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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)
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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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{{NAVCONTENT_STOP}}
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{{NAVCONTENT_START}}
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<math>\tan u=\tan v</math>
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<center> [[Image:4_4_5b-2(2).gif]] </center>
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{{NAVCONTENT_STOP}}
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is satisfied. Because
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<math>u</math>
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can be interpreted as the slope (gradient) of the line which makes an angle
 +
<math>u</math>
 +
with the positive
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<math>x</math>
 +
-axis, we see that for a fixed value of tan u, there are two angles
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<math>v</math>
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in the unit circle with this slope:
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 +
 
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<math>v=u</math>
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and
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<math>v=u+\pi </math>
 +
 
[[Image:4_4_5_b.gif|center]]
[[Image:4_4_5_b.gif|center]]
 +
 +
slope
 +
<math>=\text{ tan }u</math>
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slope
 +
<math>=\text{ tan }u</math>
 +
 +
 +
The angle
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<math>v</math>
 +
has the same slope after every half turn, so if we add multiples of
 +
<math>\pi \text{ }</math>
 +
to
 +
<math>u</math>, we will obtain all the angles
 +
<math>v</math>
 +
which satisfy the equality
 +
 +
 +
<math>v=u+n\pi </math>
 +
 +
 +
where
 +
<math>n</math>
 +
is an arbitrary integer.
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 +
If we apply this result to the equation
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 +
 +
<math>\tan x=\tan 4x</math>
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 +
 +
we see that the solutions are given by
 +
 +
 +
<math>4x=x+n\pi </math>
 +
(
 +
<math>n</math>
 +
an arbitrary integer),
 +
 +
and solving for
 +
<math>x</math>
 +
gives
 +
 +
 +
<math>x=\frac{1}{3}n\pi </math>
 +
(
 +
<math>n</math>
 +
an arbitrary integer).

Revision as of 11:06, 1 October 2008

Let's first investigate when the equality


tanu=tanv


is satisfied. Because u 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 tan u, there are two angles v in the unit circle with this slope:


v=u and v=u+


slope = tan u slope = tan 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


\displaystyle 4x=x+n\pi ( \displaystyle n an arbitrary integer),

and solving for \displaystyle x gives


\displaystyle x=\frac{1}{3}n\pi ( \displaystyle n an arbitrary integer).

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