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4.3 Exercises
From Förberedande kurs i matematik 1
(Difference between revisions)
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{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
- |width="100%" | <math>\sin{x}=\displaystyle \frac{2}{3}\,</math>,<math>\ \sin{y}=\displaystyle \frac{1}{3}\ </math> och <math>\,x\, \,y\,</math> är vinklar i första kvadranten.
+ |width="100%" | <math>\sin{x}=\displaystyle \frac{2}{3}\,</math>,<math>\ \sin{y}=\displaystyle \frac{1}{3}\ </math> och <math>\,x\, , \,y\,</math> är vinklar i första kvadranten.
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|b)
|b)
Revision as of 11:08, 3 April 2008
Övning 4.3:1
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Övning 4.3:2
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Övning 4.3:3
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Övning 4.3:4
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Övning 4.3:5
För en spetsig vinkel v i en triangel gäller att sin v = 7 5 . Bestäm cos v och tan v .
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Övning 4.3:6
a)
Bestäm \displaystyle \ \sin{v}\ och \displaystyle \ \tan{v}\ om \displaystyle \ \cos{v}=\displaystyle \frac{3}{4}\ och \displaystyle \ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\, .
b)
Bestäm \displaystyle \ \cos{v}\ och \displaystyle \ \tan{v}\ om \displaystyle \ \sin{v}=\displaystyle \frac{3}{10}\ och \displaystyle \,v\, ligger i den andra kvadranten.
c)
Bestäm \displaystyle \ \sin{v}\ och \displaystyle \ \cos{v}\ om \displaystyle \ \tan{v}=3\ och \displaystyle \ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\, .
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Övning 4.3:7
Bestäm \displaystyle \ \sin{(x+y)}\ om
a)
\displaystyle \sin{x}=\displaystyle \frac{2}{3}\, ,\displaystyle \ \sin{y}=\displaystyle \frac{1}{3}\ och \displaystyle \,x\,, \,y\, är vinklar i första kvadranten.
b)
\displaystyle \cos{x}=\displaystyle \frac{2}{5}\, , \displaystyle \ \cos{y}=\displaystyle \frac{3}{5}\ och \displaystyle \,x\, , \displaystyle \,y\, är vinklar i första kvadranten.
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Övning 4.3:8
Visa följande trigonometriska samband
a)
\displaystyle \tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}
b)
\displaystyle \displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v}
c)
\displaystyle \tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}
d)
\displaystyle \displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v
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Övning 4.3:9
Visa "Feynmans likhet"
\displaystyle \cos 20^\circ \cdot \cos 40^\circ \cdot \cos 80^\circ = \displaystyle\frac{1}{8}\,\mbox{.}
(Ledtråd: Använd formeln för dubbla vinkeln på \displaystyle \,\sin 160^\circ\, .)
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