Lösung 4.2:3c

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<center> [[Image:4_2_3c.gif]] </center>
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We can add and subtract multiples of
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<math>\text{2}\pi </math>
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to or from the argument of the sine function without changing its value. The angle
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<math>\text{2}\pi </math>
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corresponds to a whole turn in a unit circle and the sine function returns to the same value every time the angle changes by a complete revolution.
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For example, if we can subtract sufficiently many
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<math>\text{2}\pi </math>
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s from
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<math>\text{9}\pi </math>, we will obtain a more manageable argument which lies between
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<math>0</math>
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and
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<math>\text{2}\pi </math>,
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<math>\text{sin 9}\pi =\text{sin}\left( 9\pi -2\pi -2\pi -2\pi -2\pi \right)=\sin \pi </math>
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The line which makes an angle
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<math>\pi </math>
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with the positive part of the
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<math>x</math>
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-axis is the negative part of the
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<math>x</math>
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-axis
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and it cuts the unit circle at the point
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<math>\left( -1 \right.,\left. 0 \right)</math>, which is why we can see from the
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<math>y</math>
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-coordinate that
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<math>\text{sin 9}\pi =\text{sin }\pi =0</math>.
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[[Image:4_2_3_c.gif|center]]
[[Image:4_2_3_c.gif|center]]

Version vom 11:57, 28. Sep. 2008

We can add and subtract multiples of \displaystyle \text{2}\pi to or from the argument of the sine function without changing its value. The angle \displaystyle \text{2}\pi corresponds to a whole turn in a unit circle and the sine function returns to the same value every time the angle changes by a complete revolution.

For example, if we can subtract sufficiently many \displaystyle \text{2}\pi s from \displaystyle \text{9}\pi , we will obtain a more manageable argument which lies between \displaystyle 0 and \displaystyle \text{2}\pi ,


\displaystyle \text{sin 9}\pi =\text{sin}\left( 9\pi -2\pi -2\pi -2\pi -2\pi \right)=\sin \pi


The line which makes an angle \displaystyle \pi with the positive part of the \displaystyle x -axis is the negative part of the \displaystyle x -axis and it cuts the unit circle at the point \displaystyle \left( -1 \right.,\left. 0 \right), which is why we can see from the \displaystyle y -coordinate that \displaystyle \text{sin 9}\pi =\text{sin }\pi =0.


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