Övningar 4.3 - Versionshistorik

KTH.SE:u1zpa8nw den 17 juli 2007 kl. 09.46

KTH.SE:u1zpa8nw — Tue, 17 Jul 2007 09:46:06 GMT

← Äldre version		Versionen från 17 juli 2007 kl. 09.46
Rad 156:		Rad 156:
	</table>		</table>
	</div>		</div>
		+	<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>

KTH.SE:u1zpa8nw den 16 juli 2007 kl. 11.42

KTH.SE:u1zpa8nw — Mon, 16 Jul 2007 11:42:22 GMT

← Äldre version		Versionen från 16 juli 2007 kl. 11.42
Rad 1:		Rad 1:
	__NOTOC__		__NOTOC__
-	==Övning 4.3:1==	+	'''Övning 4.3:1'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	Bestäm de vinklar $\,v\,$ mellan $\,\displaystyle \frac{\pi}{2}\,$ och $\,2\pi\,$ som uppfyller		Bestäm de vinklar $\,v\,$ mellan $\,\displaystyle \frac{\pi}{2}\,$ och $\,2\pi\,$ som uppfyller
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<tr align="left">		<tr align="left">
-	<td class="ntext">a)</td>	+	<td class="ntext">a)   </td>
	<td class="ntext" width="33%">$\cos{v}=\cos{\displaystyle \frac{\pi}{5}}$</td>		<td class="ntext" width="33%">$\cos{v}=\cos{\displaystyle \frac{\pi}{5}}$</td>
-	<td class="ntext">b)</td>	+	<td class="ntext">b)   </td>
	<td class="ntext" width="33%">$\sin{v}=\sin{\displaystyle \frac{\pi}{7}}$</td>		<td class="ntext" width="33%">$\sin{v}=\sin{\displaystyle \frac{\pi}{7}}$</td>
-	<td class="ntext">c)</td>	+	<td class="ntext">c)   </td>
	<td class="ntext" width="33%">$\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}$</td>		<td class="ntext" width="33%">$\tan{v}=\tan{\displaystyle \frac{2\pi}{7}}$</td>
	</tr>		</tr>
Rad 15:		Rad 15:
	</div>		</div>

-	==Övning 4.3:2==	+	'''Övning 4.3:2'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	Bestäm de vinklar $\,v\,$ mellan 0 och $\,\pi\,$ som uppfyller		Bestäm de vinklar $\,v\,$ mellan 0 och $\,\pi\,$ som uppfyller
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<tr align="left">		<tr align="left">
-	<td class="ntext">a)</td>	+	<td class="ntext">a)   </td>
	<td class="ntext" width="50%">$\cos{v} = \cos{\displaystyle \frac{3\pi}{2}}$</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%">$\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}$</td>		<td class="ntext" width="50%">$\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}$</td>
	</tr>		</tr>
Rad 29:		Rad 29:
	</div>		</div>

-	==Övning 4.3:3==	+	'''Övning 4.3:3'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	Antag att $\,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\,$ och att $\,\sin{v} = a\,$. Uttryck med hjälp av $\,a$		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">	+	<table width="100%">
	<tr align="left">		<tr align="left">
-	<td class="ntext">a)</td>	+	<td class="ntext">a)   </td>
	<td class="ntext" width="50%">$\sin{(-v)}$</td>		<td class="ntext" width="50%">$\sin{(-v)}$</td>
-	<td class="ntext">b)</td>	+	<td class="ntext">b)   </td>
	<td class="ntext" width="50%">$\sin{(\pi-v)}$</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%">$\cos{v}$</td>		<td class="ntext" width="50%">$\cos{v}$</td>
-	<td class="ntext">d)</td>	+	<td class="ntext">d)   </td>
	<td class="ntext" width="50%">$\sin{\left(\displaystyle \frac{\pi}{2}-v\right)}$</td>		<td class="ntext" width="50%">$\sin{\left(\displaystyle \frac{\pi}{2}-v\right)}$</td>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext">e)</td>	+	<td class="ntext">e)   </td>
	<td class="ntext" width="50%">$\cos{\left( \displaystyle \frac{\pi}{2} + v\right)}$</td>		<td class="ntext" width="50%">$\cos{\left( \displaystyle \frac{\pi}{2} + v\right)}$</td>
-	<td class="ntext">f)</td>	+	<td class="ntext">f)   </td>
	<td class="ntext" width="50%">$\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}$</td>		<td class="ntext" width="50%">$\sin{\left( \displaystyle \frac{\pi}{3} + v \right)}$</td>
	</tr>		</tr>
Rad 55:		Rad 55:
	</div>		</div>

-	==Övning 4.3:4==	+	'''Övning 4.3:4'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	Antag att $\,0 \leq v \leq \pi\,$ och att $\,\cos{v}=b\,$. Uttryck med hjälp av $\,b\,$		Antag att $\,0 \leq v \leq \pi\,$ och att $\,\cos{v}=b\,$. Uttryck med hjälp av $\,b\,$
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<tr align="left">		<tr align="left">
-	<td class="ntext">a)</td>	+	<td class="ntext">a)   </td>
	<td class="ntext" width="50%">$\sin^2{v}$</td>		<td class="ntext" width="50%">$\sin^2{v}$</td>
-	<td class="ntext">b)</td>	+	<td class="ntext">b)   </td>
	<td class="ntext" width="50%">$\sin{v}$</td>		<td class="ntext" width="50%">$\sin{v}$</td>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext">c)</td>	+	<td class="ntext">c)   </td>
	<td class="ntext" width="50%">$\sin{2v}$</td>		<td class="ntext" width="50%">$\sin{2v}$</td>
-	<td class="ntext">d)</td>	+	<td class="ntext">d)   </td>
	<td class="ntext" width="50%">$\cos{2v}$</td>		<td class="ntext" width="50%">$\cos{2v}$</td>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext">e)</td>	+	<td class="ntext">e)   </td>
	<td class="ntext" width="50%">$\sin{\left( v+\displaystyle \frac{\pi}{4} \right)}$</td>		<td class="ntext" width="50%">$\sin{\left( v+\displaystyle \frac{\pi}{4} \right)}$</td>
-	<td class="ntext">f)</td>	+	<td class="ntext">f)   </td>
	<td class="ntext" width="50%">$\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}$</td>		<td class="ntext" width="50%">$\cos{\left( v-\displaystyle \frac{\pi}{3} \right)}$</td>
	</tr>		</tr>
Rad 81:		Rad 81:
	</div>		</div>

-	==Övning 4.3:5==	+	'''Övning 4.3:5'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	För en spetsig vinkel $\,v\,$ i en triangel gäller att $\,\sin{v}=\displaystyle \frac{5}{7}\,$. Bestäm $\,\cos{v}\,$ och $\,\tan{v}\,$.		För en spetsig vinkel $\,v\,$ i en triangel gäller att $\,\sin{v}=\displaystyle \frac{5}{7}\,$. Bestäm $\,\cos{v}\,$ och $\,\tan{v}\,$.
	</div>		</div>

-	==Övning 4.3:6==	+	'''Övning 4.3:6'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<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 $\ \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 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>
	</tr>		</tr>
	<tr align="left" valign="top">		<tr align="left" valign="top">
-	<td class="ntext">b)</td>	+	<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>		<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>
	<tr align="left" valign="top">		<tr align="left" valign="top">
-	<td class="ntext">c)</td>	+	<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>		<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>
Rad 105:		Rad 105:
	</div>		</div>

-	==Övning 4.3:7==	+	'''Övning 4.3:7'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	Bestäm $\ \sin{(x+y)}\ $ om		Bestäm $\ \sin{(x+y)}\ $ om
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<tr align="left">		<tr align="left">
-	<td class="ntext">a)</td>	+	<td class="ntext">a)   </td>
	<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>		<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>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext">b)</td>	+	<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>		<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>
Rad 121:		Rad 121:
	</div>		</div>

-	==Övning 4.3:8==	+	'''Övning 4.3:8'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
	Visa följande trigonometriska samband		Visa följande trigonometriska samband
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<tr align="left">		<tr align="left">
-	<td class="ntext>a)</td>	+	<td class="ntext>a)   </td>
	<td class="ntext width="100%">$\tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}$</td>		<td class="ntext width="100%">$\tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}$</td>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext>b)</td>	+	<td class="ntext>b)   </td>
	<td class="ntext width="100%">$\displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v}$</td>		<td class="ntext width="100%">$\displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v}$</td>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext>c)</td>	+	<td class="ntext>c)   </td>
	<td class="ntext width="100%">$\tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}$</td>		<td class="ntext width="100%">$\tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}$</td>
	</tr>		</tr>
	<tr align="left">		<tr align="left">
-	<td class="ntext>d)</td>	+	<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>		<td class="ntext width="100%">$\displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v$</td>
	</tr>		</tr>
Rad 144:		Rad 144:
	</div>		</div>

-	==Övning 4.3:9==	+	'''Övning 4.3:9'''
-	<div class="ovning">	+	<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
-	<table width="100%" cellspacing="10px">	+	<table width="100%">
	<tr align="left">		<tr align="left">
	<td class="ntext>		<td class="ntext>

KTH.SE:u1zpa8nw: Ny sida: NOTOC ==Övning 4.3:1==
Bestäm de vinklar $\,v\,$ mellan $\,\displaystyle \frac{\pi}{2}\,$ och $\,2\pi\,$ som uppfyller
KTH.SE:u1zpa8nw — Mon, 16 Jul 2007 08:16:43 GMT

Ny sida: NOTOC ==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...

Ny sida
NOTOC
==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>

==Övning 4.3:2==
<div class="ovning">
Bestäm de vinklar $\,v\,$ mellan 0 och $\,\pi\,$ som uppfyller
<table width="100%" cellspacing="10px">
<tr align="left">
<td class="ntext">a)</td>
<td class="ntext" width="50%">$\cos{v} = \cos{\displaystyle \frac{3\pi}{2}}$</td>
<td class="ntext">b)</td>
<td class="ntext" width="50%">$\cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}$</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 align="left">
<td class="ntext">c)</td>
<td class="ntext" width="50%">$\cos{v}$</td>
<td class="ntext">d)</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><td height="5px"/></tr>
</table>
</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>

==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>

==Övning 4.3:6==
<div class="ovning">
<table width="100%" cellspacing="10px">
<tr align="left" valign="top">
<td class="ntext">a)</td>
<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>
</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>
</div>

==Övning 4.3:7==
<div class="ovning">
Bestäm $\ \sin{(x+y)}\ $ om
<table width="100%" cellspacing="10px">
<tr align="left">
<td class="ntext">a)</td>
<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>
</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>
</div>

==Övning 4.3:8==
<div class="ovning">
Visa följande trigonometriska samband
<table width="100%" cellspacing="10px">
<tr align="left">
<td class="ntext>a)</td>
<td class="ntext width="100%">$\tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}$</td>
</tr>
<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>
</table>
</div>

==Övning 4.3:9==
<div class="ovning">
<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>