Solution 4.4:6a

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m (Lösning 4.4:6a moved to Solution 4.4:6a: Robot: moved page)
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If we move everything over to the left-hand side,
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<center> [[Image:4_4_6a.gif]] </center>
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{{NAVCONTENT_STOP}}
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<math>\sin x\cos 3x-2\sin x=0</math>
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we see that both terms have
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<math>\text{sin }x\text{ }</math>
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as a common factor which we can take out:
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 +
 
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<math>\text{sin }x\text{ }\left( \cos 3x-2 \right)=0</math>
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In this factorized version of the equation, we see the equation has a solution only when one of the factors
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<math>\text{sin }x</math>
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or
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<math>\cos 3x-2</math>
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is zero. The factor
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<math>\text{sin }x</math>
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is zero for all values of
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<math>x</math>
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that are given by
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 +
 
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<math>x=n\pi </math>
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(
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<math>n</math>
 +
an arbitrary integer)
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 +
(see exercise 3.5:2c). The other factor
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<math>\cos 3x-2</math>
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can never be zero because the value of a cosine always lies between
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<math>-\text{1 }</math>
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and
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<math>\text{1}</math>, which gives that the largest value of
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<math>\cos 3x-2</math>
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is
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<math>-\text{1 }</math>.
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The solutions are therefore
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 +
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<math>x=n\pi </math>
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(
 +
<math>n</math>
 +
an arbitrary integer).

Revision as of 11:25, 1 October 2008

If we move everything over to the left-hand side,


\displaystyle \sin x\cos 3x-2\sin x=0


we see that both terms have \displaystyle \text{sin }x\text{ } as a common factor which we can take out:


\displaystyle \text{sin }x\text{ }\left( \cos 3x-2 \right)=0


In this factorized version of the equation, we see the equation has a solution only when one of the factors \displaystyle \text{sin }x or \displaystyle \cos 3x-2 is zero. The factor \displaystyle \text{sin }x is zero for all values of \displaystyle x that are given by


\displaystyle x=n\pi ( \displaystyle n an arbitrary integer)

(see exercise 3.5:2c). The other factor \displaystyle \cos 3x-2 can never be zero because the value of a cosine always lies between \displaystyle -\text{1 } and \displaystyle \text{1}, which gives that the largest value of \displaystyle \cos 3x-2 is \displaystyle -\text{1 }.

The solutions are therefore


\displaystyle x=n\pi ( \displaystyle n an arbitrary integer).

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