Solution 4.3:1b

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

Because the sine value for an angle is equal to the angle's y-coordinate on the unit circle, two angles have the same sine value only if they have the same 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.

Because of symmetry, we have that this angle is the reflection of the angle \displaystyle \pi/7 in the y-axis, i.e. \displaystyle v = \pi - \pi/7 = 6\pi/7\,.

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