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Solution 4.4:7c

From Förberedande kurs i matematik 1

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If we want to solve the equation
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If we want to solve the equation <math>\cos 3x = \sin 4x</math>, we need an additional result which tells us for which values of ''u'' and ''v'' the equality
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<math>\text{cos 3}x=\text{sin 4}x</math>, we need an additional result which tells us for which values of
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<math>\cos u = \sin v</math> holds, but to get that we have to start with the equality <math>\cos u=\cos v</math>.
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<math>u</math>
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and
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<math>v</math>
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the equality
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<math>\text{cos }u=\text{sin }v</math>
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holds, but to get that we have to start with the equality
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<math>\cos u=\cos v</math>.
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So, we start by looking at the equality
So, we start by looking at the equality
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{{Displayed math||<math>\cos u=\cos v\,\textrm{.}</math>}}
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<math>\cos u=\cos v</math>
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We know that for fixed ''u'' there are two angles <math>v=u</math> and <math>v=-u</math> in the unit circle which have the cosine value <math>\cos u</math>, i.e. their ''x''-coordinate is equal to <math>\cos u\,</math>.
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We know that for fixed
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<math>u</math>
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there are two angles
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<math>v=u\text{ }</math>
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and
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<math>v=-\text{u}</math>
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in the unit circle which have the cosine value
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<math>\cos u</math>, i.e. their
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<math>x</math>
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-coordinate is equal to
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<math>\cos u</math>.
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[[Image:4_4_7_c1.gif|center]]
[[Image:4_4_7_c1.gif|center]]
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Imagine now that the whole unit circle is rotated anti-clockwise an angle
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Imagine now that the whole unit circle is rotated anti-clockwise an angle <math>\pi/2</math>. The line <math>x=\cos u</math> will become the line <math>y=\cos u</math> and the angles ''u'' and -''u'' are rotated to <math>u+\pi/2</math> and <math>-u+\pi/2</math>, respectively.
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<math>{\pi }/{2}\;</math>. The line
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<math>x=\cos u</math>
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will become the line
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<math>y=\cos u</math>
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and the angles
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<math>u</math>
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and
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<math>-u</math>
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are rotated to
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<math>u+{\pi }/{2}\;</math>
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and
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<math>-u+{\pi }/{2}\;</math>, respectively.
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[[Image:4_4_7_c2.gif|center]]
[[Image:4_4_7_c2.gif|center]]
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The angles
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The angles <math>u+\pi/2</math> and <math>-u+\pi/2</math> therefore have their ''y''-coordinate, and hence sine value, equal to <math>\cos u</math>. In other words, the equality
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<math>u+{\pi }/{2}\;</math>
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and
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<math>-u+{\pi }/{2}\;</math>
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therefore have their
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<math>y</math>
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-coordinate, and hence sine value, equal to
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<math>\cos u</math>. In other words, the equality
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<math>\text{cos }u=\text{sin }v</math>
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holds for fixed
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<math>u</math>
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in the unit circle when
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<math>v=\pm u+{\pi }/{2}\;</math>, and more generally when
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<math>v=\pm u+\frac{\pi }{2}+2n\pi </math>
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{{Displayed math||<math>\cos u = \sin v</math>}}
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(
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<math>n</math>
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an arbitrary integer).
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For our equation
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holds for fixed ''u'' in the unit circle when <math>v = \pm u + \pi/2</math>, and more generally when
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<math>\text{cos 3}x=\text{sin 4}x</math>, this result means that
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<math>x\text{ }</math>
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must satisfy
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{{Displayed math||<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>(''n'' is an arbitrary integer).}}
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<math>4x=\pm 3x+\frac{\pi }{2}+2n\pi </math>
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For our equation <math>\cos 3x = \sin 4x</math>, this result means that ''x'' must satisfy
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{{Displayed math||<math>4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}</math>}}
This means that the solutions to the equation are
This means that the solutions to the equation are
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{{Displayed math||<math>\left\{\begin{align}
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x &= \frac{\pi}{2} + 2n\pi\,,\\[5pt]
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x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,,
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\end{align}\right.</math>}}
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<math>\left\{ \begin{array}{*{35}l}
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where ''n'' is an arbitrary integer.
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x=\frac{\pi }{2}+2n\pi \\
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x=\frac{\pi }{14}+\frac{2}{7}\pi n \\
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\end{array} \right.</math>
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(
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<math>n</math>
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an arbitrary integer)
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Current revision

If we want to solve the equation cos3x=sin4x, we need an additional result which tells us for which values of u and v the equality cosu=sinv holds, but to get that we have to start with the equality cosu=cosv.

So, we start by looking at the equality

cosu=cosv.

We know that for fixed u there are two angles v=u and v=−u in the unit circle which have the cosine value cosu, i.e. their x-coordinate is equal to cosu.

Imagine now that the whole unit circle is rotated anti-clockwise an angle 2. The line x=cosu will become the line y=cosu and the angles u and -u are rotated to u+2 and −u+2, respectively.

The angles u+2 and −u+2 therefore have their y-coordinate, and hence sine value, equal to cosu. In other words, the equality

cosu=sinv

holds for fixed u in the unit circle when v=u+2, and more generally when

v=u+2+2n(n is an arbitrary integer).

For our equation cos3x=sin4x, this result means that x must satisfy

4x=3x+2+2n.

This means that the solutions to the equation are

xx=2+2n=14+72n

where n is an arbitrary integer.

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