Lösung 4.3:1b

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Because the sine value for an angle is equal to the angle's
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<center> [[Image:4_3_1_b.gif]] </center>
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<math>y</math>
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-coordinate on a unit circle, two angles have the same sine value only if they have the same <math>y</math>-coordinate. Therefore, if we draw in the angle
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<math>{\pi }/{7}\;</math>
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on a unit circle, we see that the only angle between
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<math>{\pi }/{2}\;</math>
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and
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<math>\pi </math>
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which has the same sine value lies in the second quadrant, where the line
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<math>{y=\sin \pi }/{7}\;</math>
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cuts the unit circle.
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<center> [[Image:4_3_1b.gif]] </center>
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FIGURE1 FIGURE2
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the line
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<math>{y=\sin \pi }/{7}\;</math>
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the line
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<math>{y=\sin \pi }/{7}\;</math>
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Because of symmetry, we have that this angle is the reflection of the angle
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<math>{\pi }/{7}\;</math>
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in the
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<math>y</math>-axis, i.e.
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<math>v=\pi -{\pi }/{7}\;={6\pi }/{7}\;</math>.

Version vom 11:59, 12. Sep. 2008

Because the sine value for an angle is equal to the angle's \displaystyle y -coordinate on a unit circle, two angles have the same sine value only if they have the same \displaystyle y-coordinate. Therefore, if we draw in the angle \displaystyle {\pi }/{7}\; on a unit circle, we see that the only angle between \displaystyle {\pi }/{2}\; and \displaystyle \pi which has the same sine value lies in the second quadrant, where the line \displaystyle {y=\sin \pi }/{7}\; cuts the unit circle.


FIGURE1 FIGURE2 the line \displaystyle {y=\sin \pi }/{7}\; the line \displaystyle {y=\sin \pi }/{7}\;


Because of symmetry, we have that this angle is the reflection of the angle \displaystyle {\pi }/{7}\; in the \displaystyle y-axis, i.e.

\displaystyle v=\pi -{\pi }/{7}\;={6\pi }/{7}\;.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_4.3:1b“