4.4 Övningar
Sommarmatte 1
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| Rad 6: | Rad 6: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$\sin{v}=\displaystyle \frac{1}{2}$</td> | + | <td class="ntext" width="50%">$\sin{v}=\displaystyle \frac{1}{2}$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$\cos{v}=\displaystyle \frac{1}{2}$</td> | + | <td class="ntext" width="50%">$\cos{v}=\displaystyle \frac{1}{2}$</td> |
| - | <td class="ntext">c)</td> | + | |
| - | <td class="ntext" width="33%">$\sin{v}=1$</td> | + | |
| </tr> | </tr> | ||
| <tr align="left"> | <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">d)</td> | ||
| - | <td class="ntext" width="33%">$\tan{v}=1$</td> | + | <td class="ntext" width="50%">$\tan{v}=1$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="33%">$\cos{v}=2$</td> | + | <td class="ntext" width="50%">$\cos{v}=2$</td> |
| <td class="ntext">f)</td> | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="33%">$\sin{v}=-\displaystyle \frac{1}{2}$</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)</td> | ||
| - | <td class="ntext" width="33%">$\tan{v}=-\displaystyle \frac{1}{\sqrt{3}}$</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> | ||
| Rad 35: | Rad 37: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$\displaystyle v=\frac{\pi}{6},\ \displaystyle v=\frac{5\pi}{6}$</td> | + | <td class="ntext" width="50%">$\displaystyle v=\frac{\pi}{6}\,$, $\,\displaystyle v=\frac{5\pi}{6}$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$\displaystyle v=\frac{\pi}{3},\ \displaystyle v=\frac{5\pi}{3}$</td> | + | <td class="ntext" width="50%">$\displaystyle v=\frac{\pi}{3}\,$, $\,\displaystyle v=\frac{5\pi}{3}$</td> |
| - | <td class="ntext">c)</td> | + | |
| - | <td class="ntext" width="33%">$\displaystyle v=\frac{\pi}{2}$</td> | + | |
| </tr> | </tr> | ||
| <tr align="left"> | <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">d)</td> | ||
| - | <td class="ntext" width="33%">$\displaystyle v=\frac{\pi}{4},\ \displaystyle v=\frac{5\pi}{4}$</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">e)</td> | ||
| - | <td class="ntext" width="33%">lösning saknas</td> | + | <td class="ntext" width="50%">lösning saknas</td> |
| <td class="ntext">f)</td> | <td class="ntext">f)</td> | ||
| - | <td class="ntext" width="33%">$\displaystyle v=\frac{11\pi}{6},\ \displaystyle v=\frac{7\pi}{6}$</td> | + | <td class="ntext" width="50%">$\displaystyle v=\frac{11\pi}{6}\,$, $\,\displaystyle v=\frac{7\pi}{6}$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">g)</td> | <td class="ntext">g)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle v=\frac{5\pi}{6},\ \displaystyle v=\frac{11\pi}{6}$</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> | ||
Versionen från 15 maj 2007 kl. 14.41
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) | $\displaystyle v=\frac{5\pi}{6}\,$, $\,\displaystyle v=\frac{11\pi}{6}$ | ||
Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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Lösning till delfråga d)
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Lösning till delfråga e)
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Lösning till delfråga f)
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Lösning till delfråga g)
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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) |
$\left\{\eqalign{ x&=\displaystyle\frac{\pi}{4}+\displaystyle\frac{2n\pi}{3}\cr x&=\displaystyle\frac{5\pi}{12}+\displaystyle\frac{2n\pi}{3}} \right.$ |
Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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Lösning till delfråga d)
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Lösning till delfråga e)
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Lösning till delfråga f)
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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) |
$\left\{\eqalign{ x&=5^\circ + n \cdot 120^\circ \cr x&= 55^\circ + n \cdot 120^\circ }\right.$ |
Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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Lösning till delfråga d)
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Övning 4.4:4
Bestäm de vinklar $\,v\,$ i intervallet $\,0^\circ \leq v \leq 360^\circ\,$ som uppfyller $\ \cos{\left(2v+10^\circ\right)}=\cos{110^\circ}\,$.
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$v_1=50^\circ$, $\ \ v_2=120^\circ$, $\ \ v_3=230^\circ\ \ $ och $\ \ v_4=300^\circ$ |
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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) |
$\left\{\eqalign{ x&=\displaystyle \frac{\pi}{20}+\displaystyle \frac{n\pi}{2}\cr x&=-\displaystyle \frac{\pi}{30}+\displaystyle \frac{n\pi}{3} }\right.$ |
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Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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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) |
$\left\{\eqalign{ x&=\displaystyle \frac{2n\pi}{3}\cr x&=\displaystyle \pi + 2n\pi\cr }\right.$ |
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Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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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) |
$\left\{ \matrix{ x=\displaystyle \frac{\pi}{2}+2n\pi\cr x=\displaystyle \frac{\pi}{14}+\displaystyle \frac{2n\pi}{7} }\right.$ |
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Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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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) |
$\left\{\eqalign{ x&=n\pi\cr x&=\displaystyle \frac{3\pi}{4}+n\pi }\right.$ |
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Lösning till delfråga a)
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Lösning till delfråga b)
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Lösning till delfråga c)
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