%">x=9 dm</td> |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | + | |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:5== | + | ==Övning 4.2:1== |
| <div class="ovning"> | | <div class="ovning"> |
| + | Bestäm längden av sidan som är markerad med \,x\, uttryckt med hjälp av de trigonometriska funktionerna. |
| <table width="100%" cellspacing="10px"> | | <table width="100%" cellspacing="10px"> |
| <tr align="left" valign="top"> | | <tr align="left" valign="top"> |
| <td class="ntext">a)</td> | | <td class="ntext">a)</td> |
- | <td class="ntext" width="100%">Bestäm ekvationen för en cirkel med medelpunkt i (1,2) och radie 2.</td> | + | <td class="ntext" width="50%">[[Bild:O4_2_1a.gif]]</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_1b.gif]]</td> |
| </tr> | | </tr> |
| <tr align="left" valign="top"> | | <tr align="left" valign="top"> |
- | <td class="ntext">b)</td> | + | <td class="ntext">c)</td> |
- | <td class="ntext" width="100%">Bestäm ekvationen för den cirkel som har medelpunkt i (2,-1) och innehåller punkten (-1,1).</td> | + | <td class="ntext" width="50%">[[Bild:O4_2_1c.gif]]</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_1d.gif]]</td> |
| + | </tr> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_1e.gif]]</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_1f.gif]]</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
Rad 253: |
Rad 329: |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Facit </div> | + | |
- | <div class=NavContent> | + | |
| Facit till alla delfrågor<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" width="100%">$(x-1)^2+(y-2)^2=4$</td> | + | <td class="ntext" width="50%"> $x=13\cdot\tan {27 ^\circ} \approx 6{,}62$</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">$x=25\cdot\cos {32 ^\circ} \approx 21{,}2$</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">x=\displaystyle\frac{14}{\tan {40 ^\circ}} \approx 16{,}7</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">x=\displaystyle\frac{16}{\cos {20 ^\circ}} \approx 17{,}0</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">x=\displaystyle\frac{11}{\sin {35 ^\circ}} \approx 19{,}2</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">x=\displaystyle\frac{19}{\tan {50 ^\circ}} \approx 15{,}9</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | ==Övning 4.2:2== |
| + | <div class="ovning"> |
| + | Bestäm en trigonometrisk ekvation som vinkeln \,v\, uppfyller. |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_2a.gif]]</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="100%">$(x-2)^2+(y+1)^2=13$</td> | + | <td class="ntext" width="50%">[[Bild:O4_2_2b.gif]]</td> |
| + | </tr> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_2c.gif]]</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_2d.gif]]</td> |
| + | </tr> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_2e.gif]]</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">[[Bild:O4_2_2f.gif]]</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%"> \tan v=\displaystyle\frac{2}{5}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">\sin v=\displaystyle\frac{7}{11}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">\cos v=\displaystyle\frac{5}{7}</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">\sin v=\displaystyle\frac{3}{5}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">v=30 ^\circ</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">\sin \displaystyle\frac{v}{2}=\displaystyle\frac{1}{3}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.2:3== |
- | <div class=NavHead>Lösning a) </div> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Bestäm |
- | Lösning till delfråga a) | + | <table width="100%" cellspacing="10px"> |
- | <table width="100%"> | + | <tr align="left"> |
- | <tr> | + | <td class="ntext">a)</td> |
- | <td align="center"> | + | <td class="ntext" width="33%"> \sin{\left(-\displaystyle \frac{\pi}{2}\right)}</td> |
- | [[Bild:4_1_5a.gif]] | + | <td class="ntext">b)</td> |
- | </td> | + | <td class="ntext" width="33%">\cos{2\pi}</td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="33%">\sin{9\pi}</td> |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="33%"> \cos{\displaystyle \frac{7\pi}{2}}</td> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="33%">\sin{\displaystyle \frac{3\pi}{4}}</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="33%">\cos{\left(-\displaystyle \frac{\pi}{6}\right)}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning b) </DIV> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr align="left"> |
- | Lösning till delfråga b) | + | <td class="ntext">a)</td> |
- | <table width="100%"> | + | <td class="ntext" width="33%">-1</td> |
- | <tr> | + | <td class="ntext">b)</td> |
- | <td align="center"> | + | <td class="ntext" width="33%">1</td> |
- | [[Bild:4_1_5b-1(2).gif]] | + | <td class="ntext">c)</td> |
- | </td> | + | <td class="ntext" width="33%">0</td> |
| </tr> | | </tr> |
- | <tr> | + | <tr align="left"> |
- | <td align="center"> | + | <td class="ntext">d)</td> |
- | [[Bild:4_1_5b-2(2).gif]] | + | <td class="ntext" width="33%">0</td> |
- | </td> | + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="33%">$\displaystyle \frac{1}{\sqrt{2}}$</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="33%">\displaystyle \frac{\sqrt{3}}{2}</td> |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:6== | + | ==Övning 4.2:4== |
| <div class="ovning"> | | <div class="ovning"> |
- | Skissera följande cirklar | + | Bestäm |
| <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" width="50%">$x^2+y^2=9$</td> | + | <td class="ntext" width="33%">$\cos{\displaystyle \frac{11\pi}{6}}$</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="50%">$(x-1)^2+(y-2)^2=3$</td> | + | <td class="ntext" width="33%">$\cos{\displaystyle \frac{11\pi}{3}}$</td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="33%">$\tan{\displaystyle \frac{3\pi}{4}}$</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="33%">\tan{\pi}</td> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="33%">\tan{\displaystyle \frac{7\pi}{6}}</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="33%">\tan{\left(-\displaystyle \frac{5\pi}{3}\right)}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="33%">\displaystyle \frac{\sqrt{3}}{2}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="33%">\displaystyle \frac{1}{2}</td> |
| <td class="ntext">c)</td> | | <td class="ntext">c)</td> |
- | <td class="ntext" width="50%">$(3x-1)^2+(3y+7)^2=10$</td> | + | <td class="ntext" width="33%">$-1$</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="33%">0</td> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="33%">\displaystyle \frac{1}{\sqrt{3}}</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="33%">$\sqrt{3}$</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
Rad 322: |
Rad 496: |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.2:5== |
- | <div class=NavHead>Facit </div> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Bestäm |
- | 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" width="50%">BILD</td> | + | <td class="ntext" width="25%">\cos{135^\circ}</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="50%">BILD</td> | + | <td class="ntext" width="25%">\tan{225^\circ}</td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="25%">\cos{330^\circ}</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="25%">\tan{495^\circ}</td> |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| <tr align="left"> | | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="25%">-\displaystyle \frac{1}{\sqrt{2}}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="25%">1</td> |
| <td class="ntext">c)</td> | | <td class="ntext">c)</td> |
- | <td class="ntext" width="50%">BILD</td> | + | <td class="ntext" width="25%">\displaystyle \frac{\sqrt{3}}{2}</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="25%">-1</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | ==Övning 4.2:6== |
| + | <div class="ovning"> |
| + | Bestäm längden av sidan som är markerad med \,x\,. |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="center"> |
| + | <td class="ntext" width="100%">[[Bild:O4_2_6.gif]]</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning a) </div> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr align="left"> |
- | Lösning till delfråga a) | + | <td class="ntext" width="100%">x= \sqrt{3}-1</td> |
- | <table width="100%"> | + | |
- | <tr> | + | |
- | <td align="center"> | + | |
- | [[Bild:4_1_6a.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| + | </div> |
| | | |
| + | ==Övning 4.2:7== |
| + | <div class="ovning"> |
| + | För att mäta upp bredden av en älv mäter vi från två punkter A och B längs den ena raka stranden vinkeln till ett träd C på motsatt sida älven. Hur bred är älven om måtten i figuren gäller? |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="center"> |
| + | <td class="ntext" width="100%">[[Bild:O_4_2_7.gif]]</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext" width="100%">Älvens bredd är \ \displaystyle\frac{100}{\sqrt{3}-1} m \approx 136{,}6 m.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | |
- | <div class=NavHead>Lösning b) </DIV> | + | ==Övning 4.2:8== |
- | <div class=NavContent> | + | <div class="ovning"> |
- | Lösning till delfråga b) | + | En stång med längd \,\ell\, är upphängd i två linor med längd \,a\, resp. $\,b\, enligt figuren. Linorna bildar vinklar \,\alpha\, resp. \,\beta\, med vertikalen. Bestäm en trigonometrisk ekvation för vinkeln \,\gamma\,$ som stången bildar med vertikalen. |
- | <table width="100%"> | + | <table width="100%" cellspacing="10px"> |
- | <tr> | + | <tr align="center"> |
- | <td align="center"> | + | <td class="ntext" width="100%">[[Bild:O_4_2_8.gif]]</td> |
- | [[Bild:4_1_6b.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext" width="100%">\ell\cos \gamma=a \cos \alpha - b\cos \beta </td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.2:9== |
- | <div class=NavHead>Lösning c) </DIV> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Bilvägen från ''A'' till ''B'' består av tre rätlinjiga delar ''AP'', ''PQ'' och ''QB'', vilka är 4,0 km, 12,0 km respektive 5,0 km. De i figuren markerade vinklarna vid ''P'' och ''Q'' är 30° respektive 90°. Beräkna avståndet fågelvägen från ''A'' till ''B''. (Uppgiften är hämtad ur Centrala provet i matematik, november 1976, men aningen modifierad.) |
- | Lösning till delfråga c) | + | <table width="100%" cellspacing="10px"> |
- | <table width="100%"> | + | <tr align="center"> |
- | <tr> | + | <td class="ntext" width="100%">[[Bild:O4_2_9.gif]]</td> |
- | <td align="center"> | + | |
- | [[Bild:4_1_6c-1(2).gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
- | <tr> | + | <tr><td height="5px"/></tr> |
- | <td align="center"> | + | </table> |
- | [[Bild:4_1_6c-2(2).gif]] | + | </div> |
- | </td> | + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext" width="100%">Avståndet är $\ \sqrt{205-48\sqrt{3}} \approx 11{,}0$ km.</td> |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | ==Övning 4.3:1== |
| + | <div class="ovning"> |
| + | Bestäm de vinklar \,v\, mellan \,\displaystyle \frac{\pi}{2}\, och \,2\pi\, som uppfyller |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="33%">\cos{v}=\cos{\displaystyle \frac{\pi}{5}}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="33%">\sin{v}=\sin{\displaystyle \frac{\pi}{7}}</td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="33%">\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}</td> |
| + | </tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:7== | + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="33%">v = \displaystyle \frac{9\pi}{5}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="33%">v = \displaystyle \frac{6\pi}{7}</td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="33%">v = \displaystyle \frac{9\pi}{7}</td> |
| + | </tr> |
| + | </table> |
| + | </div> |
| + | |
| + | ==Övning 4.3:2== |
| <div class="ovning"> | | <div class="ovning"> |
- | Skissera följande cirklar | + | Bestäm de vinklar \,v\, mellan 0 och \,\pi\, som uppfyller |
| <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" width="50%">$x^2+2x+y^2-2y=1$</td> | + | <td class="ntext" width="50%">$\cos{v} = \cos{\displaystyle \frac{3\pi}{2}}$</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="50%">$x^2+y^2+4y=0$</td> | + | <td class="ntext" width="50%">$\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}$</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">$v=\displaystyle \frac{\pi}{2}$</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">v=\displaystyle \frac{3\pi}{5}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | ==Övning 4.3:3== |
| + | <div class="ovning"> |
| + | Antag att $\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\, och att \,\sin{v} = a\,. Uttryck med hjälp av \,a$ |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">\sin{(-v)}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">$\sin{(\pi-v)}$</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| <td class="ntext">c)</td> | | <td class="ntext">c)</td> |
- | <td class="ntext" width="50%">$x^2-2x+y^2+6y=-3$</td> | + | <td class="ntext" width="50%">$\cos{v}$</td> |
| <td class="ntext">d)</td> | | <td class="ntext">d)</td> |
- | <td class="ntext" width="50%">$x^2-2x+y^2+2y=-2$</td> | + | <td class="ntext" width="50%">$\sin{\left(\displaystyle \frac{\pi}{2}-v\right)}$</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">$\cos{\left( \displaystyle \frac{\pi}{2} + v\right)}$</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">$\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}$</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
Rad 410: |
Rad 691: |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Facit </div> | + | |
- | <div class=NavContent> | + | |
- | 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" width="50%">BILD</td> | + | <td class="ntext" width="50%">-a</td> |
| <td class="ntext">b)</td> | | <td class="ntext">b)</td> |
- | <td class="ntext" width="50%">BILD</td> | + | <td class="ntext" width="50%">a</td> |
| </tr> | | </tr> |
| <tr align="left"> | | <tr align="left"> |
| <td class="ntext">c)</td> | | <td class="ntext">c)</td> |
- | <td class="ntext" width="50%">BILD</td> | + | <td class="ntext" width="50%">\sqrt{1-a^2}</td> |
| <td class="ntext">d)</td> | | <td class="ntext">d)</td> |
- | <td class="ntext" width="50%">BILD</td> | + | <td class="ntext" width="50%">\sqrt{1-a^2}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">-a</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">\displaystyle \frac{\sqrt{3}}{2}\sqrt{1-a^2}+\displaystyle \frac{1}{2}\cdot a </td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | ==Övning 4.3:4== |
| + | <div class="ovning"> |
| + | Antag att \,0 \leq v \leq \pi\, och att \,\cos{v}=b\,. Uttryck med hjälp av \,b\, |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">\sin^2{v}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">\sin{v}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">\sin{2v}</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">\cos{2v}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">\sin{\left( v+\displaystyle \frac{\pi}{4} \right)}</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning a) </div> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr align="left"> |
- | Lösning till delfråga a) | + | <td class="ntext">a)</td> |
- | <table width="100%"> | + | <td class="ntext" width="50%">1-b^2</td> |
- | <tr> | + | <td class="ntext">b)</td> |
- | <td align="center"> | + | <td class="ntext" width="50%">$\sqrt{1-b^2}$</td> |
- | [[Bild:4_1_7a-1(2).gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
- | <tr> | + | <tr align="left"> |
- | <td align="center"> | + | <td class="ntext">c)</td> |
- | [[Bild:4_1_7a-2(2).gif]] | + | <td class="ntext" width="50%">$2b\sqrt{1-b^2}$</td> |
- | </td> | + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">2b^2-1</td> |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">\sqrt{1-b^2}\cdot\displaystyle \frac{1}{\sqrt{2}} + b\cdot \displaystyle \frac{1}{\sqrt{2}} </td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">b\cdot\displaystyle \frac{1}{2}+\sqrt{1-b^2}\cdot\displaystyle \frac{\sqrt{3}}{2}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| + | </div> |
| | | |
| + | ==Övning 4.3:5== |
| + | <div class="ovning"> |
| + | För en spetsig vinkel \,v\, i en triangel gäller att \,\sin{v}=\displaystyle \frac{5}{7}\,. Bestäm \,\cos{v}\, och \,\tan{v}\,. |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext" width="100%">\cos{v}=\displaystyle \frac{2\sqrt{6}}{7}\quad och \quad\tan{v}=\displaystyle \frac{5}{2\sqrt{6}}\,.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.3:6== |
- | <div class=NavHead>Lösning b) </DIV> | + | <div class="ovning"> |
- | <div class=NavContent> | + | <table width="100%" cellspacing="10px"> |
- | Lösning till delfråga b) | + | <tr align="left" valign="top"> |
- | <table width="100%"> | + | <td class="ntext">a)</td> |
- | <tr> | + | <td class="ntext" width="100%">Bestäm \ \sin{v}\ och \ \tan{v}\ om \ \cos{v}=\displaystyle \frac{3}{4}\ och \ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,.</td> |
- | <td align="center"> | + | |
- | [[Bild:4_1_7b.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">Bestäm \ \cos{v}\ och \ \tan{v}\ om \ \sin{v}=\displaystyle \frac{3}{10}\ och \,v\, ligger i den andra kvadranten.</td> |
| + | </tr> |
| + | <tr align="left" valign="top"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="100%">Bestäm \ \sin{v}\ och \ \cos{v}\ om \ \tan{v}=3\ och \ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="100%">\sin{v}=-\displaystyle \frac{\sqrt{7}}{4}\quad och \quad\tan{v}=-\displaystyle \frac{\sqrt{7}}{3}\,.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | <tr align="left"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">\cos{v}=-\displaystyle \frac{\sqrt{91}}{10}\quad och \quad\tan{v}=-\displaystyle \frac{3}{\sqrt{91}}\,.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">\sin{v}=-\displaystyle \frac{3}{\sqrt{10}}\quad och \quad\cos{v}=-\displaystyle \frac{1}{\sqrt{10}}\,.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.3:7== |
- | <div class=NavHead>Lösning c) </DIV> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Bestäm $\ \sin{(x+y)}\ $ om |
- | Lösning till delfråga c) | + | <table width="100%" cellspacing="10px"> |
- | <table width="100%"> | + | <tr align="left"> |
- | <tr> | + | <td class="ntext">a)</td> |
- | <td align="center"> | + | <td class="ntext" width="100%">\sin{x}=\displaystyle \frac{2}{3}\,, \ \sin{y}=\displaystyle \frac{1}{3}\ och \,x\,, \,y\, är vinklar i första kvadranten.</td> |
- | [[Bild:4_1_7c.gif]] | + | |
- | </td> | + | |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">\cos{x}=\displaystyle \frac{2}{5}\,, \ \cos{y}=\displaystyle \frac{3}{5}\ och \,x\,, \,y\, är vinklar i första kvadranten.</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="100%">\sin{(x+y)}=\displaystyle \frac{4\sqrt{2}+\sqrt{5}}{9}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="100%">\sin{(x+y)}=\displaystyle \frac{3\sqrt{21}+8}{25}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.3:8== |
- | <div class=NavHead>Lösning d) </DIV> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Visa följande trigonometriska samband |
- | Lösning till delfråga d) | + | <table width="100%" cellspacing="10px"> |
- | <table width="100%"> | + | <tr align="left"> |
- | <tr> | + | <td class="ntext>a)</td> |
- | <td align="center"> | + | <td class="ntext width="100%">\tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}</td> |
- | [[Bild:4_1_7d.gif]] | + | </tr> |
- | </td> | + | <tr align="left"> |
| + | <td class="ntext>b)</td> |
| + | <td class="ntext width="100%">\displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext>c)</td> |
| + | <td class="ntext width="100%">\tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext>d)</td> |
| + | <td class="ntext width="100%">\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v</td> |
| </tr> | | </tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">Se lösningen i webmaterialet när |
| + | du loggat in till kursen</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:8== | + | ==Övning 4.3:9== |
| <div class="ovning"> | | <div class="ovning"> |
- | Hur många varv snurrar ett hjul med radie 50 cm när det rullar 10m? | + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext> |
| + | Visa "Feynmans likhet" |
| + | \cos 20^\circ \cdot \cos 40^\circ \cdot \cos 80^\circ = \displaystyle\frac{1}{8}\,\mbox{.} |
| + | (Ledtråd: Använd formeln för dubbla vinkeln på \,\sin 160^\circ\,.) |
| + | </td> |
| + | </tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Facit </div> | + | |
- | <div class=NavContent> | + | |
| <table width="100%" cellspacing="10px"> | | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">Se lösningen i webmaterialet när |
| + | du loggat in till kursen</td> |
| + | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | ==Övning 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"> | | <tr align="left"> |
- | <td class="ntext" width="100%"> | + | <td class="ntext">a)</td> |
- | $\displaystyle \frac{10}{\pi}\textrm{ varv }\approx 3,2 \textrm{ varv} $ </td> | + | <td class="ntext" width="50%">\sin{v}=\displaystyle \frac{1}{2}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">$\cos{v}=\displaystyle \frac{1}{2}$</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">$\sin{v}=1$</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">$\tan{v}=1$</td> |
| + | </tr> |
| + | <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 align="left"> |
| + | <td class="ntext">g)</td> |
| + | <td class="ntext" width="50%">$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$</td> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">\displaystyle v=\frac{\pi}{6}\,, \,\displaystyle v=\frac{5\pi}{6}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">\displaystyle v=\frac{\pi}{3}\,, \,\displaystyle v=\frac{5\pi}{3}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">\displaystyle v=\frac{\pi}{2}</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">\displaystyle v=\frac{\pi}{4}\,, \,\displaystyle v=\frac{5\pi}{4}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="50%">lösning saknas</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="50%">\displaystyle v=\frac{11\pi}{6}\,, \,\displaystyle v=\frac{7\pi}{6}</td> |
| + | </tr> |
| + | <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> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.4:2== |
- | <div class=NavHead>Lösning </div> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Lös ekvationen |
- | <table width="100%"> | + | <table width="100%" cellspacing="10px"> |
- | <tr> | + | <tr align="left"> |
- | <td align="center"> | + | <td class="ntext">a)</td> |
- | [[Bild:4_1_8.gif]] | + | <td class="ntext" width="33%">\sin{x}=\displaystyle \frac{\sqrt{3}}{2}</td> |
- | </td> | + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="33%">\cos{x}=\displaystyle \frac{1}{2} </td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="33%">\sin{x}=0</td> |
| </tr> | | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="33%">\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} </td> |
| + | <td class="ntext">e)</td> |
| + | <td class="ntext" width="33%">\sin{5x}=\displaystyle \frac{1}{2}</td> |
| + | <td class="ntext">f)</td> |
| + | <td class="ntext" width="33%">\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
| </div> | | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="33%"> |
| + | $\left\{\eqalign{ |
| + | x&=\displaystyle\frac{\pi}{3}+2n\pi\cr |
| + | x&=\displaystyle\frac{2\pi}{3}+2n\pi } \right.$ |
| + | </td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="33%"> |
| + | $\left\{\eqalign{ |
| + | x&=\displaystyle\frac{\pi}{3}+2n\pi\cr |
| + | x&=\displaystyle\frac{5\pi}{3}+2n\pi } \right.$ |
| + | </td> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="33%"> |
| + | x=n\pi</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="33%"> |
| + | $\left\{\eqalign{ |
| + | x&=\displaystyle\frac{\pi}{20}+\displaystyle\frac{2n\pi}{5}\cr |
| + | x&=\displaystyle\frac{3\pi}{20}+\displaystyle\frac{2n\pi}{5} } \right.$ |
| + | </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 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> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:9== | + | ==Övning 4.4:3== |
| <div class="ovning"> | | <div class="ovning"> |
- | På en klocka är sekundvisaren 8 cm lång. Hur stor area sveper den över på 10 sekunder? | + | Lös ekvationen |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">\cos{x}=\cos{\displaystyle \frac{\pi}{6}}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">\sin{x}=\sin{\displaystyle \frac{\pi}{5}}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%">\sin{(x+40^\circ)}=\sin{65^\circ}</td> |
| + | <td class="ntext">d)</td> |
| + | <td class="ntext" width="50%">\sin{3x}=\sin{15^\circ}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Facit </div> | + | |
- | <div class=NavContent> | + | |
| <table width="100%" cellspacing="10px"> | | <table width="100%" cellspacing="10px"> |
- | <tr><td height="5px"/></tr> | |
| <tr align="left"> | | <tr align="left"> |
- | <td class="ntext" width="100%">$\displaystyle \frac{32\pi}{3} \textrm{ cm}^2 \approx 33,5 \textrm{ cm}^2$</td> | + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%"> |
| + | $\left\{\eqalign{ |
| + | x&=\displaystyle\frac{\pi}{6}+2n\pi\cr |
| + | x&=\displaystyle\frac{11\pi}{6}+2n\pi |
| + | }\right.$ |
| + | </td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%"> |
| + | $\left\{\eqalign{ |
| + | x&=\displaystyle\frac{\pi}{5}+2n\pi\cr |
| + | x&=\displaystyle\frac{4\pi}{5}+2n\pi |
| + | }\right.$ |
| + | </td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%"> |
| + | $\left\{\eqalign{ |
| + | x&=25^\circ + n\cdot 360^\circ\cr |
| + | x&=75^\circ + n\cdot 360^\circ |
| + | }\right.$ |
| + | </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> |
| </tr> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | | |
| </div> | | </div> |
| + | |
| + | ==Övning 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}\,. |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning </div> | + | <table width="100%" cellpadding="10px"> |
- | <div class=NavContent> | + | |
- | <table width="100%"> | + | |
| <tr> | | <tr> |
- | <td align="center"> | + | <td align="left"> |
- | [[Bild:4_1_9.gif]] | + | v_1=50^\circ, \ \ v_2=120^\circ, \ \ v_3=230^\circ\ \ och \ \ v_4=300^\circ |
| </td> | | </td> |
| </tr> | | </tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |
| | | |
- | ==Övning 4.1:10== | + | ==Övning 4.4:5== |
| <div class="ovning"> | | <div class="ovning"> |
- | En 5,4 m lång tvättlina hänger mellan två vertikala träd på 4,8 m avstånd från varandra. Linans ena ände är fäst 0,6 m högre än den andra änden, och 1,2 m från trädet där linan har sin lägre infästning hänger en kavaj på en galge. Bestäm hur mycket under den nedre infästningspunkten som galgen hänger (dvs. avståndet \,x\, i figuren). | + | Lös ekvationen |
- | | + | |
| <table width="100%" cellspacing="10px"> | | <table width="100%" cellspacing="10px"> |
- | <tr><td height="5px"/></tr> | + | <tr align="left"> |
- | <tr align="center"> | + | <td class="ntext">a)</td> |
- | <td class="ntext" width="100%">BILD</td> | + | <td class="ntext" width="50%">\sin{3x}=\sin{x}</td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">\tan{x}=\tan{4x}</td> |
| + | </tr> |
| + | <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> |
| </table> | | </table> |
| + | </div> |
| | | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%"> |
| + | $\left\{\eqalign{ |
| + | x&=n\pi\cr |
| + | x&=\displaystyle \frac{\pi}{4}+\displaystyle \frac{n\pi}{2} |
| + | }\right.$ |
| + | </td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">x=\displaystyle \frac{n\pi}{3}</td> |
| + | </tr> |
| + | <tr align="left"> |
| + | <td class="ntext">c)</td> |
| + | <td class="ntext" width="50%"> |
| + | $\left\{\eqalign{ |
| + | x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr |
| + | x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3} |
| + | }\right.$ |
| + | </td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | ==Övning 4.4:6== |
- | <div class=NavHead>Facit </div> | + | <div class="ovning"> |
- | <div class=NavContent> | + | Lös ekvationen |
| <table width="100%" cellspacing="10px"> | | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">\sin x\cdot \cos 3x = 2\sin x</td> |
| + | <td class="ntext">b)</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" width="50%">\sin 2x = -\sin x</td> |
| + | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| + | </table> |
| + | </div> |
| + | |
| + | <div class="svar"> |
| + | <table width="100%" cellspacing="10px"> |
| <tr align="left"> | | <tr align="left"> |
- | <td class="ntext" width="100%">$x=9$ dm</td> | + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%"> |
| + | $x=n\pi$ |
| + | </td> |
| + | <td class="ntext">b)</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> |
| + | </tr> |
| + | <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> | | </tr> |
| <tr><td height="5px"/></tr> | | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | | |
| </div> | | </div> |
| + | |
| + | ==Övning 4.4:7== |
| + | <div class="ovning"> |
| + | Lös ekvationen |
| + | <table width="100%" cellspacing="10px"> |
| + | <tr align="left"> |
| + | <td class="ntext">a)</td> |
| + | <td class="ntext" width="50%">2\sin^2{x}+\sin{x}=1</td> |
| + | <td class="ntext">b)</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" width="50%">\cos{3x}=\sin{4x}</td> |
| + | </tr> |
| + | <tr><td height="5px"/></tr> |
| + | </table> |
| </div> | | </div> |
| | | |
- | <div class=NavFrame style="CLEAR: both"> | + | <div class="svar"> |
- | <div class=NavHead>Lösning </div> | + | <table width="100%" cellspacing="10px"> |
- | <div class=NavContent> | + | <tr align="left"> |
- | <table width="100%"> | + | <td class="ntext">a)</td> |
- | <tr> | + | <td class="ntext" width="50%"> |
- | <td align="center"> | + | $\left\{ \matrix{ |
- | [[Bild:4_1_10-1(5).gif]] | + | x=\displaystyle \frac{\pi}{6}+2n\pi\cr |
| + | x=\displaystyle \frac{5\pi}{6}+2n\pi\cr |
| + | x=\displaystyle \frac{3\pi}{2}+2n\pi |
| + | }\right.$ |
| </td> | | </td> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">x=\pm \displaystyle \frac{\pi}{3} + 2n\pi </td> |
| </tr> | | </tr> |
- | <tr> | + | <tr align="left"> |
- | <td align="center"> | + | <td class="ntext">c)</td> |
- | [[Bild:4_1_10-2(5).gif]] | + | <td class="ntext" width="50%"> |
| + | $\left\{ \matrix{ |
| + | x=\displaystyle \frac{\pi}{2}+2n\pi\cr |
| + | x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7} |
| + | }\right.$ |
| </td> | | </td> |
| </tr> | | </tr> |
- | <tr> | + | <tr><td height="5px"/></tr> |
- | <td align="center"> | + | </table> |
- | [[Bild:4_1_10-3(5).gif]] | + | </div> |
- | </td> | + | |
| + | ==Övning 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> |
- | <tr> | + | <tr align="left"> |
- | <td align="center"> | + | <td class="ntext">c)</td> |
- | [[Bild:4_1_10-4(5).gif]] | + | <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="svar"> |
| + | <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> |
| + | <td class="ntext">b)</td> |
| + | <td class="ntext" width="50%">x=\displaystyle \frac{\pi}{3}+n\pi</td> |
| </tr> | | </tr> |
- | <tr> | + | <tr align="left"> |
- | <td align="center"> | + | <td class="ntext">c)</td> |
- | [[Bild:4_1_10-5(5).gif]] | + | <td class="ntext" width="50%"> |
| + | $\left\{\eqalign{ |
| + | x&=n\pi\cr |
| + | x&=\displaystyle \frac{3\pi}{4}+n\pi |
| + | }\right.$ |
| </td> | | </td> |
| </tr> | | </tr> |
| + | <tr><td height="5px"/></tr> |
| </table> | | </table> |
- | </div> | |
| </div> | | </div> |