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Övningar 4.4

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Versionen från 16 juli 2007 kl. 08.17 (redigera)
KTH.SE:u1zpa8nw (Diskussion | bidrag)
(Ny sida: __NOTOC__ ==&Ouml;vning 4.4:1== <div class="ovning"> För vilka vinklar \,v\,, där \,0 \leq v\leq 2\pi\,, gäller att <table width="100%" cellspacing="10px"> <tr align="left"> <td clas...)
← Gå till föregående ändring		 Versionen från 16 juli 2007 kl. 11.44 (redigera) (ogör)
KTH.SE:u1zpa8nw (Diskussion | bidrag)

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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==&Ouml;vning 4.4:1==+'''&Ouml;vning 4.4:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
För vilka vinklar \,v\,, där \,0 \leq v\leq 2\pi\,, gäller att För vilka vinklar \,v\,, där \,0 \leq v\leq 2\pi\,, gäller att
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{v}=\displaystyle \frac{1}{2}</td> <td class="ntext" width="50%">\sin{v}=\displaystyle \frac{1}{2}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\cos{v}=\displaystyle \frac{1}{2}</td> <td class="ntext" width="50%">\cos{v}=\displaystyle \frac{1}{2}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{v}=1</td> <td class="ntext" width="50%">\sin{v}=1</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\tan{v}=1</td> <td class="ntext" width="50%">\tan{v}=1</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\cos{v}=2</td> <td class="ntext" width="50%">\cos{v}=2</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{v}=-\displaystyle \frac{1}{2}</td> <td class="ntext" width="50%">\sin{v}=-\displaystyle \frac{1}{2}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">g)</td>+<td class="ntext">g)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}</td> <td class="ntext" width="50%">\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}</td>
</tr> </tr>
Rad 30: Rad 30:
</div> </div>
-==&Ouml;vning 4.4:2==+'''&Ouml;vning 4.4:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
L&ouml;s ekvationen L&ouml;s ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\sin{x}=\displaystyle \frac{\sqrt{3}}{2}</td> <td class="ntext" width="33%">\sin{x}=\displaystyle \frac{\sqrt{3}}{2}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\cos{x}=\displaystyle \frac{1}{2} </td> <td class="ntext" width="33%">\cos{x}=\displaystyle \frac{1}{2} </td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\sin{x}=0</td> <td class="ntext" width="33%">\sin{x}=0</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} </td> <td class="ntext" width="33%">\sin{5x}=\displaystyle \frac{1}{\sqrt{2}} </td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\sin{5x}=\displaystyle \frac{1}{2}</td> <td class="ntext" width="33%">\sin{5x}=\displaystyle \frac{1}{2}</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}</td> <td class="ntext" width="33%">\cos{3x}=-\displaystyle\frac{1}{\sqrt{2}}</td>
</tr> </tr>
Rad 54: Rad 54:
</div> </div>
-==&Ouml;vning 4.4:3==+'''&Ouml;vning 4.4:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\cos{x}=\cos{\displaystyle \frac{\pi}{6}}</td> <td class="ntext" width="50%">\cos{x}=\cos{\displaystyle \frac{\pi}{6}}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{x}=\sin{\displaystyle \frac{\pi}{5}}</td> <td class="ntext" width="50%">\sin{x}=\sin{\displaystyle \frac{\pi}{5}}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
-<td class="ntext" width="50%">\sin{(x+40^\circ)}=\sin{65^\circ}</td>+<td class="ntext" width="50%">$\sin{(x+40^\circ)&nbsp;&nbsp;&nbsp;}=\sin{65^\circ}$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{3x}=\sin{15^\circ}</td> <td class="ntext" width="50%">\sin{3x}=\sin{15^\circ}</td>
</tr> </tr>
Rad 74: Rad 74:
</div> </div>
-==&Ouml;vning 4.4:4==+'''&Ouml;vning 4.4:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Bestäm de vinklar \,v\, i intervallet \,0^\circ \leq v \leq 360^\circ\, som uppfyller \ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,. Bestäm de vinklar \,v\, i intervallet \,0^\circ \leq v \leq 360^\circ\, som uppfyller \ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,.
</div> </div>
-==&Ouml;vning 4.4:5==+'''&Ouml;vning 4.4:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{3x}=\sin{x}</td> <td class="ntext" width="50%">\sin{3x}=\sin{x}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\tan{x}=\tan{4x}</td> <td class="ntext" width="50%">\tan{x}=\tan{4x}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\cos{5x}=\cos(x+\pi/5)</td> <td class="ntext" width="50%">\cos{5x}=\cos(x+\pi/5)</td>
</tr> </tr>
Rad 97: Rad 97:
</div> </div>
-==&Ouml;vning 4.4:6==+'''&Ouml;vning 4.4:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin x\cdot \cos 3x = 2\sin x</td> <td class="ntext" width="50%">\sin x\cdot \cos 3x = 2\sin x</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt{2}\sin{x}\cos{x}=\cos{x}</td> <td class="ntext" width="50%">\sqrt{2}\sin{x}\cos{x}=\cos{x}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin 2x = -\sin x</td> <td class="ntext" width="50%">\sin 2x = -\sin x</td>
</tr> </tr>
Rad 115: Rad 115:
</div> </div>
-==&Ouml;vning 4.4:7==+'''&Ouml;vning 4.4:7'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">2\sin^2{x}+\sin{x}=1</td> <td class="ntext" width="50%">2\sin^2{x}+\sin{x}=1</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">2\sin^2{x}-3\cos{x}=0</td> <td class="ntext" width="50%">2\sin^2{x}-3\cos{x}=0</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\cos{3x}=\sin{4x}</td> <td class="ntext" width="50%">\cos{3x}=\sin{4x}</td>
</tr> </tr>
Rad 133: Rad 133:
</div> </div>
-==&Ouml;vning 4.4:8==+'''&Ouml;vning 4.4:8'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Lös ekvationen Lös ekvationen
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{2x}=\sqrt{2}\cos{x}</td> <td class="ntext" width="50%">\sin{2x}=\sqrt{2}\cos{x}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sin{x}=\sqrt{3}\cos{x}</td> <td class="ntext" width="50%">\sin{x}=\sqrt{3}\cos{x}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}</td> <td class="ntext" width="50%">\displaystyle \frac{1}{\cos^2{x}}=1-\tan{x}</td>
</tr> </tr>

Versionen från 16 juli 2007 kl. 11.44

Övning 4.4:1

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

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

Övning 4.4:2

Lös ekvationen

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

Övning 4.4:3

Lös ekvationen

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

Övning 4.4:4

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

Övning 4.4:5

Lös ekvationen

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

Övning 4.4:6

Lös ekvationen

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

Övning 4.4:7

Lös ekvationen

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

Övning 4.4:8

Lös ekvationen

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