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Solution 4.4:5b

From Förberedande kurs i matematik 1

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m (Lösning 4.4:5b moved to Solution 4.4:5b: Robot: moved page)
Current revision (14:02, 13 October 2008) (edit) (undo)
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Let's first investigate when the equality
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<center> [[Image:4_4_5b-1(2).gif]] </center>
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{{Displayed math||<math>\tan u=\tan v</math>}}
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<center> [[Image:4_4_5b-2(2).gif]] </center>
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is satisfied. Because <math>\tan u</math> can be interpreted as the slope (gradient) of the line which makes an angle ''u'' with the positive ''x''-axis, we see that for a fixed value of <math>\tan u</math>, there are two angles ''v'' in the unit circle with this slope,
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{{NAVCONTENT_STOP}}
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{{Displayed math||<math>v=u\qquad\text{and}\qquad v=u+\pi\,\textrm{.}</math>}}
[[Image:4_4_5_b.gif|center]]
[[Image:4_4_5_b.gif|center]]
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The angle ''v'' has the same slope after every half turn, so if we add multiples of
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<math>\pi</math> to ''u'', we will obtain all the angles ''v'' which satisfy the equality
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{{Displayed math||<math>v=u+n\pi\,,</math>}}
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where ''n'' is an arbitrary integer.
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If we apply this result to the equation
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{{Displayed math||<math>\tan x=\tan 4x</math>}}
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we see that the solutions are given by
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{{Displayed math||<math>4x = x+n\pi\qquad\text{(n is an arbitrary integer),}</math>}}
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and solving for ''x'' gives
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{{Displayed math||<math>x = \tfrac{1}{3}n\pi\qquad\text{(n is an arbitrary integer).}</math>}}

Current revision

Let's first investigate when the equality

tanu=tanv

is satisfied. Because tanu can be interpreted as the slope (gradient) of the line which makes an angle u with the positive x-axis, we see that for a fixed value of tanu, there are two angles v in the unit circle with this slope,

v=uandv=u+.

The angle v has the same slope after every half turn, so if we add multiples of to u, we will obtain all the angles v which satisfy the equality

v=u+n

where n is an arbitrary integer.

If we apply this result to the equation

tanx=tan4x

we see that the solutions are given by

4x=x+n(n is an arbitrary integer),

and solving for x gives

x=31n(n is an arbitrary integer).
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