Lösung 4.2:4b

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We start by subtracting
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<center> [[Image:4_2_4b-1(2).gif]] </center>
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<math>2\pi </math>
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from
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<math>\frac{11\pi }{3}</math>, so that we get an angle between
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<center> [[Image:4_2_4b-2(2).gif]] </center>
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<math>o</math>
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and
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<math>2\pi </math>. This doesn't change the cosine value
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<math>\cos \frac{11\pi }{3}=\cos \left( \frac{11\pi }{3}-2\pi \right)=\cos \frac{5\pi }{3}</math>
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Then, by rewriting
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<math>\frac{5\pi }{3}</math>
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as a sum of
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<math>\pi </math>
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- and
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<math>\frac{\pi }{2}</math>
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-terms
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<math>\frac{5\pi }{3}=\frac{3\pi +\frac{3}{2}\pi +\frac{1}{2}\pi }{3}=\pi +\frac{\pi }{2}+\frac{\pi }{6}</math>
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we see that
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<math>\frac{5\pi }{3}</math>
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is an angle in the fourth quadrant which makes an angle
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<math>\frac{\pi }{6}</math>
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with the negative
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<math>y</math>
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-axis.
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[[Image:4_2_4b1.gif]]
[[Image:4_2_4b1.gif]]
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With the help of a triangle and a little trigonometry, we can determine the coordinates for the point on a unit circle which corresponds to the angle
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<math>\frac{5\pi }{3}</math> .
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[[Image:4_2_4_b2.gif]]
[[Image:4_2_4_b2.gif]]
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The point has coordinates
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<math>\left( \frac{1}{2} \right.,\left. -\frac{\sqrt{3}}{2} \right)</math>
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and
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<math>\cos \frac{11\pi }{3}=\cos \frac{5\pi }{3}=\frac{1}{2}</math>.

Version vom 13:05, 28. Sep. 2008

We start by subtracting \displaystyle 2\pi from \displaystyle \frac{11\pi }{3}, so that we get an angle between \displaystyle o and \displaystyle 2\pi . This doesn't change the cosine value


\displaystyle \cos \frac{11\pi }{3}=\cos \left( \frac{11\pi }{3}-2\pi \right)=\cos \frac{5\pi }{3}


Then, by rewriting \displaystyle \frac{5\pi }{3} as a sum of \displaystyle \pi - and \displaystyle \frac{\pi }{2} -terms


\displaystyle \frac{5\pi }{3}=\frac{3\pi +\frac{3}{2}\pi +\frac{1}{2}\pi }{3}=\pi +\frac{\pi }{2}+\frac{\pi }{6}

we see that \displaystyle \frac{5\pi }{3} is an angle in the fourth quadrant which makes an angle \displaystyle \frac{\pi }{6} with the negative \displaystyle y -axis.


Image:4_2_4b1.gif

With the help of a triangle and a little trigonometry, we can determine the coordinates for the point on a unit circle which corresponds to the angle \displaystyle \frac{5\pi }{3} .


Image:4_2_4_b2.gif

The point has coordinates \displaystyle \left( \frac{1}{2} \right.,\left. -\frac{\sqrt{3}}{2} \right) and \displaystyle \cos \frac{11\pi }{3}=\cos \frac{5\pi }{3}=\frac{1}{2}.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_4.2:4b“