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Solution 4.4:7b

From Förberedande kurs i matematik 1

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If we use the Pythagorean identity and write <math>\sin^2\!x</math> as <math>1-\cos^2\!x</math>, the whole equation can be written in terms of <math>\cos x</math>,
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<center> [[Image:4_4_7b-1(2).gif]] </center>
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{{Displayed math||<math>2(1-\cos^2\!x) - 3\cos x = 0\,,</math>}}
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<center> [[Image:4_4_7b-2(2).gif]] </center>
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or, in rearranged form,
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{{Displayed math||<math>2\cos^2\!x + 3\cos x - 2 = 0\,\textrm{.}</math>}}
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With the equation expressed entirely in terms of <math>\cos x</math>, we can introduce a new unknown variable <math>t=\cos x</math> and solve the equation with respect to ''t''. Expressed in terms of ''t'', the equation is
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{{Displayed math||<math>2t^2+3t-2 = 0</math>}}
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and this quadratic equation has the solutions <math>t=\tfrac{1}{2}</math> and
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<math>t=-2\,</math>.
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In terms of ''x'', this means that either <math>\cos x = \tfrac{1}{2}</math> or <math>\cos x = -2</math>. The first case occurs when
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{{Displayed math||<math>x=\pm \frac{\pi}{3}+2n\pi\qquad</math>(''n'' is an arbitrary integer),}}
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whilst the equation <math>\cos x = -2</math> has no solutions at all (the values of cosine lie between -1 and 1).
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The answer is that the equation has the solutions
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{{Displayed math||<math>x = \pm\frac{\pi}{3} + 2n\pi\,,</math>}}
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where ''n'' is an arbitrary integer.

Current revision

If we use the Pythagorean identity and write sin2x as 1−cos2x, the whole equation can be written in terms of cosx,

2(1−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 quadratic equation has the solutions t=21 and t=−2.

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

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

whilst the equation cosx=−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

where n is an arbitrary integer.

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