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Solution 4.4:7b

From Förberedande kurs i matematik 1

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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
 +
<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
 +
<math>t</math>, the equation is
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 +
 
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<math>2t^{2}+3t-2=0</math>
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 +
 
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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
 +
<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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(
 +
<math>n</math>
 +
an arbitrary integer),
 +
 
 +
whilst the equation
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<math>\text{cos }x=-\text{2 }</math>
 +
has no solutions at all (the values of cosine lie between
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<math>-\text{1 }</math>
 +
and
 +
<math>\text{1}</math>
 +
).
 +
 
 +
The answer is that the equation has the solutions
 +
 
 +
 
 +
<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 sin2x as 1−cos2x, the whole equation written in terms of cosx becomes


21−cos2x−3cosx=0 


or, in rearranged form,


2cos2x+3cosx−2=0


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


2t2+3t−2=0


and this second-degree equation has the solutions t=21 and t=−2 .

In terms of x, this means that either cosx=21 or cos x=−2. The first case occurs when


x=3+2n ( n an arbitrary integer),

whilst the equation cos x=−2 has no solutions at all (the values of cosine lie between −1 and 1 ).

The answer is that the equation has the solutions


x=3+2n ( n an arbitrary integer).

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