Solution 4.4:7b

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m (Lösning 4.4:7b moved to Solution 4.4:7b: Robot: moved page)
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If we use the Pythagorean identity and write
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<center> [[Image:4_4_7b-1(2).gif]] </center>
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<math>\sin ^{2}x</math>
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as
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<math>1-\cos ^{2}x</math>, the whole equation written in terms of
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<center> [[Image:4_4_7b-2(2).gif]] </center>
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<math>\cos x</math>
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becomes
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<math>2\left( 1-\cos ^{2}x \right)-3\cos x=0</math>
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<math></math>
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or, in rearranged form,
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<math>2\cos ^{2}x+3\cos x-2=0</math>
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With the equation expressed entirely in terms of
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<math>\cos x</math>, we can introduce a new unknown variable
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<math>t=\cos x</math>
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and solve the equation with respect to
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<math>t</math>. Expressed in terms of
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<math>t</math>, the equation is
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<math>2t^{2}+3t-2=0</math>
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and this second-degree equation has the solutions
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<math>t=\frac{1}{2}</math>
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and
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<math>t=-2</math>
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.
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In terms of
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<math>x</math>, this means that either
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<math>\cos x=\frac{1}{2}</math>
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or
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<math>\text{cos }x=-\text{2}</math>. The first case occurs when
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<math>x=\pm \frac{\pi }{3}+2n\pi </math>
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(
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<math>n</math>
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an arbitrary integer),
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whilst the equation
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<math>\text{cos }x=-\text{2 }</math>
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has no solutions at all (the values of cosine lie between
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<math>-\text{1 }</math>
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and
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<math>\text{1}</math>
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).
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The answer is that the equation has the solutions
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 +
 
 +
<math>x=\pm \frac{\pi }{3}+2n\pi </math>
 +
(
 +
<math>n</math>
 +
an arbitrary integer).

Revision as of 13:01, 1 October 2008

If we use the Pythagorean identity and write \displaystyle \sin ^{2}x as \displaystyle 1-\cos ^{2}x, the whole equation written in terms of \displaystyle \cos x becomes


\displaystyle 2\left( 1-\cos ^{2}x \right)-3\cos x=0


\displaystyle

or, in rearranged form,


\displaystyle 2\cos ^{2}x+3\cos x-2=0


With the equation expressed entirely in terms of \displaystyle \cos x, we can introduce a new unknown variable \displaystyle t=\cos x and solve the equation with respect to \displaystyle t. Expressed in terms of \displaystyle t, the equation is


\displaystyle 2t^{2}+3t-2=0


and this second-degree equation has the solutions \displaystyle t=\frac{1}{2} and \displaystyle t=-2 .

In terms of \displaystyle x, this means that either \displaystyle \cos x=\frac{1}{2} or \displaystyle \text{cos }x=-\text{2}. The first case occurs when


\displaystyle x=\pm \frac{\pi }{3}+2n\pi ( \displaystyle n an arbitrary integer),

whilst the equation \displaystyle \text{cos }x=-\text{2 } has no solutions at all (the values of cosine lie between \displaystyle -\text{1 } and \displaystyle \text{1} ).

The answer is that the equation has the solutions


\displaystyle x=\pm \frac{\pi }{3}+2n\pi ( \displaystyle n an arbitrary integer).

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