Solution 4.4:7c

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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>\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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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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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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[[Image:4_4_7_c1.gif|center]]
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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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[[Image:4_4_7_c2.gif|center]]
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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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{{Displayed math||<math>\cos u = \sin v</math>}}
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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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{{Displayed math||<math>v = \pm u + \frac{\pi}{2} + 2n\pi\qquad</math>(''n'' is an arbitrary integer).}}
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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>}}
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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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where ''n'' is an arbitrary integer.

Current revision

If we want to solve the equation \displaystyle \cos 3x = \sin 4x, we need an additional result which tells us for which values of u and v the equality \displaystyle \cos u = \sin v holds, but to get that we have to start with the equality \displaystyle \cos u=\cos v.

So, we start by looking at the equality

\displaystyle \cos u=\cos v\,\textrm{.}

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

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

The angles \displaystyle u+\pi/2 and \displaystyle -u+\pi/2 therefore have their y-coordinate, and hence sine value, equal to \displaystyle \cos u. In other words, the equality

\displaystyle \cos u = \sin v

holds for fixed u in the unit circle when \displaystyle v = \pm u + \pi/2, and more generally when

\displaystyle v = \pm u + \frac{\pi}{2} + 2n\pi\qquad(n is an arbitrary integer).

For our equation \displaystyle \cos 3x = \sin 4x, this result means that x must satisfy

\displaystyle 4x = \pm 3x + \frac{\pi}{2} + 2n\pi\,\textrm{.}

This means that the solutions to the equation are

\displaystyle \left\{\begin{align}

x &= \frac{\pi}{2} + 2n\pi\,,\\[5pt] x &= \frac{\pi}{14} + \frac{2}{7}\pi n\,, \end{align}\right.

where n is an arbitrary integer.

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