Lösung 4.4:5a

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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If we consider for a moment the equality
If we consider for a moment the equality
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{{Displayed math||<math>\sin u = \sin v</math>|(*)}}
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{{Abgesetzte Formel||<math>\sin u = \sin v</math>|(*)}}
where ''u'' has a fixed value, there are usually two angles ''v'' in the unit circle which ensure that the equality holds,
where ''u'' has a fixed value, there are usually two angles ''v'' in the unit circle which ensure that the equality holds,
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{{Displayed math||<math>v=u\qquad\text{and}\qquad v=\pi-u\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>v=u\qquad\text{and}\qquad v=\pi-u\,\textrm{.}</math>}}
[[Image:4_4_5_a.gif||center]]
[[Image:4_4_5_a.gif||center]]
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We obtain all the angles ''v'' which satisfy (*) by adding multiples of <math>2\pi</math>,
We obtain all the angles ''v'' which satisfy (*) by adding multiples of <math>2\pi</math>,
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{{Displayed math||<math>v = u+2n\pi\qquad\text{and}\qquad v = \pi-u+2n\pi\,,</math>}}
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{{Abgesetzte Formel||<math>v = u+2n\pi\qquad\text{and}\qquad v = \pi-u+2n\pi\,,</math>}}
where ''n'' is an arbitrary integer.
where ''n'' is an arbitrary integer.
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If we now go back to our equation
If we now go back to our equation
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{{Displayed math||<math>\sin 3x = \sin x</math>}}
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{{Abgesetzte Formel||<math>\sin 3x = \sin x</math>}}
the reasoning above shows that the equation is only satisfied when
the reasoning above shows that the equation is only satisfied when
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{{Displayed math||<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}</math>}}
If we make ''x'' the subject of each equation, we obtain the full solution to the equation,
If we make ''x'' the subject of each equation, we obtain the full solution to the equation,
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{{Displayed math||<math>\left\{\begin{align}
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{{Abgesetzte Formel||<math>\left\{\begin{align}
x &= 0+n\pi\,,\\[5pt]
x &= 0+n\pi\,,\\[5pt]
x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.}
x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.}
\end{align}\right.</math>}}
\end{align}\right.</math>}}

Version vom 08:59, 22. Okt. 2008

If we consider for a moment the equality

\displaystyle \sin u = \sin v (*)

where u has a fixed value, there are usually two angles v in the unit circle which ensure that the equality holds,

\displaystyle v=u\qquad\text{and}\qquad v=\pi-u\,\textrm{.}

(The only exception is when \displaystyle u = \pi/2 or \displaystyle u=3\pi/2, in which case \displaystyle u and \displaystyle \pi-u correspond to the same direction and there is only one angle v which satisfies the equality.)

We obtain all the angles v which satisfy (*) by adding multiples of \displaystyle 2\pi,

\displaystyle v = u+2n\pi\qquad\text{and}\qquad v = \pi-u+2n\pi\,,

where n is an arbitrary integer.

If we now go back to our equation

\displaystyle \sin 3x = \sin x

the reasoning above shows that the equation is only satisfied when

\displaystyle 3x = x+2n\pi\qquad\text{or}\qquad 3x = \pi-x+2n\pi\,\textrm{.}

If we make x the subject of each equation, we obtain the full solution to the equation,

\displaystyle \left\{\begin{align}

x &= 0+n\pi\,,\\[5pt] x &= \frac{\pi}{4}+\frac{n\pi}{2}\,\textrm{.} \end{align}\right.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_4.4:5a“