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Solution 4.2:4e

From Förberedande kurs i matematik 1

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m (Lösning 4.2:4e moved to Solution 4.2:4e: Robot: moved page)
Current revision (10:44, 9 October 2008) (edit) (undo)
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If we write the angle <math>\frac{7\pi}{6}</math> as
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<center> [[Image:4_2_4e.gif]] </center>
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{{Displayed math||<math>\frac{7\pi}{6} = \frac{6\pi+\pi}{6} = \pi + \frac{\pi }{6}</math>}}
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we see that the angle <math>7\pi/6</math> on the unit circle is in the third quadrant and makes an angle <math>\pi/6</math> with the negative ''x''-axis.
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[[Image:4_2_4_e1.gif|center]]
[[Image:4_2_4_e1.gif|center]]
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Geometrically, <math>\tan (7\pi/6)</math> is defined as the slope of the line having an angle <math>7\pi/6</math> and, because this line has the same slope as the line having angle <math>\pi/6</math>, we have that
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{{Displayed math||<math>\tan\frac{7\pi}{6} = \tan\frac{\pi}{6} = \frac{\sin\dfrac{\pi }{6}}{\cos\dfrac{\pi }{6}} = \frac{\dfrac{1}{2}}{\dfrac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\,\textrm{.}</math>}}
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[[Image:4_2_4_e2.gif|center]]
[[Image:4_2_4_e2.gif|center]]

Current revision

If we write the angle 67 as

67=66+=+6

we see that the angle 76 on the unit circle is in the third quadrant and makes an angle 6 with the negative x-axis.

Geometrically, tan(76) is defined as the slope of the line having an angle 76 and, because this line has the same slope as the line having angle 6, we have that

tan67=tan6=sin6cos6=2123=13.
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