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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